Publications
Most recent papers
* Yang Fu, Yuliang Jin, Deng Pan, Itamar Procaccia, Longrange angular correlations of particle displacements at a plastictoelastic transition in jammed amorphous solids, Submitted to PNAS
* Pawandeep Kaur, Itamar Procaccia, Tuhin Samanta, Selection Principle for the Screening Parameters in the Mechanical Response of Amorphous Solids
*Noemie S. Livne, Tuhin Samanta, Amit Schiller, Itamar Procaccia, Michael Moshe, Continuum mechanics of differential growth in disordered granular matter
* Yang Fu, H. George E. Hentschel, Pawandeep Kaur, Avanish Kumar, and Itamar Procaccia, Odd Dipole Screening in Radial Inflation, Submitted to Phys. Rev. E.
2024

(2024) Physical Review E. 110, 3, L033001. Abstract
The Eshelby problem refers to the response of a twodimensional elastic sheet to cutting away a circle, deforming it into an ellipse, and pushing it back. The resulting response is dominated by the socalled Eshelby kernel, which was derived for purely elastic (infinite) material, but has been employed extensively to model the redistribution of stress after plastic events in amorphous solids with finite boundaries. Here, we discuss and solve the Eshelby problem directly for amorphous solids, taking into account possible screening effects and realistic boundary conditions. We find major modifications compared to the classical Eshelby solution. These modifications are needed for modeling correctly the spatial responses to plastic events in amorphous solids.

(2024) Soft Matter. 20, 35, p. 68686888 Abstract
Soft amorphous materials are viscoelastic solids ubiquitously found around us, from clays and cementitious pastes to emulsions and physical gels encountered in food or biomedical engineering. Under an external deformation, these materials undergo a noteworthy transition from a solid to a liquid state that reshapes the material microstructure. This yielding transition was the main theme of a workshop held from January 9 to 13, 2023 at the Lorentz Center in Leiden. The manuscript presented here offers a critical perspective on the subject, synthesizing insights from the various brainstorming sessions and informal discussions that unfolded during this week of vibrant exchange of ideas. The result of these exchanges takes the form of a series of open questions that represent outstanding experimental, numerical, and theoretical challenges to be tackled in the near future.

(2024) Chaos. 34, 5, 053144. Abstract
The response of amorphous solids to mechanical loads is accompanied by plasticity that is generically associated with \u201cnonaffine\u201d quadrupolar events seen in the resulting displacement field. To develop a continuum theory, one needs to assess when these quadrupolar events have a finite density, allowing the development of a field theory. Is there a transition, as a function of the material parameters and the nature of the loads, from isolated plastic events whose density is zero to a regime governed by a finite density? And if so, what is the nature of this transition? The aim of the paper is to explore this issue. The motivation for the present study stems from recent research in which it was shown that gradients of the quadrupolar fields act as dipole charges that can screen elasticity. Analytically soluble examples of mechanical loading that lead to screening and emergent length scales (that are absent in classical elasticity) have been analyzed and tested. However, \u201cgradients of quadrupolar fields\u201d make sense only when the density of quadrupoles is finite, and hence, the issue is central to this article. The article introduces a notion of polarizability under the strain of Eshelby quadrupoles and concludes that the onset of a density of such quadrupoles with random orientations can only appear when the polarizability is finite.

(2024) Physical Review E. 109, 4, 044902. Abstract
In recent work it was shown that elasticity theory can break down in amorphous solids subjected to nonuniform static loads. The elastic fields are screened by geometric dipoles; these stem from gradients of the quadrupole field associated with plastic responses. Here we study the dynamical responses induced by oscillatory loads. The required modification to classical elasticity is described. Exact solutions for the displacement field in circular geometry are presented, demonstrating that dipole screening results in essential departures from the expected predictions of classical elasticity theory. Numerical simulations are conducted to validate the theoretical predictions and to delineate their range of validity.

(2024) Physical Review E. 109, 1, 014902. Abstract
A significant amount of attention was dedicated in recent years to the phenomenon of jamming of athermal amorphous solids by increasing the volume fraction of the microscopic constituents. At a critical value of the volume fraction, pressure shoots up from zero to finite values with a host of critical exponents discovered and discussed. In this paper, we advance evidence for the existence of a second transition, within the jammed state of twodimensional granular systems, that separates two regimes of different mechanical responses. Explicitly, highly packed systems are quasielastic with quadrupole screening, and more loosely jammed systems exhibit anomalous mechanics with dipole screening. Evidence is given for a clear transition between these two regimes, reminiscent of the intermediate hexatic phase of crystal melting in twodimensional crystals. Theoretical estimates of the screening parameters and the pressure where transition takes place are provided.

(2024) Europhysics Letters. 145, 2, 26002. Abstract
The aim of this short review is to summarize the developing theory aimed at describing the effect of plastic events in amorphous solids on its emergent mechanics. Experiments and simulations present anomalous mechanical response of amorphous solids where quadrupolar plastic events collectively induce distributed dipoles that are analogous to dislocations in crystalline solids. The novel theory is described, and a number of pertinent examples are provided, including the comparison of theoretical prediction to simulations or experiments.
2023

(2023) Physical Review E. 108, 4, L042901. Abstract
When amorphous solids are subjected to simple or pure strain, they exhibit elastic increase in stress, punctuated by plastic events that become denser (in strain) upon increasing the system size. It is customary to assume in theoretical models that the stress released in each plastic event is redistributed according to the linear Eshelby kernel, causing avalanches of additional stress release. Here we demonstrate that, contrary to the uniform affine strain resulting from simple or pure strain, each plastic event is associated with a nonuniform strain that gives rise to a displacement field that contains quadrupolar and dipolar charges that typically screen the linear elastic phenomenology and introduce anomalous length scales and influence the form of the stress redistribution. An important question that opens up is how to take this into account in elastoplastic models of shear induced phenomena like shear banding.

(2023) Europhysics Letters. 142, 3, 36001. Abstract
Applying very small purely radial strains on amorphous solids in radial geometry one observes elastic responses that break the radial symmetry. Without any plasticity involved, the responses indicate mode coupling contributions even for minute strains. We show that these symmetrybreaking responses are due to disorder, typical to amorphous configurations. The symmetrybreaking responses are quantitatively explained using the classical Michell solutions which are excited by mode coupling.

(2023) Physical Review. E. 107, 5, 055005. Abstract
In recent work, we developed a screening theory for describing the effect of plastic events in amorphous solids on its emergent mechanics. The suggested theory uncovered an anomalous mechanical response of amorphous solids where plastic events collectively induce distributed dipoles that are analogous to dislocations in crystalline solids. The theory was tested against various models of amorphous solids in two dimensions, including frictional and frictionless granular media and numerical models of amorphous glass. Here we extend our theory to screening in threedimensional amorphous solids and predict the existence of anomalous mechanics similar to the one observed in twodimensional systems. We conclude by interpreting the mechanical response as the formation of nontopological distributed dipoles that have no analog in the crystalline defects literature. Having in mind that the onset of dipole screening is reminiscent of KosterlitzThouless and hexatic transitions, the finding of dipole screening in three dimensions is surprising.
2022

(2022) Chaos, solitons, and fractals [ejournal]. 164, 112609. Abstract
The concept of mechanical screening is widely applied in solidstate systems. Examples include nucleation of defects in crystalline materials, scars and pleats in curved crystals, wrinkles in strongly confined thin sheets, and cellrearrangements in biological tissue. Available theories of such screening usually contain a crucial ingredient, which is the existence of an ordered reference state, with respect to which screening elements nucleate to release stresses. In contradistinction, amorphous materials in which a unique reference state does not exist, nevertheless exhibits plastic events that act as screening geometric charges with significant implications on the mechanical response. In a recent paper [Phys. Rev. E 104, 024904] it was proposed that mechanical strains in amorphous solids can be either weakly or strongly screened by the formation of low or high density of plastic events. At low densities the screening effect is reminiscent of the role of dipoles in dielectrics, in only renormalizing the elastic moduli. The effect of high density screening has no immediate electrostatic analog and is expected to change qualitatively the mechanical response, as seen for example in the displacement field. On the basis of experiments and simulations, we show that in granular matter, strong screening results in significant deviation from elasticity theory. The theoretical analysis, which accounts for an emergent inherent length scale, the experimental measurements and the numerical simulations of frictional granular amorphous assemblies are in agreement with each other, and provide a strong support for the novel continuum theory.

(2022) Physical Review E. 106, 3, 034906. Abstract
Applying constant tensile stress to a piece of amorphous solid results in a slow extension, followed by an eventual rapid mechanical collapse. This "creep"process is of paramount engineering concern, and as such was the subject of study in a variety of materials, for more than a century. Predictive theories for τw, the expected time of collapse, are incomplete, mainly due to its dependence on a bewildering variety of parameters, including temperature, system size, tensile force, but also the detailed microscopic interactions between constituents. The complex dependence of the collapse time on all the parameters is discussed below, using simulations of strip of amorphous material. Different scenarios are observed for ductile and brittle materials, resulting in serious difficulties in creating an allencompassing theory that could offer safety measures for given conditions. A central aim of this paper is to employ scaling concepts, to achieve data collapse for the probability distribution function (pdf) of lnτw. The scaling ideas result in a universal function which provides a prediction of the pdf of lnτw for outofsample systems, from measurements at other values of these parameters. The predictive power of the scaling theory is demonstrated for both ductile and brittle systems. Finally, we present a derivation of universal scaling function for brittle materials. The ductile case appears to be due to a plastic necking instability and is left for future research.

(2022) Physical Review E. 106, 1, 015001. Abstract
Amorphous solids under mechanical strains are prone to plastic responses. Recent work showed that in amorphous granular systems these plastic events, that are typically quadrupolar in nature, can screen the elastic response. When the density of the quadrupoles is high, the gradients of the quadrupole field act as emergent dipole sources, leading to qualitative changes in the mechanical response, as seen for example in the displacement field. In this paper we examine the effect of screening in classical glass formers. These are made of point particles that interact via binary forces. Both inverse power law forces and LennardJones interactions are examined, and it is shown that in both cases the elastic response can be strongly screened, in agreement with the novel theory. The degree of deviation from classical elasticity theory is parametrized by a proposed measure that is shown to have a functional dependence on the amount of energy lost to plastic responses.

(2022) Europhysics Letters. 137, 4, 46002. Abstract
Fatigue caused by cyclic bending of a piece of material, resulting in its mechanical failure, is a phenomenon that had been studied for ages by engineers and physicists alike. In this Letter we study such fatigue in a strip of athermal amorphous solid. On the basis of atomistic simulations we conclude that the crucial quantity to focus on is the {\em accumulated damage}. Although this quantity exhibits large sampletosample fluctuations, its dependence on the loading determines the statistics of the number of cycles to failure. Thus we can provide a scaling theory for the Wöhler plots of mean number of cycles for failure as a function of the loading amplitude.

(2022) Physical Review. E. 105, 4, L043001. Abstract
Recent progress in studying the physics of amorphous solids has revealed that mechanical strains can be strongly screened by the formation of plastic events that are typically quadrupolar in nature. The theory stipulates that gradients in the density of the quadrupoles act as emergent dipole sources, leading to strong screening and to qualitative changes in the mechanical response, as seen, for example, in the displacement field. In this Letter we first offer direct measurements of the dipole field, independently of any theoretical assumptions, and second we demonstrate detailed agreement with the recently proposed theory. These two goals are achieved by using data from both simulations and experiments. Finally, we show how measurements of the dipole fields pinpoint the theory parameters that determine the profile of the displacement field.

(2022) Physical Review E. 105, 1, 015001. Abstract
In mechanical engineering Wöhler plots serve to measure the average number of load cycles before materials break, as a function of the maximal stress in each cycle. Although such plots have been prevalent in engineering for more than 150 years, their theoretical understanding is lacking. Recently a scaling theory of Wöhler plots in the context of cyclic bending was offered [Bhowmik, arXiv:2103.03040 (2021)]. Here we elaborate further on cyclic bending and extend the considerations to cyclic tensile loads on an amorphous strip of material; the scaling theory applies to both types of cyclic loading equally well. On the basis of atomistic simulations we conclude that the crucial quantities to focus on are the accumulated damage and the average damage per cycle. The dependence of these quantities on the loading determines the statistics of the number of cycles to failure. Finally, we consider the probability distribution functions of the number of cycles to failure and demonstrate that the scaling theory allows prediction of these distributions at one value of the forcing amplitude from measurements and another value.
2021

(2021) New Zealand physical educator : journal of Physical Education New Zealand = Te ao kori Aotearoa. 104, 4, 044903. Abstract
"Remote triggering" refers to the inducement of earthquakes by weak perturbations that emanate from far away sources, typically intense earthquakes that happen at much larger distances than their nearby aftershocks, sometimes even around the globe. Here, we propose a mechanism for this phenomenon; the proposed mechanism is generic, resulting from the breaking of Hamiltonian symmetry due to the existence of friction. We allow a transition from static to dynamic friction. {\em Linearly stable} stressed systems display giant sensitivity to small perturbations of arbitrary frequency (without a need for resonance), which trigger an instability with exponential oscillatory growth. Once nonlinear effects kick in, the blow up in meansquare displacements can reach 1520 orders of magnitude. Analytic and numerical results of the proposed model are presented and discussed.

(2021) Physical Review. E. 105, 5, 054104. Abstract
The theoretical understanding of the lowfrequency modes in amorphous solids at finite temperature is still incomplete. The study of the relevant modes is obscured by the dressing of interparticle forces by collisioninduced momentum transfer that is unavoidable at finite temperatures. Recently, it was proposed that lowfrequency modes of vibrations around the thermally averaged configurations deserve special attention. In simple model glasses with bare binary interactions, these included quasilocalized modes whose density of states appears to be universal, depending on the frequencies as D(ω)∼ω4, in agreement with the similar law that is obtained with bare forces at zero temperature. In this paper, we report investigations of a model of silica glass at finite temperature; here the bare forces include binary and ternary interactions. Nevertheless, we can establish the validity of the universal law of the density of quasilocalized modes also in this richer and more realistic model glass.

(2021) Europhysics Letters. 135, 6, 66001. Abstract
A careful examination of the QuasiLocalized Modes (QLMs) that exist in a generic atomistic model of a glass former reveals at least two types of them, each exhibiting a different density of states, one depending on the frequency as and the other as . The properties of the glassy energy landscape that is responsible for the two types of modes is examined here, explaining the analytic feature responsible for the creations of (at least) two families of QLMs. It is argued that the QLMs that are revealed by diagonalizing the Hessian matrix are not related to possibly existing TwoLevel Systems (TLS).

(2021) Physical Review E. 104, 2, 024904. Abstract
Amorphous solids appear to react elastically to small external strains, but in contrast to ideal elastic media, plastic responses abound immediately at any value of the strain. Such plastic responses are quasilocalized in nature, with the "cheapest"one being a quadrupolar source. The existence of such plastic responses results in screened elasticity in which strains and stresses can either quantitatively or qualitatively differ from the unscreened theory, depending on the specific screening mechanism. Here we offer a theory of such screening effects by plastic quadrupoles, dipoles, and monopoles, explain their natural appearance, and point out the analogy to electrostatic screening by electric charges and dipoles. For low density of quadrupoles the effect is to normalize the elastic moduli without a qualitative change compared to pure elasticity theory; for higher density of quadrupoles the screening effects result in qualitative changes. Predictions for the spatial dependence of displacement fields caused by local sources of strains are provided and compared to numerical simulations. We find that anomalous elasticity is richer than electrostatics in having a screening mode that does not appear in the electrostatic analog.

(2021) Physical Review B. 103, 5, 054110. Abstract
This paper investigates whether in frictional granular packings, like in Hamiltonian amorphous elastic solids, the stress autocorrelation matrix presents long range anisotropic contributions just as elastic Green's functions. We find that in a standard model of frictional granular packing this is not the case. We prove quite generally that mechanical balance and material isotropy constrain the stress autocorrelation matrix to be fully determined by two spatially isotropic functions: the pressure and torque autocorrelations. The pressure and torque fluctuations being, respectively, normal and hyperuniform force the stress autocorrelation to decay as the elastic Green's function. Since we find the torque fluctuations to be hyperuniform, the culprit is the pressure whose fluctuations decay slower than normally as a function of the system's size. Investigating the reason for these abnormal pressure fluctuations we discover that anomalous correlations build up already during the compression of the dilute system before jamming. Once jammed these correlations remain frozen. Whether this is true for frictional matter in general or it is the consequence of the model properties is a question that must await experimental scrutiny and possible alternative models.

(2021) Physical Review Letters. 126, 8, 085502. Abstract
It has been established that the low frequency quasilocalized modes of amorphous solids at zero temperature exhibit universal density of states, depending on the frequencies as D(omega) similar to omega(4). It remains an open question whether this universal law extends to finite temperatures. In this Letter we show that well quenched model glasses at temperatures as high as Tg/3 possess the same universal density of states. The only condition required is that average particle positions stabilize before thermal diffusion destroys the cage structure of the material. The universal density of quasilocalized low frequency modes refers then to vibrations around the thermally averaged configuration of the material.

(2021) Physical Review Letters. 126, 8, 085501. Abstract
The rupture of a polymer chain maintained at temperature T under fixed tension is prototypical to a wide array of systems failing under constant external stress and random perturbations. Past research focused on analytic and numerical studies of the mean rate of collapse of such a chain. Surprisingly, an analytic calculation of the probability distribution function (PDF) of collapse rates appears to be lacking. Since rare events of rapid collapse can be important and even catastrophic, we present here a theory of this distribution, with a stress on its tail of fast rates. We show that the tail of the PDF is a power law with a universal exponent that is theoretically determined. Extensive numerics validate the offered theory. Lessons pertaining to other problems of the same type are drawn.

(2021) Physical Review Letters. 126, 7, 075501. Abstract
Theoretical treatments of frictional granular matter often assume that it is legitimate to invoke classical elastic theory to describe its coarsegrained mechanical properties. Here, we show, based on experiments and numerical simulations, that this is generically not the case since stress autocorrelation functions decay more slowly than what is expected from elasticity theory. It was theoretically shown that standard elastic decay demands pressure and torque density fluctuations to be normal, with possibly one of them being hyperuniform. However, generic compressed frictional assemblies exhibit abnormal pressure fluctuations, failing to conform with the central limit theorem. The physics of this failure is linked to correlations built in the material during compression from a dilute configuration prior to jamming. By changing the protocol of compression, one can observe different pressure fluctuations, and stress autocorrelations decay at large scales.

(2021) Physical Review E. 103, 2, 020101. Abstract
Diffusionlimited aggregation (DLA) has served for 40 years as a paradigmatic example for the creation of fractal growth patterns. In spite of thousands of references, no exact result for the fractal dimension D of DLA is known. In this Letter we announce an exact result for offlattice DLA grown on a line embedded in the plane D = 3/2. The result relies on representing DLA with iterated conformal maps, allowing one to prove selfaffinity, a proper scaling limit, and a welldefined fractal dimension. Mathematical proofs of the main results are available in N. Berger, E. B. Procaccia, and A. Turner, Growth of stationary HastingsLevitov, arXiv:2008.05792.
2020

(2020) Physical Review Letters. 125, 8, 085501. Abstract
It was recently shown that different simple models of glass formers with binary interactions define a universality class in terms of the density of states of their quasilocalized lowfrequency modes. Explicitly, once the hybridization with standard Debye (extended) modes is avoided, a number of such models exhibit a universal density of states, depending on the mode frequencies as D(omega) similar to omega(4). It is unknown, however, how wide this universality class is, and whether it also pertains to more realistic models of glass formers. To address this issue we present analysis of the quasilocalized modes in silica, a network glass that has both binary and ternary interactions. We conclude that in three dimensions silica exhibits the very same frequency dependence at low frequencies, suggesting that this universal form is a generic consequence of amorphous glassiness.

(2020) Physical Review B. 102, 1, 014202. Abstract
Lowfrequency quasilocalized modes of amorphous glass appear to exhibit universal density of states, depending on the frequencies as D(omega) similar to omega(4). To date, various models of glass formers with shortrange binary interaction and network glass with both binary and ternary interactions were shown to conform with this law. In this paper, we examine granular amorphous solids with longrange electrostatic interactions and find that they exhibit the same law. To rationalize this wide universality class, we return to a model proposed by Gurevich, Parshin, and Schober (GPS) [Phys. Rev. B 67, 094203 (2003)] and analyze its predictions for interaction laws with varying spatial decay, exploring this widerthanexpected universality class. Numerical and analytic results are provided for both the actual system with longrange interaction and for the GPS model.

(2020) Physical Review E. 102, 1, 010603. Abstract
The extreme slowing down associated with glass formation in experiments and in simulations results in serious difficulties to prepare deeply quenched, well annealed, glassy material. Recently, methods to achieve such deep quenching were proposed, including vapor deposition on the experimental side and "swap Monte Carlo" and oscillatory shearing on the simulation side. The relation between the resulting glasses under different protocols remains unclear. Here we show that oscillatory shear and swap Monte Carlo result in thermodynamically equivalent glasses sharing the same statistical mechanics and similar mechanical responses under external strain.

(2020) Physical Review E. 101, 6, 062902. Abstract
Catastrophic events in nature can be often triggered by small perturbations, with "remote triggering" of earthquakes being an important example. Here we present a mechanism for the giant amplification of small perturbations that is expected to be generic in systems whose dynamics is not derivable from a Hamiltonian. We offer a general discussion of the typical instabilities involved (being oscillatory with an exponential increase of noise) and examine in detail the normal forms that determine the relevant dynamics. The high sensitivity to external perturbations is explained for systems with and without dissipation. Numerical examples are provided using the dynamics of frictional granular matter. Finally, we point out the relationship of the presently discussed phenomenon to the highly topical issue of "exceptional points" in quantum models with nonHermitian Hamiltonians.

(2020) Physical Review E. 101, 5, 052902. Abstract
The dynamics of amorphous granular matter with frictional interactions cannot be derived in general from a Hamiltonian and therefore displays oscillatory instabilities stemming from the onset of complex eigenvalues in the stability matrix. These instabilities were discovered in the context of one and twodimensional systems, while the threedimensional case was never studied in detail. Here we fill this gap by deriving and demonstrating the presence of oscillatory instabilities in a threedimensional granular packing. We study binary assemblies of spheres of two sizes interacting via classical Hertz and Mindlin force laws for the longitudinal and tangent interactions, respectively. We formulate analytically the stability matrix in three dimensions and observe that a couple of complex eigenvalues emerge at the onset of the instability as in the case of frictional disks in two dimensions. The dynamics then shows oscillatory exponential growth in the meansquare displacement, followed by a catastrophic event in which macroscopic portions of mechanical stress and energy are lost. The generality of these results for any choice of forces that break the symplectic Hamiltonian symmetry is discussed.

Plastic instabilities in charged granular systems: Competition between elasticity and electrostatics(2020) Physical Review E. 101, 5, 052903. Abstract
Electrostatic theory preserves charges, but allows dipolar excitations. Elasticity theory preserves dipoles, but allows quadrupolar (Eshelbylike) plastic events. Charged amorphous granular systems are interesting in their own right; here we focus on their plastic instabilities and examine their mechanical response to external strain and to an external electric field, to expose the competition between elasticity and electrostatics. In this paper a generic model is offered, its mechanical instabilities are examined, and a theoretical analysis is presented. Plastic instabilities are discussed as saddlenode bifurcations that can be fully understood in terms of eigenvalues and eigenfunctions of the relevant Hessian matrix. This system exhibits moduli that describe how electric polarization and stress are influenced by strain and the electric field. Theoretical expression for these moduli are offered and compared to the measurements in numerical simulations.

(2020) Physical Review Letters. 124, 3, 030602. Abstract
The nature of an instability that controls the transition from static to dynamical friction is studied in the context of an array of frictional disks that are pressed from above on a substrate. In this case the forces are all explicit and Newtonian dynamics can be employed without any phenomenological assumptions. We show that an oscillatory instability that had been discovered recently is responsible for the transition, allowing individual disks to spontaneously reach the Coulomb limit and slide with dynamic friction. The transparency of the model allows a full understanding of the phenomenon, including the speeds of the waves that travel from the trailing to the leading edge and vice versa.
2019

(2019) Physical Review E. 100, 6, 062103. Abstract
In amorphous solids at finite temperatures the particles follow chaotic trajectories which, at temperatures sufficiently lower than the glass transition, are trapped in "cages." Averaging their positions for times shorter than the diffusion time, one can define a timeaveraged configuration. Under strain or stress, these average configurations undergo sharp plastic instabilities. In athermal glasses the understanding of plastic instabilities is furnished by the Hessian matrix and its eigenvalues and eigenfunctions. Here we propose an uplifting of Hessian methods to thermal glasses, with the aim of understanding the plastic responses in the timeaveraged configuration. We discuss a number of potential definitions of Hessians and identify which of these can provide eigenvalues and eigenfunctions which can explain and predict the instabilities of the timeaveraged configurations. The conclusion is that the nonaffine changes in the average configurations during an instability is accurately predicted by the eigenfunctions of the lowlying eigenvalues of the chosen Hessian.

(2019) Physical Review E. 100, 6, 060602. Abstract
Plastic instabilities in amorphous materials are often studied using idealized models of binary mixtures that do not capture accurately molecular interactions and bonding present in real glasses. Here we study atomicscale plastic instabilities in a threedimensional molecular dynamics model of silica glass under quasistatic shear. We identify two distinct types of elementary plastic events, one is a standard quasilocalized atomic rearrangement while the second is a bondbreaking event that is absent in simplified models of fragile glass formers. Our results show that both plastic events can be predicted by a drop of the lowest nonzero eigenvalue of the Hessian matrix that vanishes at a critical strain. Remarkably, we find very high correlation between the associated eigenvectors and the nonaffine displacement fields accompanying the bondbreaking event, predicting the locus of structural failure. Both eigenvectors and nonaffine displacement fields display an Eshelbylike quadrupolar structure for both failure modes, rearrangement, and bond breaking Our results thus clarify the nature of atomicscale plastic instabilities in silica glasses, providing useful information for the development of mesoscale models of amorphous plasticity.

(2019) Physical Review E. 100, 5, 052110. Abstract
Strained amorphous solids often fail mechanically by creating a shear band. It had been understood that the shearbanding instability is usefully described as crossing a spinodal point (with disorder) in an appropriate thermodynamic description. It remained contested, however, whether the spinodal is critical (with divergent correlation length) or not. Here we offer evidence for critical spinodal by using particle pinning. For a finite concentration of pinned particles the correlation length is bounded by the average distance between pinned particles, but without pinning it is bounded by the system size.

(2019) Physical Review B. 100, 13, 134515. Abstract
We report on a combined theoretical and numerical study of counterflow turbulence in superfluid He4 in a wide range of parameters. The energy spectra of the velocity fluctuations of both the normalfluid and superfluid components are strongly anisotropic. The angular dependence of the correlation between velocity fluctuations of the two components plays the key role. A selective energy dissipation intensifies as scales decrease, with the streamwise velocity fluctuations becoming dominant. Most of the flow energy is concentrated in a wavevector plane which is orthogonal to the direction of the counterflow. The phenomenon becomes more prominent at higher temperatures as the coupling between the components depends on the temperature and the direction with respect to the counterflow velocity.

(2019) Physical Review E. 100, 4, 042901. Abstract
It was discovered recently that frictional granular materials can exhibit an important mechanism for instabilities, i.e., the appearance of pairs of complex eigenvalues in their stability matrix. The consequence is an oscillatory exponential growth of small perturbations which are tamed by dynamical nonlinearities. The amplification can be giant, many orders of magnitude, and it ends with a catastrophic systemspanning plastic event. Here we follow up on this discovery, explore the scaling laws characterizing the onset of the instability, the scenarios of the development of the instability with and without damping, and the nature of the eventual systemspanning events. The possible relevance to earthquake physics and to the transition from static to dynamic friction is discussed.

(2019) Physical Review E. 100, 4, 042902. Abstract
We study agitated frictional disks in two dimensions with the aim of
developing a scaling theory for their diffusion over time. As a function
of the area fraction ϕ and meansquare velocity fluctuations ⟨v2⟩ the meansquare displacement of the disks ⟨d2⟩
spans four to five orders of magnitude. The motion evolves from a
subdiffusive form to a complex diffusive behavior at long times. The
statistics of ⟨dn⟩
at all times are multiscaling, since the probability distribution
function (PDF) of displacements has very broad wings. Even where a
diffusion constant can be identified it is a complex function of ϕ and ⟨v2⟩.
By identifying the relevant length and time scales and their
interdependence one can rescale the data for the meansquare
displacement and the PDF of displacements into collapsed scaling
functions for all ϕ and ⟨v2⟩. These scaling functions provide a predictive tool, allowing one to infer from one set of measurements (at a given ϕ and ⟨v2⟩) what are the expected results at any value of ϕ and ⟨v2⟩ within the scaling range. 
(2019) Granular Matter. 21, 3, 40. Abstract
Compressed frictional granular matter cannot flow without dilation. Upon forced shearing to generate flow, the amount of dilation may depend on the initial preparation and a host of material variables. On the basis of both experiments and numerical simulations we show that as a result of training by repeated compressiondecompression cycles the amount of dilation induced by shearing the system depends only on the shear rate and on the (preshearing) packing fraction. Relating the rheological response to structural properties allows us to derive a scaling law for the amount of dilation after n cycles of compressiondecompression. The resulting scaling law has a universal exponent that for trained systems is independent of the intergranules force laws, friction parameters and strain rate. The amplitude of the scaling law is analytically computable, and it depends only on the shear rate and the asymptotic packing fraction.

(2019) Physical Review Letters. 123, 9, 098003. Abstract
Frictional granular matter is shown to be fundamentally different in its plastic responses to external strains from generic glasses and amorphous solids without friction. While regular glasses exhibit plastic instabilities due to the vanishing of a real eigenvalue of the Hessian matrix, frictional granular materials can exhibit a previously unnoticed additional mechanism for instabilities, i.e., the appearance of a pair of complex eigenvalues leading to oscillatory exponential growth of perturbations that are tamed by dynamical nonlinearities. This fundamental difference appears crucial for the understanding of plasticity and failure in frictional granular materials. The possible relevance to earthquake physics is discussed.

(2019) Physical Review E. 99, 5, 050902. Abstract
Shearing with a finite shear rate a compressed granular system results in a region of grains flowing over a compact, static assembly. Perforce this region is dilated to a degree that depends on the shear rate, the loading pressure, gravity, various material parameters, and the preparation protocol. In spite of numerous studies of granular flows a predictive theory of the amount of dilation is still lacking. Here, we offer a scaling theory that is focused on such a prediction as a function of shear rate and the dissipative parameters of the granular assembly. The resulting scaling laws are universal with respect to changing the interparticle force laws.

(2019) Physical Review Letters. 122, 14, 144501. Abstract
Threedimensional anisotropic turbulence in classical fluids tends towards isotropy and homogeneity with decreasing scales, allowingeventuallythe abstract model of homogeneous and isotropic turbulence to be relevant. We show here that the opposite is true for superfluid He4 turbulence in threedimensional counterflow channel geometry. This flow becomes less isotropic upon decreasing scales, becoming eventually quasitwodimensional. The physical reason for this unusual phenomenon is elucidated and supported by theory and simulations.

(2019) Europhysics Letters. 125, 6, 68004. Abstract
When compressed frictional granular media are decompressed, generically a fragile configuration is created at low pressures. Typically this is accompanied by a giant frictional slippage as the fragile state collapses. We show that this instability is understood in terms of a scaling theory with theoretically computable amplitudes and exponents. The amplitude diverges in the thermodynamic limit hinting to the possibility of huge frictional slip events in decompressed granular media. The physics of this slippage is discussed in terms of the probability distribution functions of the tangential and normal forces on the grains which are highly correlated due to the Coulomb condition. Copyright (C) EPLA, 2019

(2019) Thin Solid Films. 669, p. 8084 Abstract
Numerical Simulations are employed to create amorphous nanofilms of a chosen thickness on a crystalline substrate which induces strain on the film. The films are grown by a vapor deposition technique. Using the exact relations between the Hessian matrix and the shear and bulk moduli we explore the mechanical properties of the nanofilms as a function of the density of the substrate and the film thickness. The existence of the substrate dominates the mechanical properties of the combined substratefilm system. Scaling concepts are then employed to achieve data collapse in a wide range of densities and film thicknesses.

(2019) Physical Review E. 99, 1, 011001. Abstract
In thermal glasses at temperatures sufficiently lower than the glass transition, the constituent particles are trapped in their cages for a sufficiently long time such that their timeaveraged positions can be determined before diffusion and structural relaxation takes place. The effective forces are those that hold these average positions in place. In numerical simulations the effective forces Fij between any pair of particles can be measured as a time average of the bare forces f(ij)(r(ij)(t)). In general, even if the bare forces come from twobody interactions, thermal dynamics dress the effective forces to contain manybody interactions. Here, we develop the effective theory for systems with generic interactions, where the effective forces are derivable from an effective potential and in turn they give rise to an effective Hessian whose eigenvalues are all positive when the system is stable. In this Rapid Communication, we offer analytic expressions for the effective theory, and demonstrate the usefulness and the predictive power of the approach.
2018

(2018) Physical Review E. 98, 6, 063001. Abstract
While perfect crystals may exhibit a purely elastic response to shear all the way to yielding, the response of amorphous solids is punctuated by plastic events. The prevalence of this plasticity depends on the number of particles N of the system, with the average strain interval before the first plastic event, (Delta gamma) over bar, scaling like Nalpha with alpha negative: larger samples are more susceptible to plasticity due to more numerous disorderinduced soft spots. In this paper we examine this scaling relation in ultrastable glasses prepared with the swap Monte Carlo algorithm, with regard to the possibility of a protocoldependent scaling exponent, which would also imply a protocol dependence in the distribution of local yield stresses in the glass. We show that, while a superficial analysis seems to corroborate this hypothesis, this is only a preasymptotic effect and in fact our data can be well explained by a simple model wherein such protocol dependence is absent.

(2018) Physical Review E. 98, 1, 012905. Abstract
We report a joint experimental and theoretical investigation of the probability distribution functions (PDFs) of the normal and tangential (frictional) forces in amorphous frictional media. We consider both the joint PDF of normal and tangential forces together, and the marginal PDFs of normal forces separately and tangential forces separately. A maximum entropy formalism is utilized for all these cases after identifying the appropriate constraints. Excellent agreements with both experimental and simulation data are reported. The proposed joint PDF predicts giant slip events at low pressures, again in agreement with observations.

(2018) Physical Review E. 97, 6, 063003. Abstract
In very recent work the mean field theory of the jamming transition in infinitedimensional hard sphere models was presented. Surprisingly, this theory predicts quantitatively the numerically determined characteristics of jamming in two and three dimensions. This is a rare and unusual finding. Here we argue that this agreement is nongeneric: only for hard sphere models does it happen that sufficiently close to the jamming density (at any temperature) the effective interactions are binary, in agreement with mean field theory, justifying the truncation of manybody interactions (which is the exact protocol in infinite dimensions). Any softening of the bare hard sphere interactions results in manybody effective interactions that are not mean field at any density, making the d=∞ results not applicable.

(2018) Physical Review B. 97, 17, 174105. Abstract
Standard approaches to magnetomechanical interactions in thermal magnetic crystalline solids involve Landau functionals in which the lattice anisotropy and the resulting magnetization easy axes are taken explicitly into account. In glassy systems one needs to develop a theory in which the amorphous structure precludes the existence of an easy axis, and in which the constituent particles are free to respond to their local amorphous surroundings and the resulting forces. We present a theory of all the mixed responses of an amorphous solid to mechanical strains and magnetic fields. Atomistic models are proposed in which we test the predictions of magnetostriction for both bulk and nanofilm amorphous samples in the paramagnetic phase. The application to nanofilms with emergent selfaffine free interfaces requires a careful definition of the film "width" and its change due to the magnetostriction effect.

(2018) EPL. 122, 3, 38003. Abstract
We report a joint experimental and theoretical investigation of cyclic training of amorphous frictional granular assemblies, with special attention to memory formation and retention. Measures of dissipation and compactification are introduced, culminating with a proposed scaling law for the reducing dissipation and increasing memory. This scaling law is expected to be universal, and insensitive to the details of the elastic and frictional interactions between the granules.

(2018) Physical Review Fluids. 3, 2, 024605. Abstract
The largescale turbulent statistics of mechanically driven superfluid He4 was shown experimentally to follow the classical counterpart. In this paper, we use direct numerical simulations to study the whole range of scales in a range of temperatures T [1.3,2.1] K. The numerics employ selfconsistent and nonlinearly coupled normal and superfluid components. The main results are that (i) the velocity fluctuations of normal and super components are well correlated in the inertial range of scales, but decorrelate at small scales. (ii) The energy transfer by mutual friction between components is particulary efficient in the temperature range between 1.8 and 2 K, leading to enhancement of smallscale intermittency for these temperatures. (iii) At low T and close to Tλ, the scaling properties of the energy spectra and structure functions of the two components are approaching those of classical hydrodynamic turbulence.

(2018) Physical Review B. 97, 1, 014508 . Abstract
Describing superfluid turbulence at intermediate scales between the intervortex distance and the macroscale requires an acceptable equation of motion for the density of quantized vortex lines L. The closure of such an equation for superfluid inhomogeneous flows requires additional inputs besides L and the normal and superfluid velocity fields. In this paper, we offer a minimal closure using one additional anisotropy parameter Il0. Using the example of counterflow superfluid turbulence, we derive two coupled closure equations for the vortex line density and the anisotropy parameter Il0 with an input of the normal and superfluid velocity fields. The various closure assumptions and the predictions of the resulting theory are tested against numerical simulations.
2017

(2017) Physical Review E. 96, 3, 032907. Abstract
The mechanical failure of amorphous media is a ubiquitous phenomenon from material engineering to geology. It has been noticed for a long time that the phenomenon is "scalefree," indicating some type of criticality. In spite of attempts to invoke "SelfOrganized Criticality," the physical origin of this criticality, and also its universal nature, being quite insensitive to the nature of microscopic interactions, remained elusive. Recently we proposed that the precise nature of this critical behavior is manifested by a spinodal point of a thermodynamic phase transition. Demonstrating this requires the introduction of an "order parameter" that is suitable for distinguishing between disordered amorphous systems. At the spinodal point there exists a divergent correlation length which is associated with the systemspanning instabilities (known also as shear bands) which are typical to the mechanical yield. The theory, the order parameter used and the correlation functions which exhibit the divergent correlation length are universal in nature and can be applied to any amorphous solid that undergoes mechanical yield. The phenomenon is seen at its sharpest in athermal systems, as is explained below; in this paper we extend the discussion also to thermal systems, showing that at sufficiently high temperatures the spinodal phenomenon is destroyed by thermal fluctuations.

(2017) Soft Matter. 13, 29, p. 50085020 Abstract
We revisit the problem of the stress distribution in a frictional sandpile with both normal and tangential (frictional) intergranular forces, under gravity, equipped with a new numerical method of generating such assemblies. Numerical simulations allow a determination of the spatial dependence of all the components of the stress field, principle stress axis, angle of repose, as a function of systems size, the coefficient of static friction and the frictional interaction with the bottom surface. We compare these results with the predictions of a theory based on continuum equilibrium mechanics. Basic to the theory of sandpiles are assumptions about the form of scaling solutions and constitutive relations for cohesiveless hard grains for which no typical scale is available. We find that these constitutive relations must be modified; moreover for smaller friction coefficients and smaller piles these scaling assumptions break down in the bulk of the sandpile due to the presence of length scales that must be carefully identified. Fortunately, for larger friction coefficient and for larger piles the breaking of scaling is weak in the bulk, allowing an approximate analytic theory which agrees well with the observations. After identifying the crucial scale, triggering the breaking of scaling, we provide a predictive theory to when scaling solutions are expected to break down. At the bottom of the pile the scaling assumption breaks always, due to the different interactions with the bottom surface. The consequences for measurable quantities like the pressure distribution and shear stress at the bottom of the pile are discussed. For example one can have a transition from no dip in the basepressure to a dip at the center of the pile as friction increases.

(2017) Physical Review B. 95, 18, 184510. Abstract
Below the phase transition temperature Tc≃103K He3B has a mixture of normal and superfluid components. Turbulence in this material is carried predominantly by the superfluid component. We explore the statistical properties of this quantum turbulence, stressing the differences from the better known classical counterpart. To this aim we study the timehonored HallVinenBekarevichKhalatnikov coarsegrained equations of superfluid turbulence. We combine pseudospectral direct numerical simulations with analytic considerations based on an integral closure for the energy flux. We avoid the assumption of locality of the energy transfer which was used previously in both analytic and numerical studies of the superfluid He3B turbulence. For T

(2017) Proceedings of the National Academy of Sciences of the United States of America. 114, 22, p. 55775582 Abstract
Amorphous solids increase their stress as a function of an applied strain until a mechanical yield point whereupon the stress cannot increase anymore, afterward exhibiting a steady state with a constant mean stress. In stresscontrolled experiments, the system simply breaks when pushed beyond this mean stress. The ubiquity of this phenomenon over a huge variety of amorphous solids calls for a generic theory that is free of microscopic details. Here, we offer such a theory: The mechanical yield is a thermodynamic phase transition, where yield occurs as a spinodal phenomenon. At the spinodal point, there exists a divergent correlation length that is associated with the systemspanning instabilities (also known as shear bands), which are typical to the mechanical yield. The theory, the order parameter used, and the correlation functions that exhibit the divergent correlation length are universal in nature and can be applied to any amorphous solids that undergo mechanical yield.

(2017) Physical Review E. 95, 3, 031001. Abstract
In recent work it was clarified that amorphous solids under strain control do not possess nonlinear elastic theory in the sense that the shear modulus exists but nonlinear moduli exhibit sampletosample fluctuations that grow without bound with the system size. More relevant, however, for experiments are the conditions of stress control. In the present Rapid Communication we show that also under stress control the shear modulus exists, but higherorder moduli show unbounded sampletosample fluctuation. The unavoidable consequence is that the characterization of stressstrain curves in experiments should be done with a stressdependent shear modulus rather than with nonlinear expansions.
2016

(2016) Physical Review E. 94, 5, 051001. Abstract
Amorphous media at finite temperatures, be them liquids, colloids, or glasses, are made of interacting particles that move chaotically due to thermal energy, continuously colliding and scattering off each other. When the average configuration in these systems relaxes only at long times, one can introduce effective interactions that keep the mean positions in mechanical equilibrium. We introduce a framework to determine the effective force laws that define an effective Hessian that can be employed to discuss stability properties and the density of states of the amorphous system. We exemplify the approach with a thermal glass of hard spheres; these experience zero forces when not in contact and infinite forces when they touch. Close to jamming we recapture the effective interactions that at temperature T depend on the gap h between spheres as T/h [C. Brito and M. Wyart, Europhys. Lett. 76, 149 (2006)]. For hard spheres at lower densities or for systems whose binary bare interactions are longer ranged (at any density), the emergent force laws include ternary, quaternary, and generally higherorder manybody terms, leading to a temperaturedependent effective Hessian.


(2016) Physical Review B. 94, 14, 146502. Abstract
This is a Reply to Nemirovskii's Comment [Phys. Rev. B 94, 146501 (2016)10.1103/PhysRevB.94.146501] on Khomenko et al. [Phys. Rev. B 91, 180504 (2015)PRBMDO1098012110.1103/PhysRevB.91.180504] in which a new form of the production term in Vinen's equation for the evolution of the vortexline density L in the thermal counterflow of superfluid He4 in a channel was suggested. To further substantiate the suggested form which was questioned in the Comment, we present a physical explanation for the improvement of the closure suggested in Khomenko et al. [Phys. Rev. B 91, 180504 (2015)PRBMDO1098012110.1103/PhysRevB.91.180504] in comparison to the form proposed by Vinen. We also discuss the closure for the flux term, which agrees well with the numerical results without any fitting parameters.

(2016) Physical Review E. 93, 6, 060601. Abstract
We consider the problem of how to determine the force laws in an amorphous system of interacting particles. Given the positions of the centers of mass of the constituent particles we propose an algorithm to determine the interparticle force laws. Having n different types of constituents we determine the coefficients in the Laurent polynomials for the n(n+1)/2 possibly different force laws. A visual providing the particle positions in addition to a measurement of the pressure is all that is required. The algorithm proposed includes a part that can correct for experimental errors in the positions of the particles. Such a correction of unavoidable measurement errors is expected to benefit many experiments in the field.

(2016) Physical Review E. 93, 6, 063003. Abstract
It is known [H. G. E. Hentschel et al., Phys. Rev. E 83, 061101 (2011)PLEEE81539375510.1103/PhysRevE.83.061101] that amorphous solids at zero temperature do not possess a nonlinear elasticity theory: besides the shear modulus, which exists, none of the higher order coefficients exist in the thermodynamic limit. Here we show that the same phenomenon persists up to temperatures comparable to that of the glass transition. The zerotemperature mechanism due to the prevalence of dangerous plastic modes of the Hessian matrix is replaced by anomalous stress fluctuations that lead to the divergence of the variances of the higher order elastic coefficients. The conclusion is that in amorphous solids elasticity can never be decoupled from plasticity: the nonlinear response is very substantially plastic.

(2016) Physical Review B. 93, 22, 224204. Abstract
Amorphous solids yield in straincontrolled protocols at a critical value of the strain. For larger strains the stress and energy display a generic complex serrated signal with elastic segments punctuated by sharp energy and stress plastic drops having a wide range of magnitudes. Here we provide a theory of the scaling properties of such serrated signals taking into account the systemsize dependence. We show that the statistics are not homogeneous: they separate sharply to a regime of "small" and "large" drops, each endowed with its own scaling properties. A scaling theory is first derived solely by data analysis, showing a somewhat complex picture. But after considering the physical interpretation one discovers that the scaling behavior and the scaling exponents are in fact very simple and universal.

(2016) Physical Review B. 93, 13, 134504. Abstract
In classical turbulence the kinematic viscosity ν is involved in two phenomena. The first is the energy dissipation and the second is the mechanical momentum flux toward the wall. In superfluid turbulence the mechanism of energy dissipation is different, and it is determined by an effective viscosity which was introduced by Vinen and is denoted as ν'. In this paper we show that in superfluid turbulence the transfer of mechanical momentum to the wall is caused by the presence of a quantum vortex tangle, giving rise to another effective "momentum" viscosity that we denote as νm(T). The temperature dependence of the second effective viscosity is markedly different from Vinen's effective viscosity ν'(T). We show that the notion of vortextension force, playing an important role in the theory of quantum turbulence, can be understood as the gradient of the Reynoldsstress tensor, which is, in fact, determined by the second newly defined kinematic viscosity νm(T).

(2016) Philosophical Magazine. 96, 14, p. 13991419 Abstract
The mechanical properties of amorphous solids like metallic glasses can be dramatically changed by adding small concentrations (as low as 0.1%) of foreign elements. The glassformingability, the ductility, the yield stress and the elastic moduli can all be greatly effected. This paper presents theoretical considerations with the aim of explaining the magnitude of these changes in light of the small concentrations involved. The theory is built around the experimental evidence that the microalloying elements organise around them a neighbourhood that differs from both the crystalline and the glassy phases of the material in the absence of the additional elements. These regions act as isotropic defects that in unstressed systems modify the shear moduli. When strained, these defects interact with the incipient plastic responses which are quadrupolar in nature. It will be shown that this interaction interferes with the creation of systemspanning shear bands and increases the yield strain. We offer experimentally testable estimates of the lengths of nanoshear bands in the presence of the additional elements.

(2016) Physical Review E. 93, 3, 033004. Abstract
Plastic events in amorphous solids can be much more than just "shear transformation zones" when the positional degrees of freedom are coupled nontrivially to other degrees of freedom. Here we consider magnetic amorphous solids where mechanical and magnetic degrees of freedom interact, leading to rather complex plastic events whose nature must be disentangled. In this paper we uncover the anatomy of the various contributions to some typical plastic events. These plastic events are seen as Barkhausen noise or other "serrated noises." Using theoretical considerations we explain the observed statistics of the various contributions to the considered plastic events. The richness of contributions and their different characteristics imply that in general the statistics of these serrated noises cannot be universal, but rather highly dependent on the state of the system and on its microscopic interactions.

(2016) Physical review letters. 116, 7, 078001. Abstract
The determination of the normal and transverse (frictional) interparticle forces within a granular medium is a longstanding, daunting, and yet unresolved problem. We present a new formalism that employs the knowledge of the external forces and the orientations of contacts between particles (of any given size), to compute all the interparticle forces. Having solved this problem, we exemplify the efficacy of the formalism showing that the force chains in such systems are determined by an expansion in the eigenfunctions of a newly defined operator.

(2016) Physical review letters. 116, 8, 085502. Abstract
Quasistatic straincontrolled measurements of stress versus strain curves in macroscopic amorphous solids result in a nonlinearlooking curve that ends up either in mechanical collapse or in a steady state with fluctuations around a mean stress that remains constant with increasing strain. It is therefore very tempting to fit a nonlinear expansion of the stress in powers of the strain. We argue here that at low temperatures the meaning of such an expansion needs to be reconsidered. We point out the enormous difference between quenched and annealed averages of the stress versus strain curves and propose that a useful description of the mechanical response is given by a stress (or strain) dependent shear modulus for which a theoretical evaluation exists. The elastic response is piecewise linear rather than nonlinear.

(2016) Physical review letters. 116, 8, 085501. Abstract
Amorphous solids yield at a critical value of the strain (in straincontrolled experiments); for larger strains, the average stress can no longer increase  the system displays an elastoplastic steady state. A longstanding riddle in the materials community is what the difference is between the microscopic states of the material before and after yield. Explanations in the literature are material specific, but the universality of the phenomenon begs a universal answer. We argue here that there is no fundamental difference in the states of matter before and after yield, but the yield is a bona fide firstorder phase transition between a highly restricted set of possible configurations residing in a small region of phase space to a vastly rich set of configurations which include many marginally stable ones. To show this, we employ an order parameter of universal applicability, independent of the microscopic interactions, that is successful in quantifying the transition in an unambiguous manner.

(2016) Physical Review B. 93, 1, 014516 . Abstract
In mechanically driven superfluid turbulence, the mean velocities of the normal and superfluid components are known to coincide: Un=Us. Numerous laboratory, numerical, and analytical studies showed that under these conditions, the mutual friction between the normal and superfluid velocity components also couples their fluctuations: un(r,t)≈us(r,t), almost at all scales. We show that this is not the case in thermally driven superfluid turbulence; here the counterflow velocity Uns≡UnUs≠0. We suggest a simple analytic model for the crosscorrelation function un(r,t)·us(r,t) and its dependence on Uns. We demonstrate that un(r,t) and us(r,t) are decoupled almost in the entire range of separations rr between the energycontaining scale and intervortex distance.
2015

(2015) Physical Review E. 92, 6, 062302. Abstract
We focus on the probability distribution function (PDF) P(Δγ;γ) where Δγ are the measured strain intervals between plastic events in a athermal strained amorphous solids, and γ measures the accumulated strain. The tail of this distribution as Δγ→0 (in the thermodynamic limit) scales like Δγη. The exponent η is related via scaling relations to the tail of the PDF of the eigenvalues of the plastic modes of the Hessian matrix P(λ) which scales like λθ, η=(θ1)/2. The numerical values of η or θ can be determined easily in the unstrained material and in the yielded state of plastic flow. Special care is called for in the determination of these exponents between these states as γ increases. Determining the γ dependence of the PDF P(Δγ;γ) can shed important light on plasticity and yield. We conclude that the PDF's of both Δγ and λ are not continuous functions of γ. In slowly quenched amorphous solids they undergo two discontinuous transitions, first at γ=0+ and then at the yield point γ=γY to plastic flow. In quickly quenched amorphous solids the second transition is smeared out due to the nonexisting stress peak before yield. The nature of these transitions and scaling relations with the system size dependence of (Δγ) are discussed.

(2015) EPL. 112, 1, 17011. Abstract
Longranged dipoledipole interactions in magnetic glasses give rise to magnetic domains having labyrinthine patterns on the scale of about 1 micron. Barkhausen Noise then results from the movement of domain boundaries which is modeled by the motion of elastic membranes with random pinning. Here we propose that on the nanoscale new sources of Barkhausen Noise can arise. We propose an atomistic model of magnetic glasses in which we measure the Barkhausen Noise which results from the creation of new domains and the movement of domain boundaries on the nanoscale. The statistics of the Barkhausen Noise found in our simulations is in striking disagreement with the expectations in the literature. In fact we find exponential statistics without any power law, stressing the fact that Barkhausen Noise can belong to very different universality classes. In the present model the essence of the phenomenon is the fact that the spin response Green's function is decaying too rapidly for having sufficiently large magnetic jumps. A theory is offered in excellent agreement with the measured data without any free parameter.

(2015) Physical Review B  Condensed Matter and Materials Physics. 92, 9, 094105. Abstract
Highly accurate numerical simulations are employed to highlight the subtle but important differences in the mechanical stability of perfect crystalline solids versus amorphous solids. We stress the difference between strain values at which the shear modulus vanishes and strain values at which a plastic instability ensues. The temperature dependence of the yield strain is computed for the two types of solids, showing different scaling laws: γY≃γY0C1T1/3 for crystals versus γY≃γY0C2T2/3 for amorphous solids.

(2015) EPL. 111, 5, 56009. Abstract
The existence of a static lengthscale that grows in accordance with the dramatic slowingdown observed at the glass transition is a subject of intense interest. Due to limitations on the relaxation times reachable by standard moleculardynamics techniques (i.e. a range of about 45 orders of magnitude) it was until now impossible to demonstrate a significant enough increase in any proposed lengthscale. In this letter we explore the typical scale at unprecedented lower temperatures. A Swap Monte Carlo approach allows us to reach a lengthscale growth by more than 500%. We conclude by discussing the relationship between the observed lengthscale and various models of the relaxation time, proposing that the associated increase in relaxation time approaches experimental values.

(2015) Physical Review B. 91, 18, 180504. Abstract
The quantization of vortex lines in superfluids requires the introduction of their density L(r,t) in the description of quantum turbulence. The space homogeneous balance equation for L(t), proposed by Vinen on the basis of dimensional and physical considerations, allows a number of competing forms for the production term P. Attempts to choose the correct one on the basis of timedependent homogeneous experiments ended inconclusively. To overcome this difficulty we announce here an approach that employs an inhomogeneous channel flow which is very suitable to distinguish the implications of the various possible forms of the desired equation. We demonstrate that the originally selected form which was extensively used in the literature is in strong contradiction with our data. We therefore present a new form of an inhomogeneous equation for L(r,t) that is in agreement with our data and propose that it should be considered for further studies of superfluid turbulence.

(2015) EPL. 110, 4, 48001. Abstract
We focus on the observed reduction in shear modulus when the stress on an amorphous solid is increased beyond the initial linear region. Careful numerical quasistatic simulations reveal an intimate relation between plastic failure and the reduction in shear modulus. The attainment of the smallest value of the shear modulus is identified with spreading of the regions that underwent a plastic event. We present an elementary "twostate" model that interpolates between failed and virgin regions and provides a simple and effective characterization of the phenomenon.

(2015) Physical Review B. 91, 14, 144501. Abstract
We discuss the energy and vorticity spectra of turbulent superfluid ^{4}He in the entire temperature range from T=0 up to the phase transition "λ point," T_{λ}≃2.17K. Contrary to classical developed turbulence in which there are only two typical scales, i.e., the energy injection L and the dissipation scales η, here, the quantization of vorticity introduces two additional scales, the vortex core radius a0 and the mean vortex spacing ℓ. We present these spectra for the super and the normalfluid components in the entire range of scales from L to a_{0} including the crossover scale ℓ where the hydrodynamic eddy cascade is replaced by the cascade of Kelvin waves on individual vortices. At this scale, a bottleneck accumulation of the energy was found earlier at T=0. We show that even very small mutual friction dramatically suppresses the bottleneck effect due to the dissipation of the Kelvin waves. Using our results for the spectra we estimate the Vinen "effective viscosity" ν in the entire temperature range and show agreement with numerous experimental observations for ν(T).

(2015) EPL. 109, 1, 16002. Abstract
The concept of a Shear Transformation Zone (STZ) refers to a region in an amorphous solid that undergoes a plastic event when the material is put under an external mechanical load. An important question that had accompanied the development of the theory of plasticity in amorphous solids for many years now is whether an STZ is a region existing in the material (which can be predicted by analyzing the unloaded material), or it is an event that depends on the loading protocol (i.e., the event cannot be predicted without following the protocol itself). In this letter we present strong evidence that the latter is the case. Infinitesimal changes of protocol result in macroscopically big jumps in the positions of plastic events, meaning that these can never be predicted from considering the unloaded material.
2014

(2014) Physical Review E. 90, 5, 052402. Abstract
Much of the progress achieved in understanding plasticity and failure in amorphous solids had been achieved using experiments and simulations in which the materials were loaded using strain control. There is paucity of results under stress control. Here we present a method that was carefully geared to allow loading under stress control either at T=0 or at any other temperature, using Monte Carlo techniques. The method is applied to a modelperfect crystalline solid, to a crystalline solid contaminated with topological defects, and to a generic glass. The highest yield stress belongs to the crystal, the lowest to the crystal with a few defects, with the glass in between. Although the glass is more disordered than the crystal with a few defects, it yields stress much higher than that of the latter. We explain this fact by considering the actual microscopic interactions that are typical of glassforming materials, pointing out the reasons for the higher cohesive nature of the glass. The main conclusion of this paper is that the instabilities encountered in stresscontrol condition are the identical saddlenode bifurcation seen in strain control. Accordingly one can use the latter condition to infer the former. Finally we discuss temperature effects and comment on the time needed to see a stresscontrolled material failure.

(2014) Physical Review E. 90, 4, 042315. Abstract
In this paper we focus on the mechanical properties of oligomeric glasses (waxes), employing a microscopic model that provides, via numerical simulations, information about the shear modulus of such materials, the failure mechanism via plastic instabilities, and the geometric responses of the oligomers themselves to a mechanical load. We present a microscopic theory that explains the numerically observed phenomena, including an exact theory of the shear modulus and of the plastic instabilities, both local and system spanning. In addition we present a model to explain the geometric changes in the oligomeric chains under increasing strains.

(2014) Physical Review B. 90, 9, 094501. Abstract
We present a comprehensive statistical study of free decay of the quantized vortex tangle in superfluid He4 at low and ultralow temperatures 0≤T≤1.1 K. Using highresolution vortex filament simulations with full BiotSavart vortex dynamics, we show that for ultralow temperatures T≲0.5 K, when the mutual friction parameters α≃α

(2014) Physical Review B. 90, 9, 094112. Abstract
The study of metallic glasses which exhibit magnetic interactions was mainly restricted to the amorphous solid regime and its properties. Here we study the glass transition in fluids where particles are endowed with spins, such that magnetic and positional degrees of freedom are coupled. Results for slowing down in the spin timecorrelation functions are described, and the effects of magnetic fields on the glass transition are studied. Aging effects in such systems and the corresponding data collapse are presented and discussed.

(2014) Journal Of Statistical MechanicsTheory And Experiment. 2014, 8, P08020. Abstract
We discuss a model metallic glass in which Barkhausen noise can be studied in exquisite detail, free of thermal effects and of the rate of ramping of the magnetic field. In this model the mechanism of the jumps in magnetic moment that cause the Barkhausen noise can be fully understood as consecutive instabilities where an eigenvalue of the Hessian matrix hits zero, leading to a magnetization jump Δm which is simultaneous with a stress and energy changes Δσ and ΔU respectively. Due to a large effect of local anisotropy, in this model Barkhausen noise is not due to movements of magnetic domain boundaries across pinning sites. There are no fractal domains, no selforganized criticality and no exact scaling behavior. We present a careful numerical analysis of the statistical properties of the phenomenon, and show that with every care taken this analysis is tricky, and easily misleading. Without a guiding theory it is almost impossible to get the right answer for the statistics of Barkhausen noise. We therefore present an analytic theory that culminates in a probability distribution function that is in excellent agreement with the simulations.

(2014) Physical Review B. 90, 2, 024508. Abstract
A nuclear capture reaction of a single neutron by ultracold superfluid He3 results in a rapid overheating followed by the expansion and subsequent cooling of the hot subregion, in a certain analogy with the big bang of the early universe. It was shown in a Grenoble experiment that a significant part of the energy released during the nuclear reaction was not converted into heat even after several seconds. It was thought that the missing energy was stored in a tangle of quantized vortex lines. This explanation, however, contradicts the expected lifetime of a bulk vortex tangle, 105104 s, which is much shorter than the observed time delay of seconds. In this paper we propose a scenario that resolves the contradiction: the vortex tangle, created by the hot spot, emits isolated vortex loops that take with them a significant part of the tangle's energy. These loops quickly reach the container walls. The dilute ensemble of vortex loops attached to the walls can survive for a long time, while the remaining bulk vortex tangle decays quickly.

(2014) EPL. 105, 3, 37006. Abstract
Amorphous magnetic solids, like metallic glasses, exhibit a novel effect: the growth of magnetic order as a function of mechanical strain under athermal conditions in the presence of a magnetic field. The magnetic moment increases in steps whenever there is a plastic event. Thus, plasticity induces the magnetic ordering, acting as the effective noise driving the system towards equilibrium. We present results of atomistic simulations of this effect in a model of a magnetic amorphous solid subjected to pure shear and a magnetic field. To elucidate the dependence on external strain and magnetic field we offer a meanfield theory that provides an adequate qualitative understanding of the observed phenomenon.

(2014) Acta Materialia. 63, p. 209215 Abstract
"Microalloying", which refers to the addition of small concentrations of a foreign metal to a given metallic glass, has been used extensively in recent years in attempts to improve the mechanical properties of these glasses. The results are haphazard and nonsystematic. In this paper we provide a microscopic theory of the effect of microalloying, exposing the delicate consequences of this procedure and the large parameter space which needs to be controlled. In particular we consider two very similar models which exhibit opposite trends for the change of the shear modulus, and explain the origins of this difference as displayed in the various microscopic structures and properties.

(2014) Physical Review B. 89, 1, 014502. Abstract
The paper presents a comprehensive characterization of welldeveloped vortex tangles in a turbulent counterflow in quantum fluids (with a laminar normal fluid component). We perform and analyze extensive numerical simulations using the vortex filament method, solving the full BiotSavart equations for the vortex dynamics in a wide range of temperatures and counterflow velocities. We start with the analysis of the macroscopic characteristics of the quantum vortex tangle such as vortex line density, its mean anisotropic and curvature parameters, the mean friction force between normal and superfluid components, the drift velocity of the vortex tangle, etc. Next we proceed to the main goal of the paper and move from the traditional macroscopic approach in terms of mean characteristics of the vortex tangle to the microscopic statistical and kinetic levels of description of quantum turbulence. These include objects that are much less studied or even totally neglected such as the vortex reconnection rates, the correlations and probability distribution functions (PDFs) of the vortex loop lengths, of the line curvature, of the mean curvatures of individual loops, the crosscorrelation function between the loop length and its mean curvature, and the autocorrelation function of the vortexline orientations. This detailed statistical information is required for a deeper understanding of quantum turbulence and for the development of its advanced theoretical description. In addition, we identify which of the studied properties are strongly affected by the choice of the reconnection criteria that are traditionally used in the vortex filament method and which of them are practically insensitive to the reconnection procedure. We conclude that the vortex filament method is sufficiently robust and wellsuited for the description of the steadystate vortex tangle in the quantum counterflow.
2013

(2013) EPL. 104, 4, 47003. Abstract
We present results of atomistic simulations of a new model of a magnetic amorphous solid subjected to mechanical strains and magnetic fields. Contrary to standard magnetic random systems which are studied on a lattice with random interaction, in the present approach all the randomness comes from the glassy nature of the material. The model employed offers new perspectives on important effects like plasticity and magnetostriction. It is shown that the plastic response in such systems exhibit singularities characterized by new exponents; the spatial structure of these plastic events requires a new coarse grained elastomagnetic theory which is provided here.

(2013) Physical review letters. 111, 16, 165701. Abstract
The dramatic dynamic slowing down associated with the glass transition is considered by many to be related to the existence of a static length scale that grows when temperature decreases. Defining, identifying, and measuring such a length is a subtle problem. Recently, two proposals, based on very different insights regarding the relevant physics, were put forward. One approach is based on the pointtoset correlation technique and the other on the scale where the lowest eigenvalue of the Hessian matrix becomes sensitive to disorder. Here we present numerical evidence that the two approaches might result in the same identical length scale. This provides mutual support for their relevance and, at the same time, raises interesting theoretical questions, discussed in the conclusion.

(2013) Physical review letters. 111, 14, 145302. Abstract
Experimental and simulational studies of the dynamics of vortex reconnections in quantum fluids showed that the distance d between the reconnecting vortices is close to a universal time dependence d=D[κt _{0}t]α with α fluctuating around 1/2 and κ=h/m is the quantum of circulation. Dimensional analysis, based on the assumption that the quantum of circulation κ=h/m is the only relevant parameter in the problem, predicts α=1/2. The theoretical calculation of the dimensionless coefficient D in this formula remained an open problem. In this Letter we present an analytic calculation of D in terms of the given geometry of the reconnecting vortices. We start from the numerically observed generic geometry on the way to vortex reconnection and demonstrate that the dynamics is well described by a selfsimilar analytic solution which provides the wanted information.

(2013) Physical Review E. 88, 3, 032401. Abstract
In this paper we extend the recent theory of shear localization in twodimensional (2D) amorphous solids to three dimensions. In two dimensions the fundamental instability of shear localization is related to the appearance of a line of displacement quadrupoles that makes an angle of 45 ^{â̂̃} with the principal stress axis. In three dimensions the fundamental plastic instability is also explained by the formation of a lattice of anisotropic elastic inclusions. In the case of pure external shear stress, we demonstrate that this is a 2D triangular lattice of similar elementary events. It is shown that this lattice is arranged on a plane that, similarly to the 2D case, makes an angle of 45 ^{â̂̃} with the principal stress axis. This solution is energetically favorable only if the external strain exceeds a yieldstrain value that is determined by the strain parameters of the elementary events and the Poisson ratio. The predictions of the theory are compared to numerical simulations and very good agreement is observed.

(2013) Physical Review E. 88, 2, 022310. Abstract
It is well known experimentally that wellquenched amorphous solids exhibit a plastic instability in the form of a catastrophic shear localization at a welldefined value of the external strain. The instability may develop to a shear band that in some cases is followed by a fracture. It is also known that the values of the yield strain (and yield stress), as well as the direction of the shear band with respect to the principal stress axis, vary considerably with variations in the external loading conditions. In this paper we present a microscopic theory of these phenomena for twodimensional athermal amorphous solids that are strained quasistatically. We present analytic formulas for the yield strains for different loading conditions, as well as for the angles of the shear bands. We explain that the external loading conditions determine the eigenvalues of the quadrupolar Eshelby inclusions which model the nonaffine displacement field. These inclusions model elementary plastic events and determine both the yield strain and the direction of the shear band. We show that the angles of the shear bands with respect to the principal stress axis are limited theoretically between 30^{} and 60^{}. Available experimental data conform to this prediction.

(2013) Applied Physics Letters. 102, 19, 191904. Abstract
The usefulness of glasses, and particularly of metallic glasses, in technological applications is often limited by their toughness, (which is defined as the area under the stress vs. strain curve before plastic yielding). Recently, toughness was found to increase significantly by the addition of small concentrations of foreign atoms that act as pinning centers. We model this phenomenon at zero temperature and quasistatic straining with randomly positioned particles that participate in the elastic deformation, but are pinned in the nonaffine return to mechanical equilibrium. We find a very strong effect on toughness via the increase of both the shear modulus and the yield stress as a function of the density of pinned particles. Understanding the results calls for analyzing separately the elastic, or Born term, and the contributions of the excess modes that result from glassy disorder. Finally, we present a scaling theory that collapses the data on one universal curve as a function of rescaled variables.

(2013) Physical Review E. 87, 4, 042310. Abstract
Rapidly cooling a liquid may result in a glass transition, creating an amorphous solid whose shear and bulk moduli are finite. Even when done with constant density, these resulting moduli depend strongly on the rate of cooling. Understanding this phenomenon calls for analyzing separately the "Born term" that exists also in perfectly ordered materials and the contributions of the "excess modes" that result from glassy disorder. We show that the Born term is very insensitive to the cooling rate, and all the variation in the shear modulus is due to the excess modes. We argue that this approach provides a quantitative understanding of the cooling rate dependence of a basic linear response coefficient, i.e., the shear modulus.

(2013) Physical Review E. 87, 2, 022810. Abstract
In recent research it was found that the fundamental shearlocalizing instability of amorphous solids under external strain, which eventually results in a shear band and failure, consists of a highly correlated array of Eshelby quadrupoles all having the same orientation and some density rho. In this paper we calculate analytically the energy E(rho, gamma) associated with such highly correlated structures as a function of the density rho and the external strain gamma. We show that for strains smaller than a characteristic strain gamma(Y) the total strain energy initially increases as the quadrupole density increases, but that for strains larger than gamma(Y) the energy monotonically decreases with quadrupole density. We identify gamma(Y) as the yield strain. Its value, derived from values of the qudrupole strength based on the atomistic model, agrees with that from the computed stressstrain curves and broadly with experimental results. DOI: 10.1103/PhysRevE.87.022810

(2013) Condensed Matter Physics. 16, 1, 13004. Abstract
The unied description of diffusion processes that cross over from a ballistic behavior at short times to normal or anomalous diffusion (sub or superdiffusion) at longer times is constructed on the basis of a nonMarkovian generalization of the FokkerPlanck equation. The necessary nonMarkovian kinetic coeffcients are determined by the observable quantities (mean andmean square displacements). Solutions of the nonMarkovian equation describing diffusive processes in the physical space are obtained. For long times, these solutions agree with the predictions of the continuous random walk theory; they are, however, much superior at shorter times when the effect of the ballistic behavior is crucial.

(2013) 4Th International Symposium On Slow Dynamics In Complex Systems: Keep Going Tohoku. 1518, p. 162169 Abstract
We present a short review of the theory of shear localization which results in shear bands in amorphous solids. As this is the main mechanism for the failure of metallic glasses, understanding the instability is invaluable in finding how to stabilize such materials against the tendency to shear localize. We explain the mechanism for shear localization under external shearstrain, which in 2dimensions is the appearance of highly correlated lines of Eshelbylike quadrupolar singularities which organize the nonaffine plastic flow of the amorphous solid into a shear band. We prove analytically that such highly correlated solutions in which N equidistant quadrupoles are aligned with equal orientations are minimum energy states when the strain is high enough. The line lies at 45 degrees to the compressive stress. We use the theory to first predict the yield strain at zero temperature and quasistatic conditions, but later generalize to the case of finite temperature and finite shear rates, deriving the JohnsonSamwer T2/3 law.

(2013) Physical Review B. 87, 2, 020101(R). Abstract
Metallic glasses are prone to fail mechanically via a shearbanding instability. In a remarkable paper Johnson and Samwer demonstrated that this failure enjoys a high degree of universality in the sense that a large group of metallic glasses appears to possess a yield strain that decreases with temperature following a T2^{/}3 law up to logarithmic corrections. In this Rapid Communication we offer a theoretical derivation of this law. We show that our formula fits very well simulation data on typical amorphous solids.

(2013) Physical Review E. 87, 1, 012801. Abstract
We employ a recently developed model that allows the study of twodimensional brittle crack propagation under fixed grip boundary conditions. The crack development highlights the importance of voids which appear ahead of the crack as observed in experiments on the nanoscale. The appearance of these voids is responsible for roughening the crack path on small scales, in agreement with theoretical expectations. With increasing speed of propagation one observes the branching instabilities that were reported in experiments. The simulations allow understanding the phenomena by analyzing the elastic stress field that accompanies the crack dynamics.

(2013) Physical review letters. 110, 1, 014502. Abstract
We consider the intermittent behavior of superfluid turbulence in He4. Because of the similarity in the nonlinear structure of the twofluid model of superfluidity and the Euler and NavierStokes equations, one expects the scaling exponents of the structure functions to be the same as in classical turbulence for temperatures close to the superfluid transition T_{λ} and also for T≪T_{λ}. This is not the case when the densities of normal and superfluid components are comparable to each other and mutual friction becomes important. Using shell model simulations, we propose that in this situation there exists a range of scales in which the effective exponents indicate stronger intermittency. We offer a bridge relation between these effective and the classical scaling exponents. Since this effect occurs at accessible temperatures and Reynolds numbers, we propose that experiments should be conducted to further assess the validity and implications of this prediction.
2012

(2012) Physical review letters. 109, 25, 255502. Abstract
The fundamental instability responsible for the shear localization which results in shear bands in amorphous solids remains unknown despite an enormous amount of research, both experimental and theoretical. As this is the main mechanism for the failure of metallic glasses, understanding the instability is invaluable in finding how to stabilize such materials against the tendency to shear localize. In this Letter we explain the mechanism for shear localization under shear, which is the appearance of highly correlated lines of Eshelbylike quadrupolar singularities which organize the nonaffine plastic flow of the amorphous solid into a shear band. We prove analytically that such highly correlated solutions in which N quadrupoles are aligned with equal orientations are minimum energy states when the strain is high enough. The line lies at 45 degrees to the compressive stress.

(2012) Physical Review E. 86, 6, 061502. Abstract
Over the past decade, computer simulations have had an increasing role in shedding light on difficult statistical physical phenomena, and in particular on the ubiquitous problem of the glass transition. Here in a wide variety of materials the viscosity of a supercooled liquid increases by many orders of magnitude upon decreasing the temperature over a modest range. A natural concern in these computer simulations is the very small size of the simulated systems compared to experimental ones, raising the issue of how to assess the thermodynamic limit. Here we turn this limitation to our advantage by performing finite size scaling on the system size dependence of the relaxation time for supercooled liquids to emphasize the importance of a growing static length scale in the theory of glass transition. We demonstrate that the static length scale that was discovered by us in Physica A0378437110.1016/j.physa.2011.11.020 391, 1001 (2012) fits the bill extremely well, allowing us to provide a finitesize scaling theory for the αrelaxation time of the glass transition, including predictions for the thermodynamic limit based on simulations in small systems.

(2012) EPL. 99, 4, 46002. Abstract
This letter is motivated by some recent experiments on pancakeshaped nanosamples of metallic glass that indicate a decline in the measured shear modulus upon decreasing the sample radius. Similar measurements on crystalline samples of the same dimensions showed a much more modest change. In this letter we offer a theory of this phenomenon; we argue that such results are generically expected for any amorphous solid, with the main effect being related to the increased contribution of surfaces with respect to the bulk when the samples get smaller. We employ exact relations between the shear modulus and the eigenvalues of the system's Hessian matrix to explore the role of surface modes in affecting the elastic moduli.

(2012) EPL. 99, 4, 46003. Abstract
Kelvin waves propagating on quantum vortices play a crucial role in the phenomenology of energy dissipation of superfluid turbulence. Previous theoretical studies have consistently focused on the zerotemperature limit of the statistical physics of Kelvinwave turbulence. In this letter, we go beyond this athermal limit by introducing a small but finite temperature in the form of nonzero mutual friction dissipative force; A situation regularly encountered in actual experiments of superfluid turbulence. In this case we show that there exists a new typical length scale separating a quasiinertial range of Kelvinwave turbulence from a fardissipation range. The letter culminates with analytical predictions for the energy spectrum of the Kelvinwave turbulence in both of these regimes.

(2012) EPL. 99, 2, 26003. Abstract
We address the crosseffects between mechanical strains and magnetic fields on the plastic response of magnetoelastic amorphous solids. It is well known that plasticity in nonmagnetic amorphous solids under external strain γ is dominated by the codimension1 saddlenode bifurcation in which an eigenvalue of the Hessian matrix vanishes at γ _{P} like √ γ _{P}γ. This squareroot singularity determines much of the statistical physics of elastoplasticity, and in particular that of the stressstrain curves under athermalquasistatic conditions. In this letter we discuss the much richer physics that can be expected in magnetic amorphous solids. Firstly, magnetic amorphous solids exhibit codimension2 plastic instabilities, when an external strain and an external magnetic field are applied simultaneously. Secondly, the phase diagrams promise a rich array of new effects that have been barely studied; this opens up a novel and extremely rich research program for magnetoplastic materials.

(2012) Physical Review E. 86, 1, 016302. Abstract
We developed a model for the enhancement of the heat flux by spherical and elongated nanoparticles in sheared laminar flows of nanofluids. Besides the heat flux carried by the nanoparticles, the model accounts for the contribution of their rotation to the heat flux inside and outside the particles. The rotation of the nanoparticles has a twofold effect: it induces a fluid advection around the particle and it strongly influences the statistical distribution of particle orientations. These dynamical effects, which were not included in existing thermal models, are responsible for changing the thermal properties of flowing fluids as compared to quiescent fluids. The proposed model is strongly supported by extensive numerical simulations, demonstrating a potential increase of the heat flux far beyond the MaxwellGarnett limit for the spherical nanoparticles. The road ahead, which should lead toward robust predictive models of heat flux enhancement, is discussed.

(2012) Physical Review E. 85, 6, 061501. Abstract
Generic glass formers exhibit at least two characteristic changes in their relaxation behavior, first to an Arrheniustype relaxation at some characteristic temperature and then at a lower characteristic temperature to a superArrhenius (fragile) behavior. We address these transitions by studying the statistics of free energy barriers for different systems at different temperatures and space dimensions. We present a clear evidence for changes in the dynamical behavior at the transition to Arrhenius and then to a superArrhenius behavior. A simple model is presented, based on the idea of competition between singleparticle and cooperative dynamics. We argue that Arrhenius behavior can take place as long as there is enough free volume for the completion of a simple T1 relaxation process. Once free volume is absent one needs a cooperative mechanism to "collect" enough free volume. We show that this model captures all the qualitative behavior observed in simulations throughout the considered temperature range.

(2012) Physical Review B. 85, 10, 104502. Abstract
In superfluid 3HeB, turbulence is carried predominantly by the superfluid component. To explore the statistical properties of this quantum turbulence and its differences from the classical counterpart, we adopt the timehonored approach of shell models. Using this approach, we provide numerical simulations of a Sabra shell model that allows us to uncover the nature of the energy spectrum in the relevant hydrodynamic regimes. These results are in qualitative agreement with analytical expressions for the superfluid turbulent energy spectra that were found using a differential approximation for the energy flux.

(2012) Physical review letters. 108, 7, 074501. Abstract
Fractal decimation reduces the effective dimensionality D of a flow by keeping only a (randomly chosen) set of Fourier modes whose number in a ball of radius k is proportional to kD for large k. At the critical dimension D _{c}=4/3 there is an equilibrium Gibbs state with a k ^{}5 ^{/}3 spectrum, as in V. L'vov et al., Phys. Rev. Lett. 89, 064501 (2002)PRLTAO0031900710.1103/PhysRevLett.89.064501. Spectral simulations of fractally decimated twodimensional turbulence show that the inverse cascade persists below D=2 with a rapidly rising Kolmogorov constant, likely to diverge as (D4/3) ^{}2 ^{/}3.

(2012) Physical Review Letters. 108, 7, 075701. Abstract
By comparing the response to external strains in metallic glasses and in LennardJones glasses we find a quantitative universality of the fundamental plastic instabilities in the athermal, quasistatic limit. Microscopically these two types of glasses are as different as one can imagine, the latter being determined by binary interactions, whereas the former is determined by multiple interactions due to the effect of the electron gas that cannot be disregarded. In spite of this enormous difference the plastic instability is the same saddlenode bifurcation. As a result, the statistics of stress and energy drops in the elastoplastic steady state are universal, sharing the same systemsize exponents.

(2012) EPL. 97, 3, 36010. Abstract
Tetrahedral liquids such as water and silicamelt show unusual thermodynamic behavior such as a density maximum and an increase in specific heat when cooled to low temperatures. Previous work had shown that Monte Carlo and meanfield solutions of a lattice model can exhibit these anomalous properties with or without a phase transition, depending on the values of the different terms in the Hamiltonian. Here we use a somewhat different approach, where we start from a very popular empirical model of tetrahedral liquids  the StillingerWeber model  and construct a coarsegrained theory which directly quantifies the local structure of the liquid as a function of volume and temperature. We compare the theory to moleculardynamics simulations and show that the theory can rationalize the simulation results and the anomalous behavior.

(2012) Physica AStatistical Mechanics And Its Applications. 391, 4, p. 10011008 Abstract
Glasses are liquids whose viscosity has increased so much that they cannot flow. Accordingly, there have been many attempts to define a static lengthscale associated with the dramatic slowing down of supercooled liquid with decreasing temperature. Here we present a simple method to extract the desired lengthscale which is highly accessible both for experiments and for numerical simulations. The fundamental new idea is that low lying vibrational frequencies come in two types, those related to elastic response and those determined by plastic instabilities. The minimal observed frequency is determined by one or the other, crossing at a typical lengthscale which is growing with the approach of the glass transition. This lengthscale characterizes the correlated disorder in the system: on longer lengthscales the details of the disorder become irrelevant, dominated by the Debye model of elastic modes. To connect the newly defined lengthscale to relaxation dynamics near the glass transition, we show that supercooled liquids in which there exist random pinning sites of density ^{ρim}∼1ξsd exhibit complete jamming of all dynamics. This is a direct demonstration that the proposed length scale is indeed the static length that was long soughtafter.

(2012) EPL. 100, 3, 36003. Abstract
Glass formers exhibit, upon an oscillatory excitation, a response function whose imaginary and real parts are known as the loss and storage moduli, respectively. The loss modulus typically peaks at a frequency known as the α frequency which is associated with the main relaxation mechanism of the supercooled liquid. In addition, the loss modulus is decorated by a smaller peak, shoulder or wing which is referred to as the βpeak. The physical origin of this secondary peak had been debated for decades, with proposed mechanisms ranging from highly localized relaxations to entirely cooperative ones. Using numerical simulations we bring an end to the debate, exposing a clear and unique cooperative mechanism for the said βpeak which is distinct from that of the αpeak.
2011

(2011) Journal Of Physical Chemistry B. 115, 48, p. 1430114310 Abstract
We extend our statistical mechanical theory of the glass transition from examples consisting of point particles to molecular liquids with internal degrees of freedom. As before, the fundamental assertion is that supercooled liquids are ergodic, although becoming very viscous at lower temperatures, and are therefore describable in principle by statistical mechanics. The theory is based on analyzing the local neighborhoods of each molecule, and a statistical mechanical weight is assigned to every possible local organization. This results in an approximate theory that is in very good agreement with simulations regarding both thermodynamical and dynamical properties.

(2011) Physical Review E. 84, 4, 046105. Abstract
Brittle materials exhibit sharp dynamical fractures when meeting Griffith's criterion, whereas ductile materials blunt a sharp crack by plastic responses. Upon continuous pulling, ductile materials exhibit a necking instability that is dominated by a plastic flow. Usually one discusses the brittle to ductile transition as a function of increasing temperature. We introduce an athermal brittle to ductile transition as a function of the cutoff length of the interparticle potential. On the basis of extensive numerical simulations of the response to pulling the material boundaries at a constant speed we offer an explanation of the onset of ductility via the increase in the density of plastic modes as a function of the potential cutoff length. Finally we can resolve an old riddle: In experiments brittle materials can be strained under grip boundary conditions and exhibit a dynamic crack when cut with a sufficiently long initial slot. Mysteriously, in molecular dynamics simulations it appeared that cracks refused to propagate dynamically under grip boundary conditions, and continuous pulling was necessary to achieve fracture. We argue that this mystery is removed when one understands the distinction between brittle and ductile athermal amorphous materials.

(2011) Physical Review B. 84, 6, 064516. Abstract
We study the statistical and dynamical behavior of turbulent Kelvin waves propagating on quantized vortices in superfluids and address the controversy concerning the energy spectrum that is associated with these excitations. Finding the correct energy spectrum is important because Kelvin waves play a major role in the dissipation of energy in superfluid turbulence at nearzero temperatures. In this paper, we show analytically that the solution proposed by enjoys existence, uniqueness, and regularity of the prefactor. Furthermore, we present numerical results of the dynamical equation that describes to leading order the nonlocal regime of the Kelvinwave dynamics. We compare our findings with the analytical results from the proposed local and nonlocal theories for Kelvinwave dynamics and show an agreement with the nonlocal predictions. Accordingly, the spectrum proposed by L'vov and Nazarenko should be used in future theories of quantum turbulence. Finally, for weaker wave forcing we observe an intermittent behavior of the wave spectrum with a fluctuating dissipative scale, which we interpreted as a finitesize effect characteristic of mesoscopic wave turbulence.

(2011) Physical Review E. 83, 6, 061101. Abstract
We study the elastic theory of amorphous solids made of particles with finite range interactions in the thermodynamic limit. For the elastic theory to exist, one requires all the elastic coefficients, linear and nonlinear, to attain a finite thermodynamic limit. We show that for such systems the existence of nonaffine mechanical responses results in anomalous fluctuations of all the nonlinear coefficients of the elastic theory. While the shear modulus exists, the first nonlinear coefficient B_{2} has anomalous fluctuations and the second nonlinear coefficient B_{3} and all the higher order coefficients (which are nonzero by symmetry) diverge in the thermodynamic limit. These results call into question the existence of elasticity (or solidity) of amorphous solids at finite strains, even at zero temperature. We discuss the physical meaning of these results and propose that in these systems elasticity can never be decoupled from plasticity: the nonlinear response must be very substantially plastic.

(2011) Physical Review E. 83, 4, 046106. Abstract
A crucially important material parameter for all amorphous solids is the yield stress, which is the value of the stress for which the material yields to plastic flow when it is strained quasistatically at zero temperature. It is difficult in laboratory experiments to determine what parameters of the interparticle potential affect the value of the yield stress. Here we use the versatility of numerical simulations to study the dependence of the yield stress on the parameters of the interparticle potential. We find a very simple dependence on the fundamental scales that characterize the repulsive and attractive parts of the potential, respectively, and offer a scaling theory that collapses the data for widely different potentials and in different space dimensions.

(2011) Physical Review E. 83, 2, 026106. Abstract
The stress field at the tip of a crack of a thin plate of elastic material that is broken due to a mode III shear tearing has a universal form with a nonuniversal amplitude, known as the stress intensity factor, which depends on the crack length and the boundary conditions. We present in this paper exact analytic results for this stress intensity factor, thus enriching the small number of exact results that can be obtained within linear elastic fracture mechanics.
2010

(2010) Physical Review E. 82, 5, 055103. Abstract
The art of making structural, polymeric, and metallic glasses is rapidly developing with many applications. A limitation is that under increasing external strain all amorphous solids (like their crystalline counterparts) have a finite yield stress which cannot be exceeded without effecting a plastic response which typically leads to mechanical failure. Understanding this is crucial for assessing the risk of failure of glassy materials under mechanical loads. Here we show that the statistics of the energy barriers ΔE that need to be surmounted changes from a probability distribution function that goes smoothly to zero as ΔE=0 to a pdf which is finite at ΔE=0. This fundamental change implies a dramatic transition in the mechanical stability properties with respect to external strain. We derive exact results for the scaling exponents that characterize the magnitudes of average energy and stress drops in plastic events as a function of system size.

(2010) Physical Review E. 82, 3, 031301. Abstract
The effect of finite temperature T and finite strain rate γ on the statistical physics of plastic deformations in amorphous solids made of N particles is investigated. We recognize three regimes of temperature where the statistics are qualitatively different. In the first regime the temperature is very low, T

(2010) Physical Review E. 82, 2, 026105. Abstract
We derive expressions for the lowest nonlinear elastic constants of amorphous solids in athermal conditions (up to third order), in terms of the interaction potential between the constituent particles. The effect of these constants cannot be disregarded when amorphous solids undergo instabilities such as plastic flow or fracture in the athermal limit; in such situations the elastic response increases enormously, bringing the system much beyond the linear regime. We demonstrate that the existing theory of thermal nonlinear elastic constants converges to our expressions in the limit of zero temperature. We motivate the calculation by discussing two examples in which these nonlinear elastic constants play a crucial role in the context of elastoplasticity of amorphous solids. The first example is the plasticityinduced memory that is typical to amorphous solids (giving rise to the Bauschinger effect). The second example is how to predict the next plastic event from knowledge of the nonlinear elastic constants. Using the results of our calculations we derive a simple differential equation for the lowest eigenvalue of the Hessian matrix in the external strain near mechanical instabilities; this equation predicts how the eigenvalue vanishes at the mechanical instability and the value of the strain where the mechanical instability takes place.

(2010) Physical Review E. 82, 2, 026104. Abstract
Amorphous solids that underwent a strain in one direction such that they responded in a plastic manner "remember" that direction also when relaxed back to a state with zero mean stress. We address the question "what is the order parameter that is responsible for this memory?" and is therefore the reason for the different subsequent responses of the material to strains in different directions. We identify such an order parameter which is readily measurable, we discuss its trajectory along the stressstrain curve, and propose that it and its probability distribution function must form a necessary component of a theory of elastoplasticity.

(2010) Physical Review E. 81, 6, 066103. Abstract
Motivated by recent experiments, we present a study of the dynamics of cracks in thin sheets. While the equations of elasticity for thin plates are well known, there remains the question of path selection for a propagating crack. We invoke a generalization of the principle of local symmetry to provide a criterion for path selection and demonstrate qualitative agreement with the experimental findings. The nature of the singularity at the crack tip is studied with and without the interference of nonlinear terms.

(2010) Physical review letters. 104, 21, 215502. Abstract
We propose a method to predict the value of the external strain where a generic amorphous solid will fail by a plastic response (i.e., an irreversible deformation), solely on the basis of measurements of the nonlinear elastic moduli. While usually considered fundamentally different, with the elastic properties describing reversible phenomena and plastic failure epitomizing irreversible behavior, we show that the knowledge of some nonlinear elastic moduli is enough to predict where plasticity sets in.

(2010) Physical Review E. 81, 3, 030105. Abstract
We address the now classical problem of a diffusion process that crosses over from a ballistic behavior at short times to a fractional diffusion (subdiffusion or superdiffusion) at longer times. Using the standard nonMarkovian diffusion equation we demonstrate how to choose the memory kernel to exactly respect the two different asymptotics of the diffusion process. Having done so we solve for the probability distribution function (pdf) as a continuous function which evolves inside a ballistically expanding domain. This general solution agrees for long times with the pdf obtained within the continuous randomwalk approach, but it is much superior to this solution at shorter times where the effect of the ballistic regime is crucial.

(2010) Physical Review B. 81, 10, 100201. Abstract
An effective temperature Teff which differs from the bath temperature is believed to play an essential role in the theory of elastoplasticity of amorphous solids. Here, we introduce a natural definition of Teff appearing naturally in a Boltzmannlike distribution of measurable structural features without recourse to any questionable assumption. The value of Teff is connected, using theory and scaling concepts, to the flow stress and the mean energy that characterize the elastoplastic flow.

(2010) Physical review letters. 104, 2, 025501. Abstract
We address the systemsize dependence of plastic flow events when an amorphous solid is put under a fixed external strain rate at a finite temperature. For small system sizes at low strain rates and low temperatures the magnitude of plastic events grows with the system size. We explain, however, that this must be a finitesize effect; for larger systems there exist two crossover length scales ξ1 and ξ2, the first determined by the elastic time scale and the second by the thermal energy scale. For systems of size L ξ there must exist (L/ξ)d uncorrelated plastic events which occur simultaneously. We present a scaling theory that culminates with the dependence of the crossover scales on temperature and strain rate. Finally, we relate these findings to the temperature and size dependence of the stress fluctuations. We comment on the importance of these considerations for theories of elastoplasticity.

(2010) Condensed Matter Physics. 13, 2, p. 2300123008 Abstract
We address the now classical problem of a diffusion process that crosses over from a ballistic behavior at short times to a fractional diffusion (sub or superdiffusion) at longer times. Using the standard nonMarkovian diffusion equation we demonstrate how to choose the memory kernel to exactly respect the two different asymptotics of the diffusion process. Having done so we solve for the probability distribution function as a continuous function which evolves inside a ballistically expanding domain. This general solution agrees for long times with the probability distribution function obtained within the continuous random walk approach but it is much superior to this solution at shorter times where the effect of the ballistic regime is crucial.

(2010) Condensed Matter Physics. 13, 2, 030105(R). Abstract
We address the, now classical problem of a diffusion process that crosses over from a ballistic behavior at short times to a fractional diffusion (sub or superdiffusion) at longer times. Using the standard nonMarkovian diffusion equation we demonstrate how to choose the memory kernel to exactly respect the two different asymptotics of the diffusion process. Having done so we solve for the probability distribution function as a continuous function which evolves inside a ballistically expanding domain. This general solution agrees for long times with the probability distribution function obtained within the continuous random walk approach but it is much superior to this solution at shorter times where the effect of the ballistic regime is crucial.
2009

(2009) Physical Review E. 80, 6, 066319. Abstract
Finitedimensional wave turbulence refers to the chaotic dynamics of interacting wave "clusters" consisting of finite number of connected wave triads with exact threewave resonances. We examine this phenomenon using the example of atmospheric planetary (Rossby) waves. It is shown that the dynamics of the clusters is determined by the types of connections between neighboring triads within a cluster; these correspond to substantially different scenarios of energy flux between different triads. All the possible cases of the energy cascade termination are classified. Free and forced chaotic dynamics in the clusters are investigated: due to the huge fluctuations of the energy exchange between resonant triads these two types of evolution have a lot in common. It is confirmed that finitedimensional wave turbulence in finite wave systems is fundamentally different from kinetic wave turbulence in infinite systems; the latter is described by wavekinetic equations that account for interactions with overlapping quasiresonances of finite amplitude waves. The present results are directly applicable to finitedimensional wave turbulence in any wave system in finite domains with threemode interactions as encountered in hydrodynamics, astronomy, plasma physics, chemistry, medicine, etc.

(2009) Journal Of Statistical MechanicsTheory And Experiment. P11010. Abstract
Using two extremely different models of glass formers in two and three dimensions we demonstrate how to encode the subtle changes in the geometric rearrangement of particles during the scenario of the glass transition. We construct a statistical mechanical description that is able to explain and predict the geometric rearrangement, the temperaturedependent thermodynamic functions and the alpharelaxation time within the measured temperature range and beyond. The theory is based on an upscaling to proper variables (quasispecies) which is validated using a simple criterion. Once constructed, the theory provides an accurate predictive tool for quantities like the specific heat or the entropy at temperatures that cannot be reached by measurements. In addition, the theory identifies a rapidly increasing typical length scale xi as the temperature decreases. This growing spatial length scale determines the alpharelaxation time as tau(alpha) similar to exp(mu xi/T), where mu is a typical chemical potential per unit length.

(2009) Physical Review B. 80, 17, 174201. Abstract
Much of the discussion in the literature of the lowfrequency part of the density of states of amorphous solids was dominated for years by comparing measured or simulated density of states to the classical Debye model. Since this model is hardly appropriate for the materials at hand, this created some amount of confusion regarding the existence and universality of the socalled "boson peak" which results from such comparisons. We propose that one should pay attention to the different roles played by different aspects of disorder, the first being disorder in the interaction strengths, the second positional disorder, and the third coordination disorder. These have different effects on the lowfrequency part of the density of states. We examine the density of states of a number of tractable models in one and two dimensions and reach a clearer picture of the softening and redistribution of frequencies in such materials. We discuss the effects of disorder on the elastic moduli and the relation of the latter to frequency softening, reaching the final conclusion that the boson peak is not universal at all.

(2009) Physical Review E. 80, 2, 026128. Abstract
Strongly correlated amorphous solids are a class of glass formers whose interparticle potential admits an approximate inverse powerlaw form in a relevant range of interparticle distances. We study the steadystate plastic flow of such systems, first in the athermal quasistatic limit and second at finite temperatures and strain rates. In all cases we demonstrate the usefulness of scaling concepts to reduce the data to universal scaling functions where the scaling exponents are determined a priori from the interparticle potential. In particular we show that the steady plastic flow at finite temperatures with efficient heat extraction is uniquely characterized by two scaled variables; equivalently, the steadystate displays an equation of state that relates one scaled variable to the other two. We discuss the range of applicability of the scaling theory, and the connection to density scaling in supercooled liquid dynamics. We explain that the description of transient states calls for additional state variables whose identity is still far from obvious.

(2009) Physical Review E. 79, 6, 066109. Abstract
A number of current theories of plasticity in amorphous solids assume at their basis that plastic deformations are spatially localized. We present in this paper a series of numerical experiments to test the degree of locality of plastic deformation. These experiments increase in terms of the stringency of the removal of elastic contributions to the observed elastoplastic deformations. It is concluded that for all our simulational protocols the plastic deformations are not localized, and their scaling is subextensive. We offer a number of measures of the magnitude of the plastic deformation, all of which display subextensive scaling characterized by nontrivial exponents. We provide some evidence that the scaling exponents governing the subextensive scaling laws are nonuniversal, depending on the degree of disorder and on the parameters of the systems. Nevertheless, understanding what determines these exponents should shed considerable light on the physics of amorphous solids.

(2009) Physical Review B. 79, 18, 180203(R). Abstract
The shear modulus and yield stress of amorphous solids are important material parameters, with the former determining the rate of increase in stress under external strain and the latter being the stress value at which the material flows in a plastic manner. It is therefore important to understand how these parameters can be related to the interparticle potential. Here a scaling theory is presented such that given the interparticle potential, the dependence of the yield stress, and the shear modulus on the density of the solid can be predicted in the athermal limit. It is explained when such prediction is possible at all densities and when it is only applicable at high densities.

(2009) Physical Review E. 79, 4, 045304(R). Abstract
In light of some recent experiments on quasi twodimensional (2D) turbulent channel flow we provide here a model of the ideal case, for the sake of comparison. The ideal 2D channel flow differs from its threedimensional (3D) counterpart by having a second quadratic conserved variable in addition to the energy and the latter has an inverse rather than a direct cascade. The resulting qualitative differences in profiles of velocity V and energy K as a function of the distance from the wall are highlighted and explained. The most glaring difference is that the 2D channel is much more energetic, with K in wall units increasing logarithmically with the Reynolds number Reτ instead of being Reτ independent in 3D channels.

(2009) Physical Review E. 79, 4, 046109. Abstract
Volume alteration in solid materials is a common cause of material failure. Here we investigate the crack formation in thin elastic layers attached to a substrate. We show that small variations in the volume contraction and substrate restraint can produce widely different crack patterns ranging from spirals to complex hierarchical networks. The networks are formed when there is no prevailing gradient in material contraction, whereas spirals are formed in the presence of a radial gradient in the contraction of a thin elastic layer.

(2009) Physical Review E. 79, 3, 031501. Abstract
The ShintaniTanaka model is a glassforming system whose constituents interact via an anisotropic potential depending on the angle of a unit vector carried by each particle. The decay of timecorrelation functions of the unit vectors exhibits the characteristics of generic relaxation functions during glass transitions. In particular it exhibits a stretched exponential form, with the stretching index β depending strongly on the temperature. We construct a quantitative theory of this correlation function by analyzing all the physical processes that contribute to it, separating a rotational from a translational decay channel. These channels exhibit different relaxation times, each with its own temperature dependence. Interestingly, the separate decay function of each of these processes is a temperatureindependent function, and is shown to scale (exhibit data collapse) at different temperatures. Taken together with temperaturedependent weights determined a priori by statistical mechanics this allows one to generate the observed correlation function in quantitative agreement with simulations at different temperatures. This underlines the danger of concluding anything about glassy relaxation functions without detailed physical scrutiny.

(2009) Physical review letters. 102, 12, 125701. Abstract
In the context of a classical example of glass formation in three dimensions, we exemplify how to construct a statisticalmechanical theory of the glass transition. At the heart of the approach is a simple criterion for verifying a proper choice of upscaled quasispecies that allow the construction of a theory with a finite number of "states." Once constructed, the theory identifies a typical scale ξ that increases rapidly with lowering the temperature and which determines the αrelaxation time τα as τα∼exp(mu;ξ/T), with μ a typical chemical potential. The theory can predict relaxation times at temperatures that are inaccessible to numerical simulations.

(2009) European Physical JournalSpecial Topics. 178, 1, p. 81122 Abstract
In this short review I summarize some progress achieved in my research group regarding the glass transition and the mechanical properties of the resulting amorphous solids. Our main concerns were on the one hand to understand the extreme slowing down over a narrow range of temperatures which results in a viscosity so high that the materials behave as solids under small mechanical strains. On the other hand we are interested in their mechanical yield to higher strains. After yielding the materials can be in an elastoplastic steady state which we want to understand and characterize.

(2009) Environmental Fluid Mechanics. 9, 3, p. 267295 Abstract
We address the dynamical and statistical description of stably stratified turbulent boundary layers with the important example of the atmospheric boundary layer with a stable temperature stratification in mind. Traditional approaches to this problem, based on the profiles of mean quantities, velocity secondorder correlations, and dimensional estimates of the turbulent thermal flux run into a wellknown difficulty, predicting the suppression of turbulence at a small critical value of the Richardson number, in contradiction with observations. Phenomenological attempts to overcome this problem suffer from various theoretical inconsistencies. Here we present a closure approach taking into full account all the secondorder statistics, which allows us to respect the conservation of total mechanical energy. The analysis culminates in an analytic solution of the profiles of all mean quantities and all secondorder correlations removing the unphysical predictions of previous theories. We propose that the approach taken here is sufficient to describe the lower parts of the atmospheric boundary layer, as long as the Richardson number does not exceed an order of unity. For much higher Richardson numbers the physics may change qualitatively, requiring careful consideration of the potential KelvinHelmoholtz waves and their interaction with the vortical turbulence.
2008

(2008) Physical review letters. 101, 26, 265701. Abstract
We present new simulation results for the specific heat in a classical model of a binary mixture glass former in two dimensions. We show that in addition to the formerly observed specific heat peak, there is a second peak at lower temperatures which was not observable in earlier simulations. This is a surprise, as most texts on the glass transition expect a single specific heat peak. We explain the physics of the two specific heat peaks by the micromelting of two types of clusters. While this physics is easily accessible, the consequences are that one should not expect any universality in the temperature dependence of the specific heat in glass formers.

(2008) Physical Review E. 78, 6, 061504. Abstract
Experimental measurements of the specific heat in glassforming systems are obtained from the linear response to either slow cooling (or heating) or to oscillatory perturbations with a given frequency about a constant temperature. The latter method gives rise to a complex specific heat with the constraint that the zero frequency limit of the real part should be identified with thermodynamic measurements. Such measurements reveal anomalies in the temperature dependence of the specific heat, including the so called "specific heat peak" in the vicinity of the glass transition. The aim of this paper is to provide theoretical explanations of these anomalies in general and a quantitative theory in the case of a simple model of glass formation. We first present interesting simulation results for the specific heat in a classical model of a binary mixture glass former. We show that in addition to the formerly observed specific heat peak there is a second peak at lower temperatures which was not observable in earlier simulations. Second, we present a general relation between the specific heat, a caloric quantity, and the bulk modulus of the material, a mechanical quantity, and thus offer a smooth connection between the liquid and amorphous solid states. The central result of this paper is a connection between the micromelting of clusters in the system and the appearance of specific heat peaks; we explain the appearance of two peaks by the micromelting of two types of clusters. We relate the two peaks to changes in the bulk and shear moduli. We propose that the phenomenon of glass formation is accompanied by a fast change in the bulk and the shear moduli, but these fast changes occur in different ranges of the temperature. Last, we demonstrate how to construct a theory of the frequency dependent complex specific heat, expected from heterogeneous clustering in the liquid state of glass formers. A specific example is provided in the context of our model for the dynamics of glycerol. We show that the frequency dependence is determined by the same α relaxation mechanism that operates when measuring the viscosity or the dielectric relaxation spectrum. The theoretical frequency dependent specific heat agrees well with experimental measurements on glycerol. We conclude the paper by stating that there is nothing universal about the temperature dependence of the specific heat in glass formersunfortunately, one needs to understand each case by itself.

(2008) Physical Review E. 78, 2, 027101. Abstract
A free material surface which supports surface diffusion becomes unstable when put under external nonhydrostatic stress. Since the chemical potential on a stressed surface is larger inside an indentation, small shape fluctuations develop because material preferentially diffuses out of indentations. When the bulk of the material is purely elastic one expects this instability to run into a finitetime cusp singularity. It is shown here that this singularity is cured by plastic effects in the material, turning the singular solution to a regular crack.

(2008) Physical Review E. 78, 2, 020501(R). Abstract
We present a quantitative theory for a relaxation function in a simple glassforming model (binary mixture of particles with different interaction parameters). It is shown that the slowing down is caused by the competition between locally favored regions (clusters) that are longlived but each of which relaxes as a simple function of time. Without the clusters, the relaxation of the background is simply determined by one typical length, which we deduce from an elementary statistical mechanical argument. The total relaxation function (which depends on time in a nontrivial manner) is quantitatively determined as a weighted sum over the clusters and the background. The "fragility" in this system can be understood quantitatively since it is determined by the temperature dependence of the number fractions of the locally favored regions.

(2008) Physica DNonlinear Phenomena. 237, 1417, p. 21672183 Abstract
We present a personal view of the state of the art in turbulence research. We summarize first the main achievements of the recent past, and then point ahead to the main challenges that remain for experimental and theoretical efforts.

(2008) Physical review letters. 101, 9, 094503. Abstract
We ask what determines the (small) angle of turbulent jets. To answer this question we first construct a deterministic vortexstreet model representing the largescale structure in a selfsimilar plane turbulent jet. Without adjustable parameters the model reproduces the mean velocity profiles and the transverse positions of the largescale structures, including their mean sweeping velocities, in a quantitative agreement with experiments. Nevertheless, the exact selfsimilar arrangement of the vortices (or any other deterministic model) necessarily leads to a collapse of the jet angle. The observed (small) angle results from a competition between vortex sweeping tending to strongly collapse the jet and randomness in the vortex structure, with the latter resulting in a weak spreading of the jet.

(2008) Physical Review E. 78, 2, 026124. Abstract
Recently, the existence and properties of unbounded cavity modes, resulting in extensive plastic deformation failure of twodimensional sheets of amorphous media, were discussed in the context of the athermal sheartransformationzones (STZ) theory. These modes pertain to perfect circular symmetry of the cavity and the stress conditions. In this paper we study the shape stability of the expanding circular cavity against perturbations, in both the unbounded and the bounded growth regimes (for the latter the unperturbed theory predicts no catastrophic failure). Since the unperturbed reference state is time dependent, the linear stability theory cannot be cast into standard timeindependent eigenvalue analysis. The main results of our study are as follows: (i) sufficiently small perturbations are stable; (ii) larger perturbations within the formal linear decomposition may lead to an instability; this dependence on the magnitude of the perturbations in the linear analysis is a result of the nonstationarity of the growth; and (iii) the stability of the circular cavity is particularly sensitive to perturbations in the effective disorder temperature; in this context we highlight the role of the rate sensitivity of the limiting value of this effective temperature. Finally we point to the consequences of the form of the stress dependence of the rate of STZ transitions. The present analysis indicates the importance of nonlinear effects that were not taken into account yet. Furthermore, the analysis suggests that details of the constitutive relations appearing in the theory can be constrained by the modes of macroscopic failure in these amorphous systems.

(2008) Physical Review E. 78, 1, 011503. Abstract
The aim of this paper is to discuss some basic notions regarding generic glassforming systems composed of particles interacting via soft potentials. Excluding explicitly hardcore interaction, we discuss the socalled glass transition in which a supercooled amorphous state is formed, accompanied by a spectacular slowing down of relaxation to equilibrium, when the temperature is changed over a relatively small interval. Using the classical example of a 5050 binary liquid of N particles with different interaction length scales, we show the following. (i) The system remains ergodic at all temperatures. (ii) The number of topologically distinct configurations can be computed, is temperature independent, and is exponential in N. (iii) Any two configurations in phase space can be connected using elementary moves whose number is polynomially bounded in N, showing that the graph of configurations has the small world property. (iv) The entropy of the system can be estimated at any temperature (or energy), and there is no Kauzmann crisis at any positive temperature. (v) The mechanism for the superArrhenius temperature dependence of the relaxation time is explained, connecting it to an entropic squeeze at the glass transition. (vi) There is no VogelFulcher crisis at any finite temperature T>0.

(2008) Physical Review E. 77, 6, 061509. Abstract
We propose that there exists a generic class of glassforming systems that have competing states (of crystalline order or not) which are locally close in energy to the ground state (which is typically unique). Upon cooling, such systems exhibit patches (or clusters) of these competing states which become locally stable in the sense of having a relatively high local shear modulus. It is in between these clusters where aging, relaxation, and plasticity under strain can take place. We demonstrate explicitly that relaxation events that lead to aging occur where the local shear modulus is low (even negative) and result in an increase in the size of local patches of relative order. We examine the aging events closely from two points of view. On the one hand we show that they are very localized in real space, taking place outside the patches of relative order, and from the other point of view we show that they represent transitions from one local minimum in the potential surface to another. This picture offers a direct relation between structure and dynamics, ascribing the slowing down in glassforming systems to the reduction in relative volume of the amorphous material which is liquidlike. While we agree with the wellknown AdamGibbs proposition that the slowing down is due to an entropic squeeze (a dramatic decrease in the number of available configurations), we do not agree with the AdamGibbs (or the VolgerFulcher) formulas that predict an infinite relaxation time at a finite temperature. Rather, we propose that generically there should be no singular crisis at any finite temperature: the relaxation time and the associated correlation length (average cluster size) increase at most superexponentially when the temperature is lowered.

(2008) Physics of Fluids. 20, 6, 065108. Abstract
We present experimental and theoretical results addressing the Reynolds number (Re) dependence of drag reduction by sufficiently large concentrations of rodlike polymers in turbulent wallbounded flows. It is shown that when Re is small the drag is enhanced. On the other hand, when Re increases, the drag is reduced and eventually, the maximal drag reduction asymptote is attained. The theory is shown to be in agreement with experiments, explaining the universal and rationalizing some of the the nonuniversal aspects of drag reduction by rodlike polymers.

(2008) Physical Review E. 77, 4, 046309. Abstract
Numerical simulations of turbulent channel flows, with or without additives, are limited in the extent of the Reynolds number (Re) and Deborah number (De). The comparison of such simulations to theories of drag reduction, which are usually derived for asymptotically high Re and De, calls for some care. In this paper we present a study of drag reduction by rodlike polymers in a turbulent channel flow using direct numerical simulation and illustrate how these numerical results should be related to the recently developed theory.

(2008) Physical Review E. 77, 3, 031507. Abstract
We address the relaxation dynamics in hydrogenbonded supercooled liquids near (but above) the glass transition, measured via broadband dielectric spectroscopy (BDS). We propose a theory based on decomposing the relaxation of the macroscopic dipole moment into contributions from hydrogenbonded clusters of s molecules, with smin ≤s≤ smax. The existence of smax is translated into a sum rule on the concentrations of clusters of size s. We construct the statistical mechanics of the supercooled liquid subject to this sum rule as a constraint, to estimate the temperaturedependent density of clusters of size s. With a theoretical estimate of the relaxation time of each cluster, we provide predictions for the real and imaginary parts of the frequencydependent dielectric response. The predicted spectra and their temperature dependence are in accord with measurements, explaining a host of phenomenological fits like the VogelFulcher fit and the stretched exponential fit. Using glycerol as a particular example, we demonstrate quantitative correspondence between theory and experiments. The theory also demonstrates that the α peak and the "excess wing" stem from the same physics in this material. The theory also shows that in other hydrogenbonded glass formers the excess wing can develop into a β peak, depending on the molecular material parameters (predominantly the surface energy of the clusters). We thus argue that α and β peaks can stem from the same physics. We address the BDS in constrained geometries (pores) and explain why recent experiments on glycerol did not show a deviation from bulk spectra. Finally, we discuss the dc part of the BDS spectrum and argue why it scales with the frequency of the α peak, providing an explanation for the remarkable data collapse observed in experiments.

(2008) Physical review letters. 100, 5, 054504. Abstract
In this Letter, we suggest a simple and physically transparent analytical model of pressure driven turbulent wallbounded flows at high but finite Reynolds numbers Re. The model provides an accurate quantitative description of the profiles of the meanvelocity and Reynolds stresses (second order correlations of velocity fluctuations) throughout the entire channel or pipe, for a wide range of Re, using only three Reindependent parameters. The model sheds light on the longstanding controversy between supporters of the centuryold loglaw theory of von Kàrmàn and Prandtl and proposers of a newer theory promoting power laws to describe the intermediate region of the mean velocity profile.

(2008) Physical Review E. 77, 2, 026606. Abstract
A classical problem in elasticity theory involves an inhomogeneity embedded in a material of given stress and shear moduli. The inhomogeneity is a region of arbitrary shape whose stress and shear moduli differ from those of the surrounding medium. In this paper we present a semianalytic method for finding the stress tensor for an infinite plate with such an inhomogeneity. The solution involves two conformal maps, one from the inside and the second from the outside of the unit circle to the inside, and respectively outside, of the inhomogeneity. The method provides a solution by matching the conformal maps on the boundary between the inhomogeneity and the surrounding material. This matching converges well only for relatively mild distortions of the unit circle due to reasons which will be discussed in the article. We provide a comparison of the present result to known previous results.

(2008) Physical Review E. 77, 2, 025101(R). Abstract
The understanding of dynamic failure in amorphous materials via the propagation of free boundaries like cracks and voids must go beyond elasticity theory, since plasticity intervenes in a crucial and poorly understood manner near the moving free boundary. We focus on failure via a cavitation instability in a radially symmetric stressed material and set up the free boundary dynamics taking both elasticity and viscoplasticity into account using the recently proposed athermal shear transformation zone theory. We demonstrate that this theory predicts the existence (in amorphous systems) of fast cavitation modes accompanied by extensive plastic deformations and discuss the revealed physics.

(2008) Iutam Symposium On Computational Physics And New Perspectives In Turbulence. 4, p. 3545 Abstract
We propose an approach to study the oldstanding problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We achieve this by constructing, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by 'Statistically Preserved Structures' which are associated with statistical conservation laws. The latter can be used for example to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the NavierStokes equations, and it demonstrates that the scaling exponents of these nonlinear models are indeed anomalous. In order to adress the universality of these nonlinear model we study the statistical properties of a semiinfinite chain of passive vectors advecting each other and study the scaling exponents at the fixed point of this chain.

(2008) Physica Scripta T. T132, 014010. Abstract
We present here an extended version of an invited talk we gave at the international conference 'Turbulent Mixing and Beyond'. The dynamical and statistical description of stably stratified turbulent boundary layers with the important example of the stable atmospheric boundary layer in mind is addressed. Traditional approaches to this problem, based on the profiles of mean quantities, velocity secondorder correlations and dimensional estimates of the turbulent thermal flux, run into a wellknown difficulty, predicting the suppression of turbulence at a small critical value of the Richardson number, in contradiction to observations. Phenomenological attempts to overcome this problem suffer from various theoretical inconsistencies. Here, we present an approach taking into full account all the secondorder statistics, which allows us to respect the conservation of total mechanical energy. The analysis culminates in an analytic solution of the profiles of all mean quantities and all secondorder correlations, removing the unphysical predictions of previous theories. We propose that the approach taken here is sufficient to describe the lower parts of the atmospheric boundary layer, as long as the Richardson number does not exceed an order of unity. For much higher Richardson numbers, the physics may change qualitatively, requiring careful consideration of the potential KelvinHelmoholtz waves and their interaction with the vortical turbulence.

(2008) Reviews of Modern Physics. 80, 1, p. 225247 Abstract
The flow of fluids in channels, pipes, or ducts, as in any other wallbounded flow (like water along the hulls of ships or air on airplanes) is hindered by a drag, which increases manyfold when the fluid flow turns from laminar to turbulent. A major technological problem is how to reduce this drag in order to minimize the expense of transporting fluids like oil in pipelines, or to move ships in the ocean. It was discovered that minute concentrations of polymers can reduce the drag in turbulent flows by up to 80%. While experimental knowledge had accumulated over the years, the fundamental theory of drag reduction by polymers remained elusive for a long time, with arguments raging whether this is a "skin" or a "bulk" effect. In this Colloquium the phenomenology of drag reduction by polymers is summarized, stressing both its universal and nonuniversal aspects, and a recent theory is reviewed that provides a quantitative explanation of all the known phenomenology. Both flexible and rodlike polymers are treated, explaining the existence of universal properties like the maximum drag reduction asymptote, as well as nonuniversal crossover phenomena that depend on the Reynolds number, on the nature of the polymer and on its concentration. Finally other agents for drag reduction are discussed with a stress on the important example of bubbles.
2007

(2007) Physical Review E. 76, 5, 052401. Abstract
We address the interesting temperature range of a glass forming system where the mechanical properties are intermediate between those of a liquid and a solid. We employ an efficient Monte Carlo method to calculate the elastic moduli, and show that in this range of temperatures the moduli are finite for short times and vanish for long times, where short and long depend on the temperature. By invoking some exact results from statistical mechanics we offer an alternative method to compute shear moduli using molecular dynamics simulations, and compare those to the Monte Carlo method. The final conclusion is that these systems are not "viscous fluids" in the usual sense, as their actual timedependence concatenates solidlike materials with varying local shear moduli.

Statistical mechanics of the glass transition in onecomponent liquids with an anisotropic potential(2007) Physical review letters. 99, 13, 135702. Abstract
We study a recently introduced model of onecomponent glassforming liquids whose constituents interact with an anisotropic potential. This system is interesting per se and as a model of liquids such as glycerol (interacting via hydrogen bonds) which are excellent glass formers. We work out the statistical mechanics of this system, encoding the liquid and glass disorder using appropriate quasiparticles (36 of them). The theory provides a full explanation of the glass transition phenomenology, including the identification of a diverging length scale and a relation between the structural changes and the diverging relaxation times.

(2007) Physical Review E. 76, 2, 025101(R). Abstract
The longrange elastic model, which is believed to describe the evolution of a selfaffine rough crack front, is analyzed to linear and nonlinear orders. It is shown that the nonlinear terms, while important in changing the front dynamics, do not change the scaling exponent which characterizes the roughness of the front. The scaling exponent thus predicted by the model is much smaller than the one observed experimentally. The inevitable conclusion is that the gap between the results of experiments and the model that is supposed to describe them is too large and some new physics has to be invoked for another model.

(2007) Physical Review E. 76, 2, 026115. Abstract
We develop an athermal sheartransformationzone (STZ) theory of plastic deformation in spatially inhomogeneous, amorphous solids. Our ultimate goal is to describe the dynamics of the boundaries of voids or cracks in such systems when they are subjected to remote, timedependent tractions. The theory is illustrated here for the case of a circular hole in an infinite twodimensional plate, a highly symmetric situation that allows us to solve much of the problem analytically. In spite of its special symmetry, this example contains many general features of systems in which stress is concentrated near free boundaries and deforms them irreversibly. We depart from conventional treatments of such problems in two ways. First, the STZ analysis allows us to keep track of spatially heterogeneous, internal state variables such as the effective disorder temperature, which determines plastic response to subsequent loading. Second, we subject the system to stress pulses of finite duration, and therefore are able to observe elastoplastic response during both loading and unloading. We compute the final deformations and residual stresses produced by these stress pulses. Looking toward more general applications of these results, we examine the possibility of constructing a boundarylayer theory that might be useful in less symmetric situations.

(2007) Nonlinearity. 20, 6, p. 14311441 006. Abstract
In a recent paper it was proposed that for some nonlinear shell models of turbulence one can construct a linear advection model for an auxiliary field such that the scaling exponents of all the structure functions of the linear and nonlinear fields coincide. The argument depended on an assumption of continuity of the solutions as a function of a parameter. The aim of this paper is to provide a rigorous proof for the validity of the assumption. In addition we clarify here when the swap of a nonlinear model by a linear one will not work.

(2007) Physical Review E. 75, 5, 050404(R). Abstract
The statistical mechanics of simple glass forming systems in two dimensions is worked out. The glass disorder is encoded via a Voronoi tesselation, and the statistical mechanics is performed directly in this encoding. The theory provides, without free parameters, an explanation of the glass transition phenomenology, including the identification of two different temperatures, Tg and Tc, the first associated with jamming and the second associated with crystallization at very low temperatures.

(2007) Physical review letters. 98, 12, 124302. Abstract
The stability of a rapid dynamic crack in a twodimensional infinite strip is studied in the framework of linear elastic fracture mechanics supplemented with a modified principle of local symmetry. It is predicted that a single crack becomes unstable by a finite wavelength oscillatory mode at a velocity vc, 0.8cR

(2007) EPL. 77, 5, 56002. Abstract
Understanding the mechanical properties of glasses remains elusive since the glass transition itself is not fully understood, even in wellstudied examples of glass formers in two dimensions. In this context we demonstrate here: i) a direct evidence for a diverging length scale at the glass transition ii) an identification of the glass transition with the disappearance of fluidlike regions and iii) the appearance in the glass state of fluidlike regions when mechanical strain is applied. These fluidlike regions are associated with the onset of plasticity in the amorphous solid. The relaxation times which diverge upon the approach to the glass transition are related quantitatively to the diverging length scale.

(2007) Physical Review E. 75, 3, 036107. Abstract
We develop an athermal version of the sheartransformationzone (STZ) theory of amorphous plasticity in materials where thermal activation of irreversible molecular rearrangements is negligible or nonexistent. In many respects, this theory has broader applicability and yet is simpler than its thermal predecessors. For example, it needs no special effort to assure consistency with the laws of thermodynamics, and the interpretation of yielding as an exchange of dynamic stability between jammed and flowing states is clearer than before. The athermal theory presented here incorporates an explicit distribution of STZ transition thresholds. Although this theory contains no conventional thermal fluctuations, the concept of an effective temperature is essential for understanding how the STZ density is related to the state of disorder of the system.

(2007) Physical Review E. 75, 3, 036108. Abstract
In the preceding paper, we developed an athermal sheartransformationzone (STZ) theory of amorphous plasticity. Here we use this theory in an analysis of numerical simulations of plasticity in amorphous silicon by Demkowicz and Argon (DA). In addition to bulk mechanical properties, those authors observed internal features of their deforming system that challenge our theory in important ways. We propose a quasithermodynamic interpretation of their observations in which the effective disorder temperature, generated by mechanical deformation well below the glass temperature, governs the behavior of other state variables that fall in and out of equilibrium with it. Our analysis points to a limitation of either the stepstrain procedure used by DA in their simulations, or the STZ theory in its ability to describe rapid transients in stressstrain curves, or perhaps to both. Once we allow for this limitation, we are able to bring our theoretical predictions into accurate agreement with the simulations.

(2007) Chaos. 17, 4, 043113. Abstract
We consider the electrical signals recorded from a subdural array of electrodes placed on the pial surface of the brain for chronic evaluation of epileptic patients before surgical resection. A simple and computationally fast method to analyze the interictal phase synchrony between such electrodes is introduced and developed with the aim of detecting and localizing the foci of the epileptic seizures. We evaluate the method by comparing the results of surgery to the localization predicted here. We find an indication of good correspondence between the success or failure in the surgery and the agreement between our identification and the regions actually operated on.
2006

(2006) Journal of Statistical Physics. 125, 56, p. 10251064 Abstract
Experiments on fracture surface morphologies offer increasing amounts of data that can be analyzed using methods of statistical physics. One finds scaling exponents associated with correlation and structure functions, indicating a rich phenomenology of anomalous scaling. We argue that traditional models of fracture fail to reproduce this rich phenomenology and new ideas and concepts are called for. We present some recent models that introduce the effects of deviations from homogeneous linear elasticity theory on the morphology of fracture surfaces, successfully reproducing the multiscaling phenomenology at least in 1+1 dimensions. For surfaces in 2+1 dimensions we introduce novel methods of analysis based on projecting the data on the irreducible representations of the SO(2) symmetry group. It appears that this approach organizes effectively the rich scaling properties. We end up proposing new experiments in which the rotational symmetry is not broken, such that the scaling properties should be particularly simple.

(2006) Europhysics Letters. 76, 2, p. 257263 Abstract
We investigate whether fractal viscous fingering and diffusionlimited aggregates are in the same scaling universality class. We bring together the largest available viscous fingering patterns and a novel technique for obtaining the conformal map from the unit circle to an arbitrary singly connected domain in two dimensions. These two Laplacian fractals appear different to the eye; in addition, viscous fingering is grown in parallel and the aggregates by a serial algorithm. Nevertheless, the data strongly indicate that these two fractal growth patterns are in the same universality class.

(2006) JETP Letters. 84, 2, p. 6267 Abstract
Turbulent boundary layers exhibit a universal structure that nevertheless is rather complex and is composed of a viscous sublayer, a buffer zone, and a turbulent loglaw region. In this letter, we present a simple analytic model of turbulent boundary layers that culminates in explicit formulas for the profiles of the mean velocity, the kinetic energy, and the Reynolds stress as a function of the distance from the wall. The resulting profiles are in close quantitative agreement with measurements over the entire structure of the boundary layer without any need of refitting in the different zones.

(2006) Journal of Fluid Mechanics. 551, p. 185195 Abstract
We employ the full FENEP model of the hydrodynamics of a dilute polymer solution to derive a theoretical approach to drag reduction in wallbounded turbulence. We recapture the results of a recent simplified theory which derived the universal maximum drag reduction (MDR) asymptote, and complement that theory with a discussion of the crossover from the MDR to the Newtonian plug when the drag reduction saturates. The FENEP model gives rise to a rather complex theory due to the interaction of the velocity field with the polymeric conformation tensor, making analytic estimates quite taxing. To overcome this we develop the theory in a computerassisted manner, checking at each point the analytic estimates by direct numerical simulations (DNS) of viscoelastic turbulence in a channel.

(2006) Physical review letters. 97, 16, 160601. Abstract
We propose a new approach to the oldstanding problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We construct, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by "statistically preserved structures" which are associated with exact conservation laws. The latter can be used, for example, to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the NavierStokes equations.

(2006) Physical review letters. 97, 13, 134301. Abstract
Dynamic fracture in a wide class of materials reveals a "fracture energy" Γ much larger than the expected nominal surface energy due to the formation of two fresh surfaces. Moreover, the fracture energy depends on the crack velocity, Γ=Γ(v). We show that a simple dynamical theory of viscoplasticity coupled to asymptotic pure linear elasticity provides a possible explanation to the above phenomena. The theory predicts tip blunting characterized by a dynamically determined crack tip radius of curvature. In addition, we demonstrate velocity selection for cracks in fixedgrip strip geometry accompanied by the identification of Γ and its velocity dependence.

(2006) Physical Review E. 73, 3, 036308. Abstract
Drag reduction by bubbles in stationary turbulent flows is sensitive to the compressibility of the bubbles. Without this dynamical effect the bubbles only renormalize the fluid density and viscosity, an effect that by itself can only lead to a small percentage of drag reduction. We show in this paper that the dynamics of bubbles and their effect on the compressibility of the mixture can lead to a much higher drag reduction.

(2006) Physical review letters. 96, 5, 055509. Abstract
Fracture paths in quasitwodimensional (2D) media (e.g., thin layers of materials or paper) are analyzed as selfaffine graphs h(x) of height h as a function of length x. We show that these are multiscaling, in the sense that nth order moments of the height fluctuations across any distance â.," scale with a characteristic exponent that depends nonlinearly on the order of the moment. Having demonstrated this, one rules out a widely held conjecture that fracture in 2D belongs to the universality class of directed polymers in random media. In fact, 2D fracture does not belong to any of the known kinetic roughening models. The presence of multiscaling offers a stringent test for any theoretical model; we show that a recently introduced model of quasistatic fracture passes this test.

(2006) Physical Review E. 74, 4, 046102. Abstract
We address the role of the nature of material disorder in determining the roughness of cracks, which grow by damage nucleation and coalescence ahead of the crack tip. We highlight the role of quenched and annealed disorders in relation to the length scales d and c associated with the disorder and the damage nucleation, respectively. In two related models, one with quenched disorder in which dâ c, the other with annealed disorder in which dâ ¡ c, we find qualitatively different roughening properties for the resulting cracks in two dimensions. The first model results in random cracks with an asymptotic roughening exponent Î¶â 0.5. The second model shows correlated roughening with Î¶â 0.66. The reasons for the qualitative difference are rationalized and explained.

Turbulent drag reduction by flexible and rodlike polymers: Crossover effects at small concentrations(2006) Physical Review E. 74, 2, 026301. Abstract
Drag reduction by polymers is bounded between two universal asymptotes, the von Kármán log law of the law and the maximum drag reduction (MDR) asymptote. It is theoretically understood why the MDR asymptote is universal, independent of whether the polymers are flexible or rodlike. The crossover behavior from the Newtonian von Kármán log law to the MDR is, however, not universal, showing different characteristics for flexible and rodlike polymers. In this paper we provide a theory for this crossover phenomenology.

(2006) Physical Review E. 73, 1, 016303. Abstract
We construct a simple analytic model for wallbounded turbulence, containing only four adjustable parameters. Two of these parameters are responsible for the viscous dissipation of the components of the Reynolds stress tensor. The other two parameters control the nonlinear relaxation of these objects. The model offers an analytic description of the profiles of the mean velocity and the correlation functions of velocity fluctuations in the entire boundary region, from the viscous sublayer, through the buffer layer, and further into the loglaw turbulent region. In particular, the model predicts a very simple distribution of the turbulent kinetic energy in the loglaw region between the velocity components: the streamwise component contains a half of the total energy whereas the wallnormal and crossstream components contain a quarter each. In addition, the model predicts a very simple relation between the von Kármán slope κ and the turbulent velocity in the loglaw region v+ (in wall units): v+ =6κ. These predictions are in excellent agreement with direct numerical simulation data and with recent laboratory experiments.
2005


(2005) Physical review letters. 95, 25, 255503. Abstract
Structure functions of rough fracture surfaces in isotropic materials exhibit complicated scaling properties due to the broken isotropy in the fracture plane generated by a preferred propagation direction. Decomposing the structure functions into the even order irreducible representations of the SO(2) symmetry group indexed by (m=0,2,4,...) results in a lucid and quickly convergent description. The scaling exponent of the isotropic sector (m=0) dominates at small length scales. One can reconstruct the anisotropic structure functions using only the isotropic and the first nonvanishing anisotropic sector (m=2) [or at most the next one (m=4)]. The scaling exponent of the isotropic sector should be observed in a proposed, yet unperformed, experiment.

(2005) Europhysics Letters. 72, 6, p. 943949 Abstract
A celebrated universal aspect of wallbounded turbulent flows is the von Kármán loglawofthewall, describing how the mean velocity in the streamwise direction depends on the distance from the wall. Although the loglaw is known for more than 75 years, the von Kármán constant governing the slope of the loglaw was not determined theoretically. In this letter we show that the von Kármán constant can be estimated from homogeneous turbulent data, i. e, without information from wallbounded flows.

(2005) Physical Review E. 72, 5, 055103(R). Abstract
The apparent similarity of microbranching instabilities in different brittle materials gave rise to a widely held belief that many aspects of the postinstability physics were universal. We propose that the physics determining the typical length and time scales characterizing the postinstability patterns differ greatly from material to material. We offer a scaling theory connecting the pattern characteristics to material properties (like molecular weight) in brittle plastics like PMMA, and stress the fundamental differences with patterns in glass which are crucially influenced by threedimensional dynamics. In both cases the present ab initio theoretical models are still too far from reality, disregarding some fundamental physics of the phenomena.

(2005) Physical review letters. 95, 19, 194502. Abstract
Drag reduction by polymers in wall turbulence is bounded from above by a universal maximal drag reduction (MDR) velocity profile that is a log law, estimated experimentally by Virk as V+(y+) 11.7log y+17. Here V+(y) and y+ are the mean streamwise velocity and the distance from the wall in "wall" units. In this Letter we propose that this MDR profile is an edge solution of the NavierStokes equations (with an effective viscosity profile) beyond which no turbulent solutions exist. This insight rationalizes the universality of the MDR and provides a maximum principle which allows an ab initio calculation of the parameters in this law without any viscoelastic experimental input.

(2005) Physics ReportsReview Section Of Physics Letters. 414, 23, p. 43164 Abstract
The problem of anisotropy and its effects on the statistical theory of high Reynolds number (Re) turbulence (and turbulent transport) is intimately related and intermingled with the problem of the universality of the (anomalous) scaling exponents of structure functions. Both problems had seen tremendous progress in the last 5 years. In this review we present a detailed description of the new tools that allow effective data analysis and systematic theoretical studies such as to separate isotropic from anisotropic aspects of turbulent statistical fluctuations. Employing the invariance of the equations of fluid mechanics to all rotations, we show how to decompose the (tensorial) statistical objects in terms of the irreducible representation of the SO (d) symmetry group (with d being the dimension, d = 2 or 3). This device allows a discussion of the scaling properties of the statistical objects in welldefined sectors of the symmetry group, each of which is determined by the "angular momenta" sector numbers (j, m). For the case of turbulent advection of passive scalar or vector fields, this decomposition allows rigorous statements to be made: (i) the scaling exponents are universal, (ii) the isotropic scaling exponents are always leading, (iii) the anisotropic scaling exponents form a discrete spectrum which is strictly increasing as a function of j. This emerging picture offers a complete understanding of the decay of anisotropy upon going to smaller and smaller scales. Next, we explain how to apply the SO (3) decomposition to the statistical NavierStokes theory. We show how to extract information about the scaling behavior in the isotropic sector. Doing so furnishes a systematic way to assess the universality of the scaling exponents in this sector, clarifying the anisotropic origin of the many measurements that claimed the opposite. A systematic analysis of direct numerical simulations (DNS) of the NavierStokes equations and of experiments provides a strong support to the proposition that also for the nonlinear problem there exists foliation of the statistical theory into sectors of the symmetry group. The exponents appear universal in each sector, and again strictly increasing as a function of j. An approximate calculation of the anisotropic exponents based on a closure theory is reviewed. The conflicting experimental measurements on the rate of decay of anisotropy upon reducing the scales are explained and systematized, showing that isotropy is eventually recovered at small scales.

(2005) Physical Review E. 72, 1, 016305. Abstract
We address the "additive equivalence" discovered by Virk and coworkers: drag reduction affected by flexible and rigid rodlike polymers added to turbulent wallbounded flows is limited from above by a very similar maximum drag reduction (MDR) asymptote. Considering the equations of motion of rodlike polymers in wallbounded turbulent ensembles, we show that although the microscopic mechanism of attaining the MDR is very different, the macroscopic theory is isomorphic, rationalizing the interesting experimental observations.

(2005) Physical Review E. 71, 6, 066127. Abstract
Slow crack propagation in ductile, and in certain brittle materials, appears to take place via the nucleation of voids ahead of the crack tip due to plastic yields, followed by the coalescence of these voids. Postmortem analysis of the resulting fracture surfaces of ductile and brittle materials on the Îmmm and the nm scales, respectively, reveals selfaffine cracks with anomalous scaling exponent Î¶â0.8 in 3 dimensions and Î¶â 0.65 in 2 dimensions. In this paper we present an analytic theory based on the method of iterated conformal maps aimed at modelling the void formation and the fracture growth, culminating in estimates of the roughening exponents in 2 dimensions. In the simplest realization of the model we allow one void ahead of the crack, and address the robustness of the roughening exponent. Next we develop the theory further, to include two voids ahead of the crack. This development necessitates generalizing the method of iterated conformal maps to include doubly connected regions (maps from the annulus rather than the unit circle). While mathematically and numerically feasible, we find that the employment of the stress field as computed from elasticity theory becomes questionable when more than one void is explicitly inserted into the material. Thus further progress in this line of research calls for improved treatment of the plastic dynamics.

(2005) Physical review letters. 94, 17, 174502. Abstract
Drag reduction by microbubbles is a promising engineering method for improving ship performance. A fundamental theory of the phenomenon is lacking, however, making actual design quite haphazard. We offer here a theory of drag reduction by microbubbles in the limit of very small bubbles, when the effect of the bubbles is mainly to normalize the density and the viscosity of the carrier fluid. The theory culminates with a prediction of the degree of drag reduction given the concentration profile of the bubbles. Comparisons with experiments are discussed and the road ahead is sketched.

(2005) Physical Review E. 71, 5, 056118. Abstract
We propose a theoretical model for branching instabilities in 2dimensional fracture, offering predictions for when crack branching occurs, how multiple cracks develop, and what is the geometry of multiple branches. The model is based on equations of motion for crack tips which depend only on the time dependent stress intensity factors. The latter are obtained by invoking an approximate relation between static and dynamic stress intensity factors, together with an essentially exact calculation of the static ones. The results of this model are in qualitative agreement with a number of experiments in the literature.

(2005) 11th International Conference on Fracture 2005, ICF11. p. 52405245 Abstract
We will discuss how the structure and stress fields around quasistatically growing fractures can be generated and analysed using iterated conformal mapping techniques. Iterated conformal maps provide a powerful new analytic tool for the study of free boundary problems in two dimensions generated by harmonic or biharmonic fields resulting in branched patterns with nontrivial geometries. In all cases the object of interest is the function which conformally maps the exterior of the unit circle to the exterior of the growing fracture. The complexity of the evolving interface is described by the dynamics of the conformal map. The latter is obtained as an niterate of a fundamental conformal map. This method allows for an efficient and accurate solution of the Lamé equations without resorting to lattice models. Typical fracture patterns exhibit increased ramification due to the increase of the stress at the tips. We find the roughness exponent of the experimentally relevant backbone of the fracture pattern crosses over from about 0.5 for small scales to about 0.75 for large scales, in excellent agreement with experiments.

(2005) Physical Review E. 71, 1, 016305. Abstract
The interaction of polymers with turbulent shear flows is examined. We focus on the structure of the elastic stress tensor, which is proportional to the polymer conformation tensor. We examine this object in turbulent flows of increasing complexity. First is isotropic turbulence, then anisotropic (but homogenous) shear turbulence, and finally wall bounded turbulence. The main result of this paper is that for all these flows the polymer stress tensor attains a universal structure in the limit of large Deborah number De≫ 1. We present analytic results for the suppression of the coilstretch transition at large Deborah numbers. Above the transition the turbulent velocity fluctuations are strongly correlated with the polymer's elongation: there appear highquality "hydroelastic" waves in which turbulent kinetic energy turns into polymer potential energy and vice versa. These waves determine the trace of the elastic stress tensor but practically do not modify its universal structure. We demonstrate that the influence of the polymers on the balance of energy and momentum can be accurately described by an effective polymer viscosity that is proportional to the crossstream component of the elastic stress tensor. This component is smaller than the streamwise component by a factor proportional to De ^{2}. Finally we tie our results to wall bounded turbulence and clarify some puzzling facts observed in the problem of drag reduction by polymers.
2004

(2004) Europhysics Letters. 68, 6, p. 825831 Abstract
Drag reduction by polymers in turbulent wallbounded flows exhibits universal and nonuniversal aspects. The universal maximal moanvelocity profile was explained in a recent theory. The saturation of this profile and the crossover back to the Newtonian plug are nonuniversal, depending on Reynolds number Re, concentration of polymer cp and the degree of polymerization N _{p}. We explain the mechanism of saturation stemming from the finiteness of extensibility of the polymers, predict its dependence on c _{p} and N in the limit of small c_{p} and large R_{e}, and present the excellent comparison of our predictions to experiments on drag reduction by DNA.

(2004) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 70, 5, p. 4 Abstract
Drag reduction by polymers in turbulent flows raises an apparent contradiction: the stretching of the polymers must increase the viscosity, so why is the drag reduced? A recent theory proposed that drag reduction, in agreement with experiments, is consistent with the effective viscosity growing linearly with the distance from the wall. With this selfconsistent solution the reduction in the Reynolds stress overwhelms the increase in viscous drag. In this Rapid Communication we show, using direct numerical simulations, that a linear viscosity profile indeed reduces the drag in agreement with the theory and in close correspondence with direct simulations of the FENEP model at the same flow conditions.

(2004) Physical Review E. 70, 5 2, p. 05530110553014 055301(R). Abstract
The drag reduction by polymers in turbulent flows was analyzed. The drag reduction, in a recent theory, was found to be consistent with the effective viscocity growing linearly with the distance from the wall. the reduction in the Reynolds stress overwhelmed the increase in the viscous drag by the selfconsistent solution. By using the direct numerical simulations it was shown that a linear viscosity profile reduced the drag in agreement with the theory and in close correspondence with direct simulations of the FENEP model at the same flow conditions.

(2004) Europhysics Letters. 68, 2, p. 310315 Abstract
We address the effect of polymer additives on twodimensional turbulence, an issue that was studied recently in experiments and direct numerical simulations. We show that the same simple shell model that reproduced drag reduction in threedimensional turbulence reproduces all the reported effects in the twodimensional case. The simplicity of the model offers a straightforward simulation of all the major effects under consideration.

(2004) Physical Review E. 70, 4 , 046107. Abstract
The connection between velocity fluctuations and the putative existence of microcracks ahead of the crack tip was studied. The dynamics of one macrocrack and one microcrack were analyzed. It was demonstrated that the interaction results in a large and rapid velocity fluctuation. An approximate methodology was proposed that will allow one to write ordinary differential equations for the positions of the tips of both macrocrack and microcrack.

(2004) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 70, 2, 026304. Abstract
We demonstrate, by using suitable shell models, that drag reduction in homogeneous turbulence is usefully discussed in terms of a scaledependent effective viscosity. The essence of the phenomenon of drag reduction found in models that couple the velocity field to the polymers can be recaptured by an \u201cequivalent\u201d equation of motion for the velocity field alone, with a judiciously chosen scaledependent effective viscosity that succinctly summarizes the important aspects of the interaction between the velocity and the polymer fields. Finally, we clarify the differences between drag reduction in homogeneous and in wall bounded flows.

(2004) Physical review letters. 92, 24, p. 24450312445034 244503. Abstract
The mechanism of drag reduction by polymers in turbulent wallbounded flows was investigated. A new logarithmic law for the mean velocity was derived with a slope that fit existing data. The phenomenonology of the maximum drag reduction asymptote which is the maximum drag reduction attained by polymers was developed on the basis of the equations of fluid mechanics. It was observed that the mechanism of drag reduction is the suppression of the Reynolds stress in the elastic sublayer.

(2004) Physical Review Letters. 92, 24, 245505. Abstract
Post mortem analysis of fracture surfaces of ductile and brittle materials on the mummm and the nm scales, respectively, reveal selfaffine cracks with anomalous scaling exponent zetaapproximate to0.8 in three dimensions and zetaapproximate to0.65 in two dimensions. Attempts to use elasticity theory to explain this result failed, yielding exponent zetaapproximate to0.5 up to logarithms. We show that when the cracks propagate via plastic void formations in front of the tip, followed by void coalescence, the void positions are positively correlated to yield exponents higher than 0.5.

(2004) Physical Review E. 69, 3, 031401. Abstract
We report an algorithm to generate Laplacian growth patterns using iterated conformal maps. The difficulty of growing a complete layer with local width proportional to the gradient of the Laplacian field is overcome. The resulting growth patterns are compared to those obtained by the best algorithms of direct numerical solutions. The fractal dimension of the patterns is discussed.

(2004) Physical Review E. 69, 2 2, p. 02612710261277 026127. Abstract
The stress intensity factors for arbitrarily shaped cracks in an infinite twodimensional elastic medium were investigated. The conformal map was constructed from the exterior of the unit circle to exterior of the arbitrary crack using the method of iterated conformal maps. The method, when compared with the SchwartzCristoffel transformation, was found to be better. The conformal method allowed an accurate estimate of the stress intensity factors or of the subleading terms such as the Tstress and simplified the calculation of the full stress field.

(2004) Physical review letters. 92, 7, 078302. Abstract
A simple model of the effect of polymer concentration on the amount of drag reduction in turbulence is presented, simulated, and analyzed. The qualitative phase diagram of drag coefficient versus Reynolds number (Re) is recaptured in this model, including the theoretically elusive onset of drag reduction and the maximum drag reduction (MDR) asymptote. The Redependent drag and the MDR are analytically explained, and the dependence of the amount of drag on material parameters is rationalized.
2003

(2003) Physical Review E. 68, 3 2, p. 363031363039 036303. Abstract
The Eulerian statistically preserved structures in passive scalar advection were presented. The timedependent linear operators that govern the dynamics of Eulerian correlation functions was analyzed numerically. The ways to naturally discuss the dynamics in terms of effective compact operators that display Eulerian statistically preserved structures which determine the anomalous scaling of the correlation functions were shown.

(2003) Europhysics Letters. 63, 5, p. 708714 Abstract
An exact integrodifferential equation for the conformal map from the unit circle to the boundary of an evolving cavity in a stressed 2dimensional solid is derived. This equation provides an accurate description of the dynamics of precursors to fracture when surface diffusion is important. The solution predicts the creation of sharp grooves that eventually lead to material failure via rapid fracture. Solutions of the new equation are demonstrated for the dynamics of an elliptical cavity and the stability of a circular cavity under biaxial stress, including the effects of surface stress.

(2003) Physical Review E. 68, 2 2, 025303(R). Abstract
A simple model of the effect of polymeric additives to Newtonian fluids, is introduced, with the aim of understanding in simple mathematical terms some of the prominent features associated with the phenomenon of drag reduction. The model reveals that arguments concerning the turbulent cascade process do not appear essential. Drag reduction is a phenomenon that appears on the scales of the system size, involving energy containing modes. Furthermore, the main point appears to be the space dependence of the stretching of the polymer.

(2003) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 68, 1, p. 10 Abstract
Recent direct numerical simulations of the finiteextensibility nonlinear elastic dumbbell model with the Peterlin approximation of nonNewtonian hydrodynamics revealed that the phenomenon of drag reduction by polymer additives exists (albeit in reduced form) also in homogeneous turbulence. We use here a simple shell model for homogeneous viscoelastic flows, which recaptures the essential observations of the full simulations. The simplicity of the shell model allows us to offer a transparent explanation of the main observations. It is shown that the mechanism for drag reduction operates mainly on large scales. Understanding the mechanism allows us to predict how the amount of drag reduction depends on the various parameters in the model. The main conclusion is that drag reduction is not a universal phenomenon; it peaks in a window of parameters such as the Reynolds number and the relaxation rate of the polymer.

(2003) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 67, 2, 026312. Abstract
In anisotropic turbulence, the correlation functions are decomposed in the irreducible representations of the SO(3) symmetry group (with different \u201cangular momenta\u201d [Formula presented] For different values of [Formula presented] the secondorder correlation function is characterized by different scaling exponents [Formula presented] In this paper, we compute these scaling exponents in a closure approximation. By linearizing the closure equations in small anisotropy we set up a linear operator and find its zero modes in the inertial interval of scales. Thus the scaling exponents in each [Formula presented] sector follow from solvability condition, and are not determined by dimensional analysis. The main result of our calculation is that the scaling exponents [Formula presented] form a strictly increasing spectrum at least until [Formula presented] guaranteeing that the effects of anisotropy decay as power laws when the scale of observation diminishes. The results of our calculations are compared to available experiments and simulations.

(2003) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 67, 5, p. 11 Abstract
We address the phenomenon of drag reduction by a dilute polymeric additive to turbulent flows, using direct numerical simulations (DNS) of the FENEP model of viscoelastic flows. It had been amply demonstrated that these model equations reproduce the phenomenon, but the results of DNS were not analyzed so far with the goal of interpreting the phenomenon. In order to construct a useful framework for the understanding of drag reduction we initiate in this paper an investigation of the most important modes that are sustained in the viscoelastic and Newtonian turbulent flows, respectively. The modes are obtained empirically using the KarhunenLoéve decomposition, allowing us to compare the most energetic modes in the viscoelastic and Newtonian flows. The main finding of the present study is that the spatial profile of the most energetic modes is hardly changed between the two flows. What changes is the energy associated with these modes, and their relative ordering in the decreasing order from the most energetic to the least. Modes that are highly excited in one flow can be strongly suppressed in the other, and vice versa. This dramatic energy redistribution is an important clue to the mechanism of drag reduction as is proposed in this paper. In particular, there is an enhancement of the energy containing modes in the viscoelastic flow compared to the Newtonian one; drag reduction is seen in the energy containing modes rather than the dissipative modes, as proposed in some previous theories.

(2003) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 67, 6, p. 9 Abstract
The weak version of universality in turbulence refers to the independence of the scaling exponents of the nth order structure functions from the statistics of the forcing. The strong version includes universality of the coefficients of the structure functions in the isotropic sector, once normalized by the mean energy flux. We demonstrate that shell models of turbulence exhibit strong universality for both forced and decaying turbulence. The exponents and the normalized coefficients are time independent in decaying turbulence, forcing independent in forced turbulence, and equal for decaying and forced turbulence. We conjecture that this is also the case for NavierStokes turbulence.

(2003) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 67, 2, p. 11 Abstract
Motivated by the large effect of turbulent drag reduction by minute concentrations of polymers, we study the effects of a weakly spacedependent viscosity on the stability of hydrodynamic flows. In a recent paper [Phys. Rev. Lett. 87, 174501, (2001)], we exposed the crucial role played by a localized region where the energy of fluctuations is produced by interactions with the mean flow (the \u201ccritical layer\u201d). We showed that a layer of a weakly spacedependent viscosity placed near the critical layer can have a very large stabilizing effect on hydrodynamic fluctuations, retarding significantly the onset of turbulence. In this paper we extend these observations in two directions: first we show that the strong stabilization of the primary instability is also obtained when the viscosity profile is realistic (inferred from simulations of turbulent flows with a small concentration of polymers). Second, we analyze the secondary instability (around the timedependent primary instability) and find similar strong stabilization. Since the secondary instability develops around a timedependent solution and is three dimensional, this brings us closer to the turbulent case. We reiterate that the large effect is not due to a modified dissipation (as is assumed in some theories of drag reduction), but due to reduced energy intake from the mean flow to the fluctuations. We propose that similar physics act in turbulent drag reduction.

(2003) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 67, 4, p. 4 042402. Abstract
An early (and influential) scaling relation in the multifractal theory of diffusion limited aggregation (DLA) is the TurkevichScher conjecture that relates the exponent [Formula presented] that characterizes the \u201chottest\u201d region of the harmonic measure and the fractal dimension D of the cluster, i.e., [Formula presented] Due to lack of accurate direct measurements of both D and [Formula presented] this conjecture could never be put to a serious test. Using the method of iterated conformal maps, D was recently determined as [Formula presented] In this paper, we determine [Formula presented] accurately with the result [Formula presented] We thus conclude that the TurkevichScher conjecture is incorrect for DLA.

(2003) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 68, 3, p. 3660113660113 036601. Abstract
We address the theory of quasistatic crack propagation in a strip of glass that is pulled from a hot oven towards a cold bath. This problem had been carefully studied in a number of experiments that offer a wealth of data to challenge the theory. We improve upon previous theoretical treatments in a number of ways. First, we offer a technical improvement of the discussion of the instability towards the creation of a straight crack. This improvement consists in employing Padé approximants to solve the relevant WienerHopf factorization problem that is associated with this transition. Next we improve the discussion of the onset of oscillatory instability towards an undulating crack. We offer a way of considering the problem as a sum of solutions of a finite strip without a crack and an infinite medium with a crack. This allows us to present a closed form solution of the stress intensity factors in the vicinity of the oscillatory instability. Most importantly we develop a dynamical description of the actual trajectory in the regime of oscillatory crack. This theory is based on the dynamical law for crack propagation proposed by Hodgdon and Sethna. We show that this dynamical law results in a solution of the actual crack trajectory in postcritical conditions; we can compute from first principles the critical value of the control parameters, and the characteristics of the solution such as the wavelength of the oscillations. We present detailed comparison with experimental measurements without any free parameters. The comparison appears quite excellent. Finally we show that the dynamical law can be translated to an equation for the amplitude of the oscillatory crack; this equation predicts correctly the scaling exponents observed in experiments.

(2003) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 67, 1, 016304. Abstract
We have recently proposed that the statistics of active fields (which affect the velocity field itself) in welldeveloped turbulence are also dominated by the statistically preserved structures of auxiliary passive fields which are advected by the same velocity field. The statistically preserved structures are eigenmodes of eigenvalue 1 of an appropriate propagator of the decaying (unforced) passive field, or equivalently, the zero modes of a related operator. In this paper we investigate further this surprising finding via two examples of shell models, one akin to turbulent convection in which the temperature is the active scalar, and the other akin to magnetohydrodynamics in which the magnetic field is the active vector. In the first example, all the even correlation functions of the active and passive fields exhibit identical scaling behavior. The second example appears at first sight to be a counterexample: the statistical objects of the active and passive fields have entirely different scaling exponents. We demonstrate, nevertheless, that the statistically preserved structures of the passive vector dominate again the statistics of the active field, except that due to a dynamical conservation law the amplitude of the leading zero mode cancels exactly. The active vector is then dominated by the subleading zero mode of the passive vector. Our work thus suggests that the statistical properties of active fields in turbulence can be understood with the same generality as those of passive fields.
2002

(2002) Physical Review E. 66, 6, p. 11 066122. Abstract
The method of iterated conformal maps is developed for quasistatic fracture of brittle materials, for all modes of fracture. Previous theory, that was relevant for mode III only, is extended here to modes I and II. The latter require the solution of the biLaplace rather than the Laplace equation. For all cases we can consider quenched randomness in the brittle material itself, as well as randomness in the succession of fracture events. While mode III calls for the advance (in time) of one analytic function, modes I and II call for the advance of two analytic functions. This fundamental difference creates different stress distribution around the cracks. As a result the geometric characteristics of the cracks differ, putting mode III in a different class compared to modes I and II.

(2002) Europhysics Letters. 60, 3, p. 369375 Abstract
The anomalous scaling of correlation functions in the turbulent statistics of active scalars (like temperature in turbulent convection) is understood in terms of an auxiliary passive scalar which is advected by the same turbulent velocity field. The evenorder correlation functions of the two fields are the same to leading order (up to a trivial multiplicative factor). The leading correlation functions are statistically preserved structures of the passive scalar decaying problem; thus the universality of the scaling exponents of the evenorder correlations of the active scalar is demonstrated.

(2002) FractalsComplex Geometry Patterns And Scaling In Nature And Society. 10, 3, p. 291296 Abstract
We introduce a model of hydrodynamic turbulence with a tunable parameter ε, which represents the ratio between deterministic and random components in the coupling between N identical copies of the turbulent field. To compute the anomalous scaling exponents ζ_{n} (of the nth order structure functions) for chosen values of ε, we consider a systematic closure procedure for the hierarchy of equations for the norder correlation functions, in the limit N → ∞. The parameter ε regularizes the closure procedure, in the sense that discarded terms are of higher order in ε compared to those retained. It turns out that after the terms of O(1), the first nonzero terms are O(ε^{4}). Within this εcontrolled procedure, we have a finite and closed set of scaleinvariant equations for the 2nd and 3rd order statistical objects of the theory. This set of equations retains all terms of O(1) and O(ε^{4}) and neglects terms of O(ε^{6}). On this basis, we expect anomalous corrections δζ_{n} in the scaling exponents ζ_{n} to increase with ε_{n}. This expectation is confirmed by extensive numerical simulations using up to 25 copies and 28 shells for various values of ε_{n}. The simulations demonstrate that in the limit N → ∞, the scaling is normal for ε cr with ε_{cr} ≈ 0.6. We observed the birth of anomalous scaling at ε = ε_{cr} with δζ_{n} ∝ ε^{4}  ε_{cr}^{4} according to our expectation.

(2002) Physical Review Letters. 89, 6, p. 064501/1064501/4 064501. Abstract
The statistics of twodimensional turbulence exhibit a riddle: the scaling exponents in the regime of inverse energy cascade agree with the K41 theory of turbulence far from equilibrium, but the probability distribution functions are close to Gaussianlike in equilibrium. The skewness Sequivalent toS(3)(R)/S2(3/2)(R) was measured as S(exp)approximate to0.03. This contradiction is lifted by understanding that twodimensional turbulence is not far from a situation with equipartition of enstrophy, which exists as true thermodynamic equilibrium with K41 exponents in space dimension of d=4/3. We evaluate the skewness S(d) for 4/3 less than or equal todless than or equal to2, showing that S(d)=0 at d=4/3, and that it remains as small as Sexp in two dimensions.

(2002) Physical Review E. 66, 1, 016308. Abstract
We study the fractal and multifractal properties (i.e., the generalized dimensions of the harmonic measure) of a twoparameter family of growth patterns that result from a growth model that interpolates between diffusionlimited aggregation (DLA) and Laplacian growth patterns in two dimensions. The two parameters are [formula presented] that determines the size of particles accreted to the interface, and [formula presented] that measures the degree of coverage of the interface by each layer accreted to the growth pattern at every growth step. DLA and Laplacian growth are obtained at [formula presented] and [formula presented] respectively. The main purpose of this paper is to show that there exists a line in the [formula presented] phase diagram that separates fractal [formula presented] from nonfractal [formula presented] growth patterns. Moreover, Laplacian growth is argued to lie in the nonfractal part of the phase diagram. Some of our arguments are not rigorous, but together with the numerics they indicate this result rather strongly. We first consider the family of models obtained for [formula presented] and derive for them a scaling relation [formula presented] We then propose that this family has growth patterns for which [formula presented] for some [formula presented] where [formula presented] may be zero. Next we consider the whole [formula presented] phase diagram and define a line that separates twodimensional growth patterns from fractal patterns with [formula presented] We explain that Laplacian growth lies in the region belonging to twodimensional growth patterns, motivating the main conjecture of this paper, i.e., that Laplacian growth patterns are two dimensional. The meaning of this result is that the branches of Laplacian growth patterns have finite (and growing) area on scales much larger than any ultraviolet cutoff length.

(2002) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 65, 4, p. 4 Abstract
We study the geometrical characteristic of quasistatic fractures in brittle media, using iterated conformal maps to determine the evolution of the fracture pattern. This method allows an efficient and accurate solution of the Lamé equations without resorting to lattice models. Typical fracture patterns exhibit increased ramification due to the increase of the stress at the tips. We find the roughness exponent of the experimentally relevant backbone of the fracture pattern, it crosses over from about 0.5 for small scales to about 0.75 for large scales. We propose that this crossover reflects the increased ramification of the fracture pattern.

(2002) Physical Review E. 65, 4, p. 045101/1045101/4 045101(R). Abstract
We study the geometrical characteristic of quasistatic fractures in brittle media, using iterated conformal maps to determine the evolution of the fracture pattern. This method allows an efficient and accurate solution of the Lamé equations without resorting to lattice models. Typical fracture patterns exhibit increased ramification due to the increase of the stress at the tips. We find the roughness exponent of the experimentally relevant backbone of the fracture pattern, it crosses over from about 0.5 for small scales to about 0.75 for large scales. We propose that this crossover reflects the increased ramification of the fracture pattern.

(2002) Physical Review E. 65, 4, p. 046109/1046109/8 046109. Abstract
The method of iterated conformal maps allows one to study the harmonic measure of diffusionlimited aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as 10 ^{35}, and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions D_{q} are infinite for q

(2002) Physical Review E. 65, 2, 026314. Abstract
It was conjectured recently that statiscally preserved structures underlie the statistical physics of turbulent transport processes. We analyze here in detail the timedependent (noncompact) linear operator that governs the dynamics of correlation functions in the case of shell models of passive scalar advection. The problem is generic in the sense that the driving velocity field is neither Gaussian nor delta correlated in time. We show how to naturally discuss the dynamics in terms of an effective compact operator that displays "zero modes," which determine the anomalous scaling of the correlation functions. Since shell models have neither a Lagrangian structure nor "shape dynamics," this example differs significantly from standard passive scalar advection. Nevertheless, with the necessary modifications, the generality and efficacy of the concept of statistically preserved structures are further exemplified. In passing we point out a bonus of the present approach, in providing analytic predictions for the timedependent correlation functions in decaying turbulent transport.

(2002) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 65, 4, 046144. Abstract
The method of iterated conformal maps for the study of diffusion limited aggregates (DLA) is generalized to the study of Laplacian growth patterns and related processes. We emphasize the fundamental difference between these processes: DLA is grown serially with constant size particles, while Laplacian patterns are grown by advancing each boundary point in parallel, proportional to the gradient of the Laplacian field. We introduce a twoparameter family of growth patterns that interpolates between DLA and a discrete version of Laplacian growth. The ultraviolet putative finitetime singularities are regularized here by a minimal tip size, equivalently for all the models in this family. With this we stress that the difference between DLA and Laplacian growth is not in the manner of ultraviolet regularization, but rather in their deeply different growth rules. The fractal dimensions of the asymptotic patterns depend continuously on the two parameters of the family, giving rise to a \u201cphase diagram\u201d in which DLA and discretized Laplacian growth are at the extreme ends. In particular, we show that the fractal dimension of Laplacian growth patterns is higher than the fractal dimension of DLA, with the possibility of dimension 2 for the former not excluded.

(2002) Physical review letters. 89, 23, 234501. Abstract
Experiments in quasitwodimensional geometry (HeleShaw cells) in which a fluid is injected into a viscoelastic medium (foam, clay, or associating polymers) show patterns akin to fracture in brittle materials, very different from standard Laplacian growth patterns of viscous fingering. An analytic theory is lacking since a prerequisite to describing the fracture of elastic material is the solution of the biLaplace rather than the Laplace equation. In this Letter we close this gap, offering a theory of biLaplacian growth patterns based on the method of iterated conformal maps.

(2002) Physical review letters. 89, 7, 074501 . Abstract
We consider shell models that display an inverse energy cascade similar to twodimensional turbulence (together with a direct cascade of an enstrophylike invariant). Previous attempts to construct such models ended negatively, stating that shell models give rise to a \u201cquasiequilibrium\u201d situation with equipartition of the energy among the shells. We show analytically that the quasiequilibrium state predicts its own disappearance upon changing the model parameters in favor of the establishment of an inverse cascade regime with Kolmogorov scaling. The latter regime is found where predicted, offering a useful model to study inverse cascades.
2001

(2001) Physical Review Letters. 87, 16, 164502. Abstract
We address the statistical theory of fields that are transported by a turbulent velocity field, both in forced and in unforced (decaying) experiments. With very few provisos on the transporting velocity field, correlation functions of the transported field in the forced case are dominated by statistically preserved structures. In decaying experiments we identify infinitely many statistical constants of the motion, which are obtained by projecting the decaying correlation functions on the statistically preserved functions. We exemplify these ideas and provide numerical evidence using a simple model of turbulent transport. This example is chosen for its lack of Lagrangian structure, to stress the generality of the ideas.

(2001) Physical review letters. 87, 17, 174501. Abstract
Motivated by turbulent drag reduction by minute concentrations of polymers we study the effects of minor viscosity contrasts on the stability of hydrodynamic flows. The key player is a localized region where fluctuations are produced by interactions with the mean flow (the \u201ccritical layer\u201d). We show that a layer of weakly spacedependent viscosity placed near the critical layer has a very large stabilizing effect on hydrodynamic fluctuations, retarding significantly the onset of turbulence. The effect is not due to a modified dissipation (as is assumed in theories of drag reduction) but is due to reduced energy intake from the mean flow to the fluctuations. Similar physics may act in turbulent drag reduction.

(2001) Physical review letters. 87, 16, p. 164101 Abstract
We study the nature of the phase transition in the multifractal formalism of the harmonic measure of diffusion limited aggregates. Contrary to previous work that relied on random walk simulations or ad hoc models to estimate the low probability events of deep fjord penetration, we employ the method of iterated conformal maps to obtain an accurate computation of the probability of the rarest events. We resolve probabilities as small as 10^{−35}. We show that the generalized dimensions D_{q} are infinite for q

(2001) Journal of Statistical Physics. 103, p. 9731007 Abstract
We study the dynamics of "finger" formation in Laplacian growth without surface tension in a channel geometry (the Saffman Taylor problem). We present a pedagogical derivation of the dynamics of the conformal map from a strip in the complex plane to the physical channel. In doing so we pay attention to the boundary conditions (no flux rather than periodic) and derive a field equation of motion for the conformal map. We first consider an explicit analytic class of conformal maps that form a basis for solutions in infinitely long channels, characterized by meromorphic derivatives. The great bulk of these solutions can lose conformality due to finite time singularities. By considerations of the nature of the analyticity of these solutions, we show that those solutions which are free of such singularities inevitably result in a single asymptotic "finger" whose width is determined by initial conditions. This is in contradiction with the experimental results that indicate selection of a finger of width 1/2. In the last part of this paper we show that such a solution might be determined by the boundary conditions of a finite body of fluid, e.g. finiteness can lead to pattern selection.

(2001) Physical Review E. 63, 5 II, p. 5611815611810 056118. Abstract
The detailed dynamic scaling properties of the natural phenomena and prediction of the probability of their occurrence was discussed. The dynamical scaling form is characterized by static and dynamic exponents whose values were determined by analyzing the isolated events in short time horizons. The probability density functions were used to demonstrate the existence of scaling properties of extreme events.

(2001) Nature. 409, 6823, p. 993995 Abstract
Traditional devices for measuring turbulence have been unable to keep up with the latest developments in theory. But detectors derived from highenergy physics may narrow the gap between experiment and theory.

(2001) Physical review letters. 87, 13, 134501. Abstract
It had been conjectured that diffusion limited aggregates and Laplacian growth patterns (with small surface tension) are in the same universality class. Using iterated conformal maps we construct a oneparameter family of fractal growth patterns with a continuously varying fractal dimension. This family can be used to bound the dimension of Laplacian growth patterns from below. The bound value is higher than the dimension of diffusion limited aggregates, showing that the two problems belong to two different universality classes.

(2001) Physical Review E. 63, 5, 056302. Abstract
We discuss the scaling exponents characterizing the powerlaw behavior of the anisotropic components of correlation functions in turbulent systems with pressure. The anisotropic components are conveniently labeled by the angular momentum index (Formula presented) of the irreducible representation of the SO(3) symmetry group. Such exponents govern the rate of decay of anisotropy with decreasing scales. It is a fundamental question whether they ever increase as (Formula presented) increases, or they are bounded from above. The equations of motion in systems with pressure contain nonlocal integrals over all space. One could argue that the requirement of convergence of these integrals bounds the exponents from above. It is shown here on the basis of a solvable model (the \u201clinear pressure model\u201d) that this is not necessarily the case. The model introduced here is of a passive vector advection by a rapidly varying velocity field. The advected vector field is divergent free and the equation contains a pressure term that maintains this condition. The zero modes of the secondorder correlation function are found in all the sectors of the symmetry group. We show that the spectrum of scaling exponents can increase with (Formula presented) without bounds while preserving finite integrals. The conclusion is that contributions from higher and higher anisotropic sectors can disappear faster and faster upon decreasing the scales also in systems with pressure.

Anomalous scaling in passive scalar advection and Lagrangian shape dynamics(2001) Iutam Symposium On Geometry And Statistics Of Turbulence. 59, p. 175184 Abstract
The problem of anomalous scaling in passive scalar advection, especially with delta correlated velocity field (the Kraichnan model) has attracted a lot of interest since the exponents can be computed analytically in certain limiting cases. In this paper we focus, rather than on the evaluation of the exponents, on elucidating the physical mechanism responsible for the anomaly. We show that the anomalous exponents zeta (n) stem from the Lagrangian dynamics of shapes which characterize configurations of n points in space. Using the shapetoshape transition probability, we define an operator whose eigenvalues determine the anomalous exponents for all n, in all the sectors of the SO(3) symmetry group.
2000

(2000) Physica A. 288, 14, p. 280307 Abstract
We present a short review of the work conducted by our group on the subject of anomalous scaling in anisotropic turbulence. The basic idea that unifies all the applications discussed here is that the equations of motion for correlation functions are always linear and invariant to rotations, and therefore the solutions foliate into sectors of the symmetry group of all rotations (SO(3)). We have considered models of passive scalar and passive vector advections by a rapidly changing turbulent velocity field (Kraichnantype models) for which we find a discrete spectrum of universal anomalous exponents, with a different exponent characterizing the scaling behavior in every sector. Generically the correlation functions and structure functions appear as sums over all these contributions, with nonuniversal amplitudes which are determined by the anisotropic boundary conditions. In addition we considered NavierStokes turbulence by analyzing simulations and experiments, and reached some interesting conclusions regarding the scaling exponents in the anisotropic sectors. The theory presented here clarifies questions like the restoration of local isotropy upon decreasing scales. We explain when the local isotropy is fully restored and when the lingering effects of the anisotropic forcing appear for arbitrarily small scales.

(2000) Physical review letters. 85, 17, p. 36083611 Abstract
A conformal theory of fractal growth processes in two dimensions was considered, including diffusion limited aggregation (DLA) as a special case. The computation of the dimension from early stage dynamics was demonstrated. Further, a novel notion was introduced that the fractal dimension appears as a dynamical exponent already at early stages of the growth in which the cluster has not yet developed a fractal geometry.

(2000) Europhysics Letters. 50, 4, p. 473479 Abstract
The major difficulty in developing theories for anomalous scaling in hydrodynamic turbulence is the lack of a small parameter. In this letter we introduce a shell model of turbulence that exhibits anomalous scaling with a tunable parameter ∈, 0 ≤ ∈ ≤ 1, representing the ratio between deterministic and random components in the coupling between N identical copies of the turbulent field. Our numerical experiments give strong evidence that in the limit N → ∞ anomalous scaling sets in proportional to ∈^{4}. This result shows consistency with the nonperturbative closure proposed by the authors in Phys. Fluids, 12 (2000) 803. In this procedure closed equations of motion for the loworder correlation and response functions are obtained, keeping terms proportional to ∈^{0} and ∈^{4}, discarding terms of orders ∈^{6} and higher. Moreover we give strong evidences that the birth of anomalous scaling appears at a finite critical ∈, being ∈_{c} ≈ 0.6.

(2000) Physics of Fluids. 12, 4, p. 803821 Abstract
We present a model of hydrodynamic turbulence for which the program of computing the scaling exponents from first principles can be developed in a controlled fashion. The model consists of N suitably coupled copies of the "Sabra" shell model of turbulence. The couplings are chosen to include two components: random and deterministic, with a relative importance that is characterized by a parameter called ∈. It is demonstrated, using numerical simulations of up to 25 copies and 28 shells that in the N→∞ limit but for 0

(2000) Physical Review E. 61, 3, p. 26542662 Abstract
We address the scaling behavior of the covariance of the magnetic field in the threedimensional kinematic dynamo problem when the boundary conditions and/or the external forcing are not isotropic. The velocity field is Gaussian, space homogeneous, and [Formula Presented] correlated in time, and its structure function scales with a positive exponent [Formula Presented] The covariance of the magnetic field is naturally computed as a sum of contributions proportional to the irreducible representations of the SO(3) symmetry group. The amplitudes are nonuniversal, determined by boundary conditions. The scaling exponents are universal, forming a discrete, strictly increasing, spectrum indexed by the sectors of the symmetry group. When the initial mean magnetic field is zero, no dynamo effect is found, irrespective of the anisotropy of the forcing. The rate of isotropization with decreasing scales is fully understood from these results.

(2000) Physical Review E. 62, 4, p. 49044919 Abstract
Kraichnans model of passive scalar advection in which the driving (Gaussian) velocity field has fast temporal decorrelation is studied as a case model for understanding the anomalous scaling behavior in the anisotropic sectors of turbulent fields. We show here that the solutions of the Kraichnan equation for the norder correlation functions foliate into sectors that are classified by the irreducible representations of the [Formula Presented] symmetry group. We find a discrete spectrum of universal anomalous exponents, with a different exponent characterizing the scaling behavior in every sector. Generically the correlation functions and structure functions appear as sums over all these contributions, with nonuniversal amplitudes that are determined by the anisotropic boundary conditions. The isotropic sector is always characterized by the smallest exponent, and therefore for sufficiently small scales local isotropy is always restored. The calculation of the anomalous exponents is done in two complementary ways. In the first they are obtained from the analysis of the correlation functions of gradient fields. The theory of these functions involves the control of logarithmic divergences that translate into anomalous scaling with the ratio of the inner and the outer scales appearing in the final result. In the second method we compute the exponents from the zero modes of the Kraichnan equation for the correlation functions of the scalar field itself. In this case the renormalization scale is the outer scale. The two approaches lead to the same scaling exponents for the same statistical objects, illuminating the relative role of the outer and inner scales as renormalization scales. In addition we derive exact fusion rules, which govern the small scale asymptotics of the correlation functions in all the sectors of the symmetry group and in all dimensions.

(2000) Physical Review E. 61, 1, p. 407421 Abstract
The statistical objects characterizing turbulence in real turbulent flows differ from those of the ideal homogeneous isotropic model. They contain contributions from various two and threedimensional aspects, and from the superposition of inhomogeneous and anisotropic contributions. We employ the recently introduced decomposition of statistical tensor objects into irreducible representations of the SO(3) symmetry group (characterized by j and m indices, where [Formula Presented] to disentangle some of these contributions, separating the universal and the asymptotic from the specific aspects of the flow. The different j contributions transform differently under rotations, and so form a complete basis in which to represent the tensor objects under study. The experimental data are recorded with hotwire probes placed at various heights in the atmospheric surface layer. Time series data from single probes and from pairs of probes are analyzed to compute the amplitudes and exponents of different contributions to the second order statistical objects characterized by [Formula Presented] 1, and 2. The analysis shows the need to make a careful distinction between longlived quasitwodimensional turbulent motions (close to the ground) and relatively shortlived threedimensional motions. We demonstrate that the leading scaling exponents in the three leading sectors [Formula Presented] 1, and 2) appear to be different but universal, independent of the positions of the probe, the tensorial component considered, and the large scale properties. The measured values of the scaling exponent are [Formula Presented] [Formula Presented] and [Formula Presented] We present theoretical arguments for the values of these exponents using the Clebsch representation of the Euler equations; neglecting anomalous corrections, the values obtained are 2/3, 1, and 4/3, respectively. Some enigmas and questions for the future are sketched.

(2000) Physical Review E. 62, 5, p. R5919R5922 Abstract
Diffusion limited aggregation (DLA) is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes. We present a convergent calculation of the fractal dimension D of DLA based on a renormalization scheme for the first Laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster. The theory is applicable from very small (23 particles) to asymptotically large [Formula Presented] clusters. The computed dimension is [Formula Presented].

(2000) Physical Review E. 62, 6, p. 80378057 Abstract
The main difficulty of statistical theories of fluid turbulence is the lack of an obvious small parameter. In this paper we show that the formerly established fusion rules can be employed to develop a theory in which Kolmogorovs statistics of 1941 (K41) acts as the zero order, or background statistics, and the anomalous corrections to the K41 scaling exponents [Formula Presented] of the [Formula Presented]order structure functions can be computed analytically. The crux of the method consists of renormalizing a fourpoint interaction amplitude on the basis of the fusion rules. This amplitude includes a small dimensionless parameter, which is shown to be of the order of the anomaly of [Formula Presented] [Formula Presented] Higherorder interaction amplitudes are shown to be even smaller. The corrections to K41 to [Formula Presented] result from standard logarithmically divergent ladder diagrams in which the fourpoint interaction acts as a \u201crung.\u201d The theory allows a calculation of the anomalous exponents [Formula Presented] in powers of the small parameter [Formula Presented] The n dependence of the scaling exponents [Formula Presented] stems from pure combinatorics of the ladder diagrams. In this paper we calculate the exponents [Formula Presented] up to [Formula Presented] Previously derived bridge relations allow a calculation of the anomalous exponents of correlations of the dissipation field and of dynamical correlations in terms of the same parameter [Formula Presented] The actual evaluation of the small parameter [Formula Presented] from first principles requires additional developments that are outside the scope of this paper.

(2000) Physical Review E. 62, 2, p. 17061715 Abstract
Many models of fractal growth patterns (such as diffusion limited aggregation and dielectric breakdown models) combine complex geometry with randomness; this double difficulty is a stumbling block to their elucidation. In this paper we introduce a wide class of fractal growth models with highly complex geometry but without any randomness in their growth rules. The models are defined in terms of deterministic itineraries of iterated conformal maps, generating the function [Formula Presented] which maps the exterior of the unit circle to the exterior of an nparticle growing aggregate. The complexity of the evolving interfaces is fully contained in the deterministic dynamics of the conformal map [Formula Presented] We focus attention on a class of growth models in which the itinerary is quasiperiodic. Such itineraries can be approached via a series of rational approximants. The analytic power gained is used to introduce a scaling theory of the fractal growth patterns and to identify the exponent that determines the fractal dimension.
1999

(1999) Europhysics Letters. 48, 5, p. 547553 Abstract
We employ the recently introduced conformal iterative construction of DiffusionLimited Aggregates (DLA) to study the multifractal properties of the harmonic measure. The support of the harmonic measure is obtained from a dynamical process which is complementary to the iterative cluster growth. We use this method to establish the existence of a series of random scaling functions that yield, via the thermodynamic formalism of multifractals, the generalized dimensions D_{q} of DLA for q ≥ 1. The scaling function is determined just by the last stages of the iterative growth process which are relevant to the complementary dynamics. Using the scaling relation D_{3} = D_{0}/2, we estimate the fractal dimension of DLA to be D_{0} = 1.69±0.03.

(1999) Physical Review E. 60, 4 A, p. 41754184 Abstract
The Taylor hypothesis, which allows surrogating spatial measurements requiring many experimental probes by time series from one or two probes, is examined on the basis of a simple analytic model of turbulent statistics. The main points are as follows: (i) The Taylor hypothesis introduces systematic errors in the evaluation of scaling exponents. (ii) When the mean wind V̄_{0} is not infinitely larger than the rootmeansquare longitudinal turbulent fluctuations v_{T}, the effective Taylor advection velocity V_{ad} should take the latter into account. (iii) When two or more probes are employed the application of the Taylor hypothesis and the optimal choice of the effective advecting wind V_{ad} need extra care. We present practical considerations for minimizing the errors incurred in experiments using one or two probes. (iv) Analysis of the Taylor hypothesis when different probes experience different mean winds is offered.

(1999) Physical Review Letters. 82, 25, p. 50405043 Abstract
We address scaling in inhomogeneous and anisotropic turbulent flows by decomposing structure functions into their irreducible representation of the SO(3) symmetry group which are designated by j, m indices. Employing simulations of channel flows with Relambda approximate to 70 we demonstrate that different components characterized by different j display different scaling exponents, but for a given j these remain the same at different distances ham the wall. The j = 0 exponent agrees extremely well with high Re measurements of the scaling exponents, demonstrating the vitality of the SO(3) decomposition.

(1999) Europhysics Letters. 46, 5, p. 609612 Abstract
We show that the Sabra shell model of turbulence, which was introduced recently, displays a Hamiltonian structure for given values of the parameters. The requirement of scale independence of the flux of this Hamiltonian allows us to compute exactly a oneparameter family of anomalous scaling exponents associated with 4thorder correlation functions.

(1999) Physical Review E. 59, 6, p. 67536765 Abstract
The theory of fully developed turbulence is usually considered in an idealized homogeneous and isotropic state. Real turbulent flows exhibit the effects of anisotropic forcing. The analysis of correlation functions and structure functions in isotropic and anisotropic situations is facilitated and made rational when performed in terms of the irreducible representations of the relevant symmetry group which is the group of all rotations SO(3). In this paper we first consider the needed general theory, and explain why we expect different (universal) scaling exponents in the different sectors of the symmetry group. We exemplify the theory context of isotropic turbulence (for third order tensorial structure functions) and in weakly anisotropic turbulence (for the second order structure function). The utility of the resulting expressions for the analysis of experimental data is demonstrated in the context of high Reynolds number measurements of turbulence in the atmosphere.

(1999) Physical Review E. 59, 3, p. 25872593 Abstract
We consider flame front propagation in channel geometries. The steadystate solution in this problem is space dependent and therefore the linear stability analysis is described by a partial integrodifferential equation with a spacedependent coefficient. Accordingly, it involves complicated eigenfunctions. We show that the analysis can be performed using a finiteorder dynamical system in terms of the dynamics of singularities in the complex plane, yielding a detailed understanding of the physics of the eigenfunctions and eigenvalues.

(1999) Physical Review E. 59, 2, p. 13681378 Abstract
The creation of fractal clusters by diffusion limited aggregation (DLA) is studied by using iterated stochastic conformal maps following the method proposed recently by Hastings and Levitov. The object of interest is the function [Formula Presented] which conformally maps the exterior of the unit circle to the exterior of an nparticle DLA. The map [Formula Presented] is obtained from n stochastic iterations of a function [Formula Presented] that maps the unit circle to the unit circle with a bump. The scaling properties usually studied in the literature on DLA appear in a new light using this language. The dimension of the cluster is determined by the linear coefficient in the Laurent expansion of [Formula Presented] which asymptotically becomes a deterministic function of n. We find new relationships between the generalized dimensions of the harmonic measure and the scaling behavior of the Laurent coefficients.

(1999) Physical Review E. 60, 6, p. 66566662 Abstract
It was shown recently that the anomalous scaling of simultaneous correlation functions in turbulence is intimately related to the breaking of temporal scale invariance, which is equivalent to the appearance of infinitely many times scales in the time dependence of timecorrelation functions. In this paper we derive a continued fraction representation of turbulent time correlation functions which is exact and in which the multiplicity of time scales is explicit. We demonstrate that this form yields precisely the same scaling laws for time derivatives and time integrals as the \u201cmultifractal\u201d representation that was used before. Truncating the continued fraction representation yields the \u201cbest\u201d estimates of time correlation functions if the given information is limited to the scaling exponents of the simultaneous correlation functions up to a certain, finite order. It is worth noting that the derivation of a continued fraction representation obtained here for a time evolution operator which is not Hermitian or antiHermitian may be of independent interest.
1998

(1998) Journal of Statistical Physics. 93, 34, p. 797832 Abstract
We develop a consistent closure procedure for the calculation of the scaling exponents ζ_{n} of the nthorder correlation functions in fully developed hydrodynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζ_{n}. This hierarchy was discussed in detail in a recent publication by V. S. L'vov and I. Procaccia. The scaling exponents in this set of equations cannot be found from power counting. In this paper we present in detail the lowest nontrivial closure of this infinite set of equations, and prove that this closure leads to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integrodifferential equations, reflecting the nonlinearity of the original NavierStokes equations. Nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L. We demonstrate that the solvability condition of our equations leads to nonKolmogorov values of the scaling exponents ζ_{n}. Finally, we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence.

(1998) Europhysics Letters. 43, 3, p. 277283 Abstract
A recent theoretical development in the understanding of the smallscale structure of NavierStokes turbulence has been the proposition that the scales η_{n}(R) that separate inertial from viscous behavior of manypoint correlation functions depend on the order n and on the typical separations R of points in the correlation. This is of fundamental significance in itself but it also has implications for the scaling behaviour of various correlation functions. This dependence has never been observed directly in laboratory experiments. In order to observe it, turbulence data which both display a welldeveloped scaling range with clean scaling behaviour and are wellresolved in the small scales to well within the viscous range is required. The data of the experiments performed in the laboratory of P. Tabeling of NavierStokes turbulence in a helium cell with counterrotating disks approach these criteria, and provide supporting evidence for the existence of the predicted scaling of the viscous scale.

(1998) Physica A. 257, 4Jan, p. 165196 Abstract
We propose a scheme for the calculation from the NavierStokes equations of the scaling exponents in of the nth order correlation functions in fully developed hydrodynamic turbulence. The scheme is nonperturbative and constructed to respect the fundamental rescaling symmetry of the Euler equation. It constitutes an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents zeta(n). As a consequence the scaling exponents are determined by solvability conditions and not from power counting. It is argued that in order to achieve such a formulation one must recognize that the manypoint spacetime correlation functions are not scale invariant in their time arguments, The assumption of full scale invariance leads unavoidably to Kolmogorov exponents. It is argued that the determination of all the scaling exponents in requires equations for infinitely many renormalized objects. One can however proceed in controlled successive approximations by successive truncations of the infinite hierarchy of equations. Clues as to how to truncate without reintroducing power counting can be obtained from renormalized perturbation theory. To this aim we show that the fully resummed perturbation theory is equivalent in its contents to the exact hierarchy of equations obeyed by the nth order correlation functions and Green's function. In light of this important result we can safely use finite resummations to construct successive closures of the infinite hierarchy of equations. This paper presents the conceptual and technical details of the scheme. The analysis of the highorder closure procedures which do not destroy the rescaling symmetry and the actual calculations for truncated models will be presented in a forthcoming paper in collaboration with V. Belinicher. (C) 1998 Elsevier Science B.V. All rights reserved.

(1998) Physica A. 254, 12, p. 215230 Abstract
In this short paper we describe the essential ideas behind a new consistent closure procedure for the calculation of the scaling exponents ζ_{n} of the nth order correlation functions in fully developed hydrodynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζ_{n}. This hierarchy was discussed in detail in a recent publication [V.S. L'vov and I. Procaccia, Physica A (1998), in press, chaodyn/9707015]. The scaling exponents in this set of equations cannot be found from power counting. In this short paper we discuss in detail low order nontrivial closures of this infinite set of equations, and prove that these closures lead to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integrodifferential equations, reflecting the nonlinearity of the original NavierStokes equations. Nevertheless, they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L.

(1998) Physical Review E. 57, 4, p. 41064134 Abstract
The onset of spacetime chaos is studied on the basis of a Galilean invariant model that exhibits the essential characteristics of the phenomenon. By keeping the linear part of the model extremely simple, one has better than usual control of the classes of available stationary solutions. These stationary solutions include not only spatially periodic but also a large set of spatially chaotic solutions that can be characterized by words of a symbolic language. The main proposition of this paper is that spacetime chaos in Galilean invariant models can be understood in a qualitative fashion as an orbit in the space of functions that visits words in this language in a random fashion. The appearance of topological defects and other \u201csignatures\u201d of spacetime chaos are a natural consequence of this dynamics. Finally, we construct a simple demonstration of this scenario.

(1998) Physical review letters. 81, 24, p. 53305333 Abstract
We analyze turbulent velocity signals in the atmospheric surface layer, obtained by pairs of probes separated by inertialrange distances parallel to the ground and (nominally) orthogonal to the mean wind. The Taylor microscale Reynolds number ranges up to 20 000. Choosing a suitable coordinate system with respect to the mean wind, we derive theoretical forms for second order structure functions and fit them to experimental data. The effect of flow anisotropy is small for the longitudinal component but significant for the transverse component. The data provide an estimate for a universal exponent from among a hierarchy that governs the decay of flow anisotropy with the scale size.

(1998) Physical review letters. 80, 11, p. 24772480 Abstract
Flame propagation is used as a prototypical example of expanding fronts that wrinkle without limit in radial geometries but reach a simple shape in channel geometry. We show that the relevant scaling laws that govern the radial growth can be inferred once the simpler channel geometry is understood in detail. In radial geometries (in contrast to channel geometries) the effect of external noise is crucial in accelerating and wrinkling the fronts. Nevertheless, once the interrelations between system size, velocity of propagation, and noise level are understood in channel geometry, the scaling laws for radial growth follow.

(1998) Physical Review E. 57, 6, p. 69136916 Abstract
We show that both analytic and numerical evidence points to the existence of a critical angle of [formula presented][formula presented] in viscous fingers and diffusionlimited aggregates growing in a wedge. The significance of this angle is that it is the typical angular spread of a major finger. For wedges with an angle larger than [formula presented], two fingers can coexist. Thus a finger with this angular spread is a kind of building block for viscous fingering patterns and diffusionlimited aggregation clusters in radial geometry.

(1998) Physical Review E. 58, 2, p. 18111822 Abstract
We introduce a shell model of turbulence that exhibits improved properties in comparison to the standard (and very popular) Gledzer, Ohkitani, and Yamada (GOY) model. The nonlinear coupling is chosen to minimize correlations between different shells. In particular, the secondorder correlation function is diagonal in the shell index and the thirdorder correlation exists only between three consecutive shells. Spurious oscillations in the scaling regime, which are an annoying feature of the GOY model, are eliminated by our choice of nonlinear coupling. We demonstrate that the model exhibits multiscaling similar to the GOY model. The scaling exponents are shown to be independent of the viscous mechanism as is expected for NavierStokes turbulence and other shell models. These properties of the model make it optimal for further attempts to achieve understanding of multiscaling in nonlinear dynamics.

(1998) Physical review letters. 81, 4, p. 802805 Abstract
In a series of recent works it was proposed that shell models of turbulence exhibit inertial range scaling exponents that depend on the nature of the dissipative mechanism. If true, and if one could imply a similar phenomenon to NavierStokes turbulence, this finding would cast strong doubts on the universality of scaling in turbulence. In this Letter we propose that these \u201cnonuniversalities\u201d are just corrections to scaling that disappear when the Reynolds number goes to infinity.

(1998) Physical review letters. 80, 25, p. 55365539 Abstract
We present a numerical method which is used to calculate anomalous scaling exponents of structure functions in the Kraichnan passive scalar advection model [R. H. Kraichnan, Phys. Fluids 11, 945 (1968)]. This Monte Carlo method, which is applicable in any space dimension, is based on the Lagrangian path interpretation of passive scalar dynamics, and uses the recently discovered equivalence between scaling exponents of structure functions and relaxation rates in the stochastic shape dynamics of groups of Lagrangian particles. We calculate third and fourth order anomalous exponents for several dimensions, comparing with the predictions of perturbative calculations in large dimensions. We find that Kraichnan's closure appears to give results in close agreement with the numerics. The third order exponents are compatible with our own previous nonperturbative calculations.
1997

(1997) Physical review letters. 79, 21, p. 41664169 Abstract
We present results from direct numerical simulations of the Kraichnan model for passive scalar advection by a rapidly varying random scaling velocity field for intermediate values of the velocity scaling exponent. These results are compared with the scaling exponents predicted for this model by Kraichnan. Further, we test the recently proposed fusion rules which govern the scaling properties of multipoint correlations, and present results on the linearity of the conditional statistics of the Laplacian operator on the scalar field.

(1997) Physical review letters. 79, 11, p. 20502052 Abstract
The phenomenology of the scaling behavior of higher order structure functions of velocity differences across a scale R in turbulence should be built around the irreducible representations of the rotation symmetry group. Every irreducible representation is associated with a scalar function of R which may exhibit different scaling exponents. The common practice of using moments of longitudinal and transverse fluctuations mixes different scalar functions and therefore may mix different scaling exponents. It is shown explicitly how to extract pure scaling exponents for correlation functions of arbitrary orders.

(1997) Physical Review E. 55, 3, p. 26492663 Abstract
The problem of flame propagation is studied as an example of unstable fronts that wrinkle on many scales. The analytic tool of pole expansion in the complex plane is employed to address the interaction of the unstable growth process with random initial conditions and perturbations. We argue that the effect of random noise is immense and that it can never be neglected in sufficiently large systems. We present simulations that lead to scaling laws for the velocity and acceleration of the front as a function of the system size and the level of noise, and analytic arguments that explain these results in terms of the noisy pole dynamics.

(1997) Physical Review E. 55, 6, p. 70307035 Abstract
On the basis of the NavierStokes equations, we develop the high Reynolds number statistical theory of differenttime, manypoint spatial correlation functions of velocity differences. We find that their time dependence is not scale invariant: norder correlation functions exhibit infinitely many distinct decorrelation times that are characterized by anomalous dynamical scaling exponents. We derive exact scaling relations that bridge all these dynamical exponents to the static anomalous exponents [Formula Presented] of the standard structure functions. We propose a representation of the time dependence using the Legendretransform formalism of multifractals that automatically reproduces all the newly found bridge relationships.

Hydrodynamic turbulence: a 19th century problem with a challenge for the 21st century(1997) Turbulence Modeling And Vortex Dynamics. 491, p. 116 Abstract
The theoretical calculation of the scaling exponents that characterize the statistics of fully developed turbulence is one of the major open problems of statistical physics. We review the subject, explain some of the recent developments, and point out the road ahead.

Exact Result for the 3rd Order Correlations of Velocity in Turbulence with Helicity(1997) arXiv. 9705016. Abstract
All statistical models of turbulence take into account Kolmogorov's exact result known as the "4/5 law" which stems from energy conservation. This law states that the energy flux expressed as a spatial derivative of the 3rd order velocity correlator equals the rate of energy dissipation. We have found an additional exact result which stems from the conservation of helicity in turbulence without inversion symmetry. It equates the flux of helicity expressed as a second spatial derivative of the 3rd order velocity correlator with the rate of helicity dissipation. This exact result must be incorporated to all statistical theories of turbulence with helicity. After submitting this paper for publication we learned that the main result was independently found by Otto Chkhetiani in JETP Lett . 63, 808 (1996).

(1997) Physical review letters. 79, 17, p. 31743177 Abstract
We present the first experimental tests of the recently derived fusion rules for NavierStokes turbulence. The fusion rules address the asymptotic properties of manypoint correlation functions as some of the coordinates coalesce, and form an important ingredient of the nonperturbative statistical theory of turbulence. Here we test the fusion rules when the spatial separations lie within the inertial range, and find good agreement between experiment and theory. For inertialrange separations and for velocity increments which are not too large, a simple linear relation appears to exist for the Laplacian of the velocity fluctuation conditioned on velocity increments.

Nonperturbative zero modes in the Kraichnan model for turbulent advection(1997) Physical Review E. 55, 4, p. R3836R3839 Abstract
The anomalous scaling behavior of the nth order correlation functions [formula presented] of the Kraichnan model of turbulent passive scalar advection is believed to be dominated by the homogeneous solutions (zero modes) of the Kraichnan equation B[formula presented][formula presented]=0. Previous analysis found zero modes in perturbation theory with respect to a small parameter. We present a computerassisted analysis of the simplest nontrivial case of n=3: we demonstrate nonperturbatively the existence of anomalous scaling, and compare the results with the perturbative predictions.

(1997) Physical Review E. 56, 1 SUPPL. A, p. 406416 Abstract
The anomalous scaling behavior of the nthorder correlation functions F_{n} of the Kraichnan model of turbulent passive scalar advection is believed to be dominated by the homogeneous solutions (zero modes) of the Kraichnan equation B_{n}F_{n} = 0. In this paper we present an extensive analysis of the simplest (nontrivial) case of n = 3 in the isotropic sector. The main parameter of the model, denoted as ζ_{h}, characterizes the eddy diffusivity and can take values in the interval 0≤ζ_{h}≤2. After choosing appropriate variables we can present nonperturbative numerical calculations of the zero modes in a projective two dimensional circle. In this presentation it is also very easy to perform perturbative calculations of the scaling exponent ζ_{3} of the zero modes in the limit ζ_{h}→0, and we display quantitative agreement with the nonperturbative calculations in this limit. Another interesting limit is ζ_{h}→2. This second limit is singular, and calls for a study of a boundary layer using techniques of singular perturbation theory. Our analysis of this limit shows that the scaling exponent ζ_{3} vanishes as √ζ_{2}/Inζ_{2}, where ζ_{2} is the scaling exponent of the secondorder correlation function. In this limit as well, perturbative calculations are consistent with the nonperturbative calculations.
1996

(1996) Physics of Fluids. 8, 10, p. 25652567 Abstract
Correlation functions of nonscalar fields in isotropic hydrodynamic turbulence are characterized by a set of universal exponents. These exponents also characterize the rate of decay of the effects of anisotropic forcing in developed turbulence. These exponents are important for the general theory of turbulence, and for modeling anisotropic flows. We propose methods for measuring these exponents by designing new laboratory experiments.


(1996) Physical review letters. 76, 21, p. 39633966 Abstract
It is shown that the description of anomalous scaling in turbulent systems requires the simultaneous use of two normalization scales. This phenomenon stems from the existence of two independent (infinite) sets of anomalous scaling exponents that appear in leading order, one set due to infrared anomalies and the other due to ultraviolet anomalies. To expose this clearly we introduce here a set of local fields whose correlation functions depend simultaneously on the two sets of exponents. Thus the Kolmogorov picture of \u201cinertial range\u201d scaling is shown to fail because of anomalies that are sensitive to the two ends of this range.

(1996) Physical Review E. 53, 4, p. 34683490 Abstract
Elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described. We focus here on the structure functions of velocity differences that satisfy inertial range scaling laws Sn(R)similar to R(zeta n), and the correlation of energy dissipation Kepsilon epsilon(R)similar to R(mu). The goal is to understand from first principles what is the mechanism that is responsible for changing the exponents zeta(n) and mu from their classical Kolmogorov values. In paper II of this series [V. S. L'vov and I. Procaccia, Phys. Rev. E 52, 3858 (1995)] it was shown that the existence of an ultraviolet scale (the dissipation scale eta) is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields. The leading scaling exponent in this family was denoted Delta. The exact resummation of ladder diagrams resulted in a ''bridging relation,'' which determined Delta in terms of zeta(2): Delta = 2zeta(2). In this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling (i.e., zeta(n) not linear in n) with the renormalization scale being the infrared outer scale of turbulence L. It is shown that deviations from the classical Kolmogorov 1941 theory scaling of Sn(R) (zeta(n) not equal n/3) must appear if the correlation of dissipation is mixing (i.e., mu>0). We suggest possible scenarios for multiscaling, and discuss the implication of these scenarios on the values of the scaling exponents zeta(n) and their ''bridge'' with mu.

(1996) Physical review letters. 76, 11, p. 18281831 Abstract
In turbulent flows the nth order structure functions S_{n}(R) scale like R^{ζ}_{n} when R is in the \u201cinertial range\u201d. Extended SelfSimilarity refers to the substantial increase in the range of power law behaviour of S_{n}(R) when they are plotted as a function of S_{2}(R) or S_{3}(R) In this Letter we demonstrate this phenomenon analytically in the context of the \u201cmultiscaling\u201d turbulent advection of a passive scalar. This model gives rise to a series of differential equations for the structure functions S_{n}(R) which can be solved and shown to exhibit extended self similarity. The phenomenon is understood by comparing the equations for S_{n} to those for S_{n}(S_{2}).

(1996) Physical Review E. 54, 6, p. 63646371 Abstract
We consider turbulent advection of a scalar field [Formula Presented], passive or active, and focus on the statistics of gradient fields conditioned on scalar differences [Formula Presented] across a scale [Formula Presented]. In particular we focus on two conditional averages [Formula Presented] and [Formula Presented]. We find exact relations between these averages, and with the help of the fusion rules we propose a general representation for these objects in terms of the probability density function [Formula Presented] of [Formula Presented]. These results offer a way to analyze experimental data that is presented in this paper. The main question that we ask is whether the conditional average [Formula Presented] is linear in [Formula Presented]. We show that there exists a dimensionless parameter which governs the deviation from linearity. The data analysis indicates that this parameter is very small for passive scalar advection, and is generally a decreasing function of the Rayleigh number for the convection data.

(1996) Physical review letters. 77, 17, p. 35413544 Abstract
It is shown that the idea that scaling behavior in turbulence is limited by one outer length L and one inner length η is untenable. Every nth order correlation function of velocity differences F_{n}\(R_{1}, R_{2}.,\) exhibits its own crossover length η_{n} to dissipative behavior as a function of R_{1}. This depends on n and on the remaining separations R_{2}, R_{3}., One result is that when separations are of the same order R, this scales as η_{n}\(R\)∼η\(R/L\)^{xn} with x_{n} = \(ζ_{m} ζ_{n+1} + ζ_{3} ζ_{2}\)/\(2  ζ_{2}), ζ_{n} the scaling exponent of the nth order structure function. We derive an infinite set of scaling relations that bridge the exponents of correlations of gradient fields to the exponents ζ_{n}, including the \u201cbridge relation\u201d for the scaling exponent of dissipation fluctuations μ = 2ζ_{6}.

Exact resummations in the theory of hydrodynamic turbulence. III. Scenarios for anomalous scaling and intermittency(1996) Physical Review E  Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 53, 4, p. 34683490 Abstract
The lectures presented by one of us (IP) at the Les Houches summer school dealt with the scaling properties of high Reynolds number turbulence in fluid flows. The results presented are available in the literature and there is no real need to reproduce them here. Quite on the contrary, some of the basic tools of the field and theoretical techniques are not available in a pedagogical format, and it seems worthwhile to present them here for the benefit of the interested student. We begin with a detailed exposition of the naive perturbation theory for the ensemble averages of hydrodynamic observables (the mean velocity, the response functions and the correlation functions). The effective expansion parameter in such a theory is the Reynolds number (Re); one needs therefore to perform infinite resummations to change the effective expansion parameter. We present in detail the DysonWyld line resummation which allows one to dress the propagators, and to change the effective expansion parameter from Re to O(1). Next we develop the ``dressed vertex" representation of the diagrammatic series. Lastly we discuss in full detail the pathintegral formulation of the statistical theory of turbulence, and show that it is equivalent order by order to the DysonWyld theory. On the basis of the material presented here one can proceed smoothly to read the recent developments in this field.

(1996) Physical Review E. 54, 5, p. 51225133 Abstract
We examine the dynamic interplay between vorticity magnitude and vortex line geometry, and its relevance for curbing potential finitetime singularities in incompressible NavierStokes flows. We present direct numeri cal simulations of flows with various low and midrange Reynolds numbers and different types of forcing. The central conclusion is that the vortex lines in regions of high vorticity tend to be straight and well aligned. Such an organization indicates the existence of a selfcorrecting mechanism that cancels the quadratic nonlinearity inherent in the vorticity equation. We consider several relevant effects, including the observation of straight ening of vortex lines by stretching.

(1996) Physical Review E. 54, 6, p. 62686284 Abstract
In this paper we address nonperturbative aspects of the analytic theory of hydrodynamic turbulence. Of paramount importance for this theory are the "fusion rules" that describe the asymptotic properties of [Formula Presented]point correlation functions when some of the coordinates tend toward one other. We first derive here, on the basis of two fundamental assumptions, a set of fusion rules for correlations of velocity differences when all the separations are in the inertial interval. Using this set of fusion rules we consider the standard hierarchy of equations relating the [Formula Presented]order correlations (originating from the viscous term in the NavierStokes equations) to [Formula Presented] order (originating from the nonlinear term) and demonstrate that for fully unfused correlations the viscous term is negligible. Consequently the hierarchic chain of equations is decoupled in the sense that the correlations of [Formula Presented] order satisfy a homogeneous equation that may exhibit anomalous scaling solutions. Using the same hierarchy of equations when some separations go to zero we derive, on the basis of the NavierStokes equations, a second set of fusion rules for correlations with differences in the viscous range. The latter includes gradient fields. We demonstrate that every [Formula Presented]order correlation function of velocity differences [Formula Presented] exhibits its own crossover length [Formula Presented] to dissipative behavior as a function of, say, [Formula Presented]. This length depends on [Formula Presented] and on the remaining separations [Formula Presented] When all these separations are of the same order [Formula Presented] this length scales as [Formula Presented] with [Formula Presented], with [Formula Presented] being the scaling exponent of the [Formula Presented]order structure function. We derive a class of exact scaling relations bridging the exponents of correlations of gradient fields to the exponents [Formula Presented] of the [Formula Presented]order structure functions. One of these relations is the well known "bridge relation" for the scaling exponent of dissipation fluctuations [Formula Presented].

(1996) Physical review letters. 76, 1, p. 146149 Abstract
The roughening of expanding flame fronts by the accretion of cusplike singularities is a fascinating example of the interplay between instability, noise, and nonlinear dynamics that is reminiscent of selffractalization in Laplacian growth patterns. The nonlinear integrodifferential equation that describes the dynamics of expanding flame fronts is amenable to analytic investigations using pole decomposition. This powerful technique allows the development of a satisfactory understanding of the qualitative and some quantitative aspects of the complex geometry that develops in expanding flame fronts.

(1996) Physical review letters. 76, 16, p. 28982901 Abstract
Fusion rules in turbulence specify the analytic structure of manypoint correlation functisons of the turbulent field when a group of coordinates coalesce. We show that the existence of universal flux equilibrium in fully developed turbulent systems combined with a direct cascade induces universal fusion rules. In certain examples these fusion rules suffice to compute the multiscaling exponents exactly, and in other examples they give rise to an infinite number of scaling relations that constrain enormously the structure of the allowed theory.


Anomalous Scaling in Turbulence: a Field Theoretic Approach(1996) Nonlinear Dynamics, Chaotic and Complex Systems. Abstract

Fusion rules and conditional statistics in turbulent advection(1996) Physical Review E. 54, 5, p. R4520R4523 Abstract
Fusion rules in turbulence address the asymptotic properties of manypoint correlation functions when some of the coordinates are very close to each other. Here we put to the experimental test some nontrivial consequences of the fusion rules for scalar correlations in turbulence. To this aim we examine passive turbulent advection as well as convective turbulence. Adding one assumption to the fusion rules, one obtains a prediction for universal conditional statistics of gradient fields. We examine the conditional average of the scalar dissipation field [Formula Presented] for [Formula Presented] in the inertial range and find that it is linear in [Formula Presented] with a fully determined proportionality constant. The implications of these findings for the general scaling theory of scalar turbulence are discussed.

(1996) Physica Scripta. 67, p. 131135 Abstract
In this short note we present a brief overview of our recent progress in understanding the universal statistics of fully developed turbulence, with a stress on anomalous scaling.

(1996) Physical Review E. 53, 4, p. 35183535 Abstract
Kraichnans model of passive scalar advection in which the driving velocity field has fast temporal decorrelation is studied as a case model for understanding the appearance of anomalous scaling in turbulent systems. We demonstrate how the techniques of renormalized perturbation theory lead (after exact resummations) to equations for the statistical quantities that also reveal nonperturbative effects. It is shown that ultraviolet divergences in the diagrammatic expansion translate into anomalous scaling with the inner length acting as the renormalization scale. In this paper, we compute analytically the infinite set of anomalous exponents that stem from the ultraviolet divergences. Notwithstanding these computations, nonperturbative effects furnish a possibility of anomalous scaling based on the outer renormalization scale. The mechanism for this intricate behavior is examined and explained in detail. We show that in the language of Lvov, Procaccia, and Fairhall [Phys. Rev. E 50, 4684, (4684)], the problem is "critical," i.e., the anomalous exponent of the scalar primary field [Formula Presented]. This is precisely the condition that allows for anomalous scaling in the structure functions as well, and we prove that this anomaly must be based on the outer renormalization scale. Finally, we derive the scaling laws that were proposed by Kraichnan for this problem and show that his scaling exponents are consistent with our theory.
1995

(1995) EPL. 29, 4, p. 291296 Abstract
We discuss the theoretical implications of the experimental results for the cross correlations between velocity differences and dissipative fields which are reported in the companion (preceding) letter (Europhys. Lett., 28 (1994) 635). The first implication is that 3d hydrodynamic turbulence has no conformal symmetry. Secondly, the experiment confirms the nonconformal scaling behaviour of such correlations as predicted by the analytical theory of the present authors. The results of the measurements lend support to the subcritical scenario that was suggested recently as an explanation of the nonKolmogorov scaling of the structure functions in large but finite Reynolds number turbulence.

(1995) Physical Review E. 52, 4, p. 34023414 Abstract
We present a theoretical framework for the discussion of the scaling properties of interfaces advancing in systems with quenched disorder. In all such systems there are critical conditions at which the interface gains scale invariance for sufficiently slow growth. There are two fundamental concepts, the ''blocking surfaces'' and the ''associated processes,'' whose nature determines the scaling properties of the advancing interfaces at criticality. The associated processes define a network whose scaling properties determine all the exponents (static and dynamic) that characterize the critical growing interface via universal scaling relations. We point out in this paper that most of the physical rules that can be used to advance the interface also incorporate noncritical elements; as a result, the roughness exponent of the growing interface may deviate from that of the critical interface in a ruledependent way. We illustrate the wide applicability of the universal scaling relations with diverse models, such as the EdwardsWilkinson (EW) model with quenched noise, the randomfield Ising model, and the KardarParisiZhang (KPZ) model with quenched noise. It is shown that the last model is characterized by bounded slopes, whereas in the EW model the slopes are unbounded. This fact makes the KPZ model equivalent to the selforganized interface depinning model of Buldyrev and Sneppen.

(1995) Physical review letters. 74, 14, p. 26902693 Abstract
A physical interpretation of a recent NavierStokes based theory for scaling in developed hydrodynamic turbulence is presented. It is proposed that corrections to the normal Kolmogorov scaling behavior of the nth order velocity structure functions are finite Reynolds number effects which disappear when the inertial interval exceeds 56 decades. These corrections originate from the correlation between the velocity differences and energy dissipation which are characterized by an anomalous (subcritical) exponent. The values of the experimentally observed scaling indices for the nth order structure functions for n between 4 and 14 are in agreement with our findings.

(1995) Physical Review E. 51, 4, p. 32073222 Abstract
We examine the possibility of a blowup of the vorticity due to selfstretching and mechanisms for its prevention. We first estimate directly from the NavierStokes equations the length scale of coherence in the direction of the vortex lines to be of the order of the Kolmogorov length. Alignment of vortex lines is seen to be a viscous phenomenon and may prevent some scenarios for blowup. Next we derive equations for the curvature and torsion of vortex lines. We show that the same stretching that amplifies the vorticity also tends to straighten out the vortex lines. Then we show that in wellaligned vortex tubes, the selfstretching rate of the vorticity is proportional to the ratio of the vorticity and the radius of curvature. Thus blowup of the vorticity in such tubes can be prevented by the growth of the vorticity being balanced by the straightening of the vortex lines. Implications for vorticitystrain alignment and the scaling theory of turbulence are noted. Finally, we examine the effects of viscous diffusion on the vorticity field and see how viscosity can lead to organization and alignment of vortex lines.

(1995) Physics of Plasmas. 2, 4, p. 11481156 Abstract
The structures in magnetohydrodynamic (MHD) flow, flux tubes in particular, are investigated with respect to coherence in the direction of the magnetic field. A length scale, which is interpreted as the diameter of the tubes, is derived from the MHD equations. This scale implies that the tendency towards alignment of flux lines in tubes is a diffusion driven phenomenon. The dynamics of the tubes is also investigated; the major conclusion is that stronger tubes are expected to be straighter. These ideas are tested out on data from numerical simulations of turbulent MHD convection. It is also seen that alignment of flux lines increases with the strength of the tube. Possible reasons for this effect are discussed.

(1995) Physical Review E. 52, 4, p. 38403857 Abstract
This paper is the first in a series of papers that aim at understanding the scaling behavior of hydrodynamic turbulence. We present in this paper a perturbative theory for the structure functions and the response functions of the hydrodynamic velocity field in real space and time. Starting from the NavierStokes equations (at high Reynolds number Re) we show that the standard perturbative expansions that suffer from infrared divergences can be exactly resummed using the BelinicherL'vov transformation. After this exact (partial) resummation it is proven that the resulting perturbation theory is free of divergences, both in large and in small spatial separations. The hydrodynamic response and the correlations have contributions that arise from mediated interactions which take place at some spacetime coordinates. It is shown that the main contribution arises when these coordinates lie within a shell of a ''ball of locality'' that is defined and discussed. We argue that the real spacetime formalism that is developed here offers a clear and intuitive understanding of every diagram in the theory, and of every element in the diagrams. One major consequence of this theory is that none of the familiar perturbative mechanisms may ruin the classical 1941 Kolmogorov (K41) scaling solution for the structure functions. Accordingly, corrections to the K41 solutions should be sought in nonperturbative effects. These effects are the subjects of paper II (the following paper) and a future paper in this series that will propose a mechanism for anomalous scaling in turbulence, which in particular allows a multiscaling of the structure functions.

(1995) Physical Review E. 52, 4, p. 38583875 Abstract
In paper I of this series on fluid turbulence we showed that exact resummations of the perturbative theory of the structure functions of velocity differences result in a finite (order by order) theory. These findings exclude any known perturbative mechanism for anomalous scaling of the velocity structure functions. In this paper we continue to build the theory of turbulence and commence the analysis of nonperturbative effects that form the analytic basis of anomalous scaling. Starting from the NavierStokes equations (at high Reynolds number Re) we discuss the simplest examples of the appearance of anomalous exponents in fluid mechanics. These examples are the nonlinear (fourpoint) Green's function and related quantities. We show that the renormalized perturbation theory for these functions contains ''ladder'' diagrams with (convergent) logarithmic terms that sum up to anomalous exponents. Using a sum rule that is derived here we calculate the leading anomalous exponent and show that it is critical. This result opens up the possibility of multiscaling of the structure functions with the outer scale of turbulence as the renormalization length. This possibility will be discussed in detail in a concluding paper of this series.

(1995) Physical Review E. 51, 2, p. 11481154 Abstract
We present an analytic and numerical analysis of the GledzerOhkitani Yamada (GOY) cascade model for turbulence. We concentrate on the dynamic correlations, and demonstrate both numerically and analytically, using resummed perturbation theory, that the correlations do not follow a dynamic scaling ansatz. The basic reason for this is the existence of a second quadratic invariant, in addition to energy. This implies the breakdown of the Kolmogorovtype scaling law, in a manner different from the conventional mechanisms proposed for NavierStokes intermittency. By modifying the model equation so as to eliminate the spurious invariant, we recover to good accuracy both dynamic scaling and the Kolmogorov exponents. We conclude that intermittency in the GOY model may be attributed to the effects of the spurious invariant which does not exist in the threedimensional NavierStokes flow.
1994

Anomalous scaling in fluid mechanics: The case of the passive scalar: The case of the passive scalar(1994) Physical Review E. 50, 6, p. 46844704 Abstract
A mechanism for anomalous scaling in turbulent advection of passive scalars is identified as being similar to a recently discovered mechanism in NavierStokes dynamics [V. V. Lebedev and V. S. Lvov, JETP Lett. 59, 577 (1994)]. This mechanism is demonstrated in the context of a passive scalar field that is driven by a rapidly varying velocity field. The mechanism is not perturbative, and its demonstration within renormalized perturbation theory calls for a resummation of infinite sets of diagrams. For the example studied here we make use of a small parameter, the ratio of the typical time scales of the passive scalar vs that of the velocity field, to classify the diagrams of renormalized perturbation theory such that the relevant ones can be resummed exactly. The main observation here, as in the NavierStokes counterpart, is that the dissipative terms lead to logarithmic divergences in the diagrammatic expansion, and these are resummed to an anomalous exponent. The anomalous exponent can be measured directly in the scaling behavior of the dissipation twopoint correlation function, and it also affects the scaling laws of the structure functions. It is shown that when the structure functions exhibit anomalous scaling, the dissipation correlation function does not decay on length scales that are in the scaling range. The implication of our findings is that the concept of an inertial range in which the dissipative terms can be ignored is untenable. The consequences of this mechanism for other cases of possible anomalous scaling in turbulence are discussed.

(1994) Physical Review E. 49, 2, p. 12321237 Abstract
We present a theoretical analysis of a recently introduced interface growth model with a global search of optimal growth sites, realized by quenched random forces. The interest in this model lies in the fact that it yields a variety of longranged correlated phenomena, which are characterized by scaling exponents that differ from the KardarParisiZhang universality class [Phys. Rev. Lett. 56, 889 (1986)], in closer correspondence with experimental observations. It is shown that all the phenomenological findings can be recovered from the theory, and all the exponents found can be computed from the knowledge of one exponent of the directed percolation problem.

(1994) Physical Review E. 49, 5, p. 40444051 Abstract
In the inertial interval of turbulence one asserts that the velocity structure functions Sn(r) scale like rnnζ. Recent experiments indicate that Sn(r) has a more general universal form [rf(r/η)]nnζ, where η is the Kolmogorov viscous scale. This form seems to be obeyed on a range of scales that is larger than power law scaling. It is shown here that this extended universality stems from the structure of the NavierStokes equations and from the property of the locality of interactions. The approach discussed here allows us to estimate the range of validity of the universal form. In addition, we examine the possibility that the observed deviations from the classical values of ζn=1/3 are due to the finite values of the Reynolds numbers and the anisotropy of the excitation of turbulence.

(1994) Physical review letters. 72, 2, p. 307 Abstract
A Comment on the Letter by C. Jayaprakash, F. Hayot, and R. Pandit, Phys. Rev. Lett. 71, 12 (1993).

(1994) Nonlinearity. 7, 3, p. 10451054 014. Abstract
Lower bounds on the fractal dimension of level sets of advecting passive scalars in turbulent fields are derived, in the limit that the scalar diffusivity kappa goes to zero. The main result is as follows: denote the Holder exponent of the velocity field u by zeta (u), with 0

(1994) Physical review letters. 73, 3, p. 432435 Abstract
We study analytically and numerically the corrections to scaling in turbulence which arise due to finite size effects as anisotropic forcing or boundary conditions at large scales. We find that the deviations δζm from the classical Kolmogorov scaling ζm=m/3 of the velocity moments u(k)mkζm decrease like δζm(Re)=cmRe3/10. If, on the contrary, anomalous scaling in the inertial subrange can experimentally be verified in the large Re limit, this will support the suggestion that small scale structures should be responsible, originating form viscous effects either in the bulk (vortex tubes or sheets) or from the boundary layers (plumes or swirls), as both are underestimated in our reduced wave vector set approximation of the NavierStokes dynamics.
1993

(1993) Physics Letters A. 182, 1, p. 9398 Abstract
A new conceptual framework for understanding multiscaling in classical field theories in general, and in turbulence in particular, is proposed. Multiscaling refers to the phenomenon that structure functions of different order are characterized by independent scaling exponents. In our approach the time dependence of the fields is taken explicitly into account. The fields are assumed to be Hölder continuous in space with a range of Hölder exponents, with the worst singularities having a shorter lifetime. As a result, the time averaged structure functions exhibit a spectrum of scaling exponents as is seen in experiments on turbulence. By taking higher order structure functions one exposes again the more singular spatial contributions. We expect that this framework reflects the equations of motion better than the currently popular fractal and multifractal models.

(1993) Europhysics Letters. 22, 9, p. 689694 Abstract
Recent experiments on the dispersal of a dye by chaotic surface waves indicate that the isoconcentration curves exhibit fractal geometry on a significant range of scales. In this letter we offer a rigorous theory in which the dimension of the isoconcentration curves is calculated. It is shown that the results are universal for all types of surface waves without mean advection, and are in good agreement with the experimental measurements.

(1993) Physical Review Letters. 70, 22, p. 34163419 Abstract
Scaling behavior in turbulence is studied on the basis of its relation to the wrinkling, or fractalization, of the graph of the velocity field in 3 + 3 dimensions. We propose a novel mechanism for deviations from the Kolmogorov exponents, which is realized if the fine structure of turbulence tends locally towards two dimensionality. It is argued that some of the popular fractal and multifractal models of intermittency in turbulence are not consistent with fluid mechanics, and miss some essential physics.

(1993) Physical Review E. 47, 5, p. 33073315 Abstract
We develop a theory that is nonperturbative and free of uncontrolled approximations to understand scaling behavior in turbulence. The main tool is a connection between the dimension of the graphs of the hydrodynamic fields and the scaling exponents of their structure functions. The connection is developed in some generality for both scalar and vector fields, in terms of the geometric invariants of the gradient tensor. We show that fluid mechanics is consistent with fractal graphs for both the scalar and the vector fields, and explain how this leads to the scaling behavior of the structure functions. We derive scaling relations between various scaling exponents, and show that in the case of "strong scaling" (which is defined below) the Kolmogorov solution is unique. Our theory allows additional solutions in which a weaker version of scaling results in a spectrum of scaling exponents. In particular, we identify the dimensionless (but Reynoldsnumberdependent) contributions which can lead to deviations from the Kolmogorov exponents (which are derived using dimensional analysis). Results for the dimensions of fractal level sets in hydrodynamic turbulence which are measured in experiments and simulations follow immediately from this theory.

(1993) Physical Review E. 47, 5, p. 30993121 Abstract
The scaling properties of three nontrivial onedimensional avalanche models are analyzed. The first two of them are the local limited model with one open, one closed, and with periodic boundary conditions, respectively. A theory for the scaling properties of these models based on the existence of two fundamental length scales, which diverge in the thermodynamic limit, is developed. The third model studied is a trapless version of the nonperiodic local limited model. We find that it is scale invariant. Our theoretical predictions are compared with extensive computer simulations in all three cases.

(1993) Nonlinearity. 6, 1, p. 2547 002. Abstract
Under the assumption that the KardarParisiZhang model (KPZ) possesses scale invariant solutions, there exists an exact calculation of the dynamic scaling exponent z=3/2. The authors prove that both KPZ and the related KuramotoSivashinsky model indeed possess scale invariant solutions in 1+1 dimensions which are in fact the same for both models. The proof entails an examination of the higher order diagrams in the perturbation theory in terms of the dressed Green function and the correlator. Although each higher order diagram contains logarithmic divergences, endangering the existence of the scale invariant solution, the authors show that these divergences cancel in each order. The proof uses a fluctuationdissipation theorem (FDT), which is an exact result for KPZ in 1+1 dimensions.
1992

(1992) Europhysics Letters. 19, 3, p. 183187 Abstract
The fractal dimension of isovorticity surfaces is estimated from a 3dimensional simulation of homogeneous turbulence at moderate Reynolds numbers, performed by Vincent and Meneguzzi. The results are found to be compatible with a recently proposed theory which predicts a crossover from a 2dimensional geometry at small scales to a fractal geometry at larger scales, with a dimension D = 2.5 + zeta/2, with zeta being the exponent characterizing the scaling of velocity differences.

(1992) Physical review letters. 69, 24, p. 35433546 Abstract
It is shown that the scale invariant solutions of the KS and KPZ models of surface roughening are identical for dimensions d2 in the strong coupling limit. For d>2, these models posses two different scaling solutions, one with dindependent scaling exponents y=z=2, and the other with ddependent nontrivial exponents. The first of these solutions is realizable in one of these models, but not the other. These conclusions are valid to all orders in renormalized perturbation theory.

(1992) Physical Review A. 46, 8, p. 47364741 Abstract
The multifractal model of turbulence assumes a concentration of the field of vorticity magnitude on a set of dimension smaller than 3. We examine the mechanism for the exponential growth of the vorticity magnitude, and estimate the fractal dimenison of its level sets on the basis of the equations of fluid mechanics. We propose a model of multifractal turbulence in which there is a connection between the fractal dimension of the level sets and the information dimension D1 of the carrier of the field of vorticity magnitude. Under the stated conditions we can estimate the dimension D1 and evaluate, within the framework of the multifractal model of turbulence, the values of the scaling exponents of velocity and temperature structure functions. We find exponents that are consistent with the available experimental information and which differ from the ones obtained by the Kolmogorov dimensional analysis.

(1992) Physical Review A. 46, 6, p. 32203224 Abstract
The longwavelength properties of the KuramotoSivashinsky equation are studied in 2+1 dimensions using numerical and analytic techniques. It is shown that this equation is not in the universality class of the KardarParisiZhang model. Its roughening exponents are (up to logarithmic corrections) like those of the freefield theory, with dimension 2 being the marginal dimension for roughening. Assuming that the solution has logarithmic corrections, we derive a scaling relation for the exponents of the logarithmic terms. This solution is consistent order by order with the DysonWyld diagrams. We explain why previous renormalizationgroup treatments failed.

(1992) Physical Review A. 46, 8, p. 48194828 Abstract
The fractal nature of level sets and the multifractal nature of various scalar and vector fields in hydromagnetic and hydrodynamic turbulence are investigated using data of direct simulations. It turns out that fields whose evolution is governed by stretching terms (vortex stretching, magneticfield line stretching) exhibit ''near singularities'' that result in a multifractal scaling. Such stretching terms can lead to a rapid increase in the local value of the field. Fields without rapid local increase have no multifractal scaling. Furthermore, the simulations support recent theoretical suggestions that the fractal properties of the level sets of various fields are quite insensitive to the existence of stretching. Indeed, all the fields under study (temperature, vorticity magnitude, magneticfield magnitude) show rather universal behavior in the geometry of their level sets, consistent with a twodimensional geometry at small scales, with a crossover to a universal fractal geometry at large scales. The dimension at large scales is compatible with the theoretical prediction of about 2.7. The most surprising result of the simulations is that it appears that the ''near singularities'' are not efficiently eliminated by viscous dissipation, but rather seem to be strongest at the Kolmogorov cutoff. The effects of the singularities do not quite penetrate into the inertial range. We offer a simple analytic model to account for this behavior. We conclude that our findings may be due to the relatively small Reynolds numbers, but may also be indicative of generic behavior at larger Reynolds numbers. We offer some thoughts about the expected scaling behavior in the inertial range in light of our findings.

(1992) Physical Review A. 45, 8, p. 60956098 Abstract
A scaling description is given for the longwavelength behavior of avalanche models. Two examples are analyzed in detail: the twostate model and the locallimited model. Scaling estimates for the longwavelength behavior of coherence length, relaxation time, and diffusion coefficient are derived for both models. The results agree well with the estimates obtained from previous simulations.
1991

(1991) Physical review letters. 66, 7, p. 891894 Abstract
Combining methods and ideas of dynamical systems theory with the usual stability analysis for extended hydrodynamic systems we show that defectmediated turbulence is a generic consequence of a set of physical properties which are shared by many systems. We show how the interplay between broken continuous symmetries and the dynamics of patterns leads to a universal scenario for the onset of this type of turbulence.

(1991) Physical Review A. 44, 2, p. R805R808 Abstract
A mechanism for nucleating pairs of topological defects in non equilibrium patternforming systems in two space dimensions plus time is studied. Its relation to the creation of phase slips in spacetime plane is revealed, stressing its deterministric origin from smooth dynamics. The role of this mechanism in the context of spatiotemporal chaos is discussed.

(1991) EPL. 14, 1, p. 5560 Abstract
We present an experimental study of the dynamics and interactions of laminar plumes emitted from a localized heat source. The observations are explained by a simple model of the flow structure around a plume. Using sources and sinks in a uniform flow, we reproduce the experimental shapes and extract the scaling behavior of the size of the plume. The model describes the initial stage of the interaction between plumes.

(1991) Nonlinearity. 4, 2, p. 567581 016. Abstract
By studying typical hydrodynamic systems in the large aspect ratio limit, the authors show that spatiotemporal chaos is a natural consequence of the availability of secondary hydrodynamic instabilities. The study of such systems beyond the onset of secondary instabilities shows that, within perturbation theory, there exist stationary, spatially biperiodic solutions. Going beyond perturbation theory, it is found that a KAM mechanism is responsible for creating stationary, spatially chaotic solutions. Second, these solutions are shown to have stable and unstable manifolds in function space. They therefore conjecture that, typically, the time evolution is attracted to one of these spatially chaotic states and is then repelled along an unstable direction. In this second stage, topological defects are created if the system is sufficiently wide and couples to twodimensional modes. Thus, they identify spatiotemporal chaos with a mechanism of generating a random array of defects in an already spatially disordered system.

(1991) Physical review letters. 67, 13, p. 17391742 Abstract
We present a Lagrangian calculation of the area of isothermal (or isoconcentration) surfaces in a medium of fluid turbulence. We argue that such surfaces appear fractal above some inner scale which depends on the Reynolds number. The fractal dimension is estimated theoretically. The theory is compared to experiments in which a dye is used as a passive scalar in various turbulent flows. We find excellent agreement in many details.

(1991) Physical Review A. 44, 12, p. 80918102 Abstract
Experiments on convective turbulence have revealed a number of unexpected changes in measured signals. Most notable are (i) the transition from soft to hard turbulence at Rayleigh number (Ra) of about 108, which is seen as a qualitative change in the statistics of temperature fluctuations; (ii) a transition at Ra of about 1011, seen in the power spectra of temperature fluctuations measured at the center of the experimental cell; and (iii) a change, at Ra of about 1013, in the power spectra of a probe placed 0.2 cm from the bottom boundary. In this paper we present experimental evidence for the last two changes and offer a unified mechanism for all these transitions. The main ingredient of our theoretical analysis is a calculation suggesting that isothermal surfaces wrinkle, or appear fractal, above an inner scale. This inner scale and the Hausdorff dimension of the isothermal surfaces are estimated theoretically. This scale diminishes upon increasing Rayleigh number. We argue that as it diminishes it goes through the relevant scales of this experiment, i.e., the height of the central probe, the size of the mixing zone, and the height of the bottom probe. Thus it is suggested that all these transitions may be but different manifestations of the very same physics.
1990

(1990) Nonlinearity. 3, 2, p. 555 Abstract
The conditions under which one expects universal symbolic dynamics for strange attractor on wrinkled tori are restated and clarified.

(1990) Physical Review A. 42, 2, p. 821830 Abstract
The scaling exponents of twopoint correlation functions in nonisotropic convective turbulence are evaluated on the basis of the Boussinesq equations. The scaling indices differ significantly from Kolmogorovs scaling for isotropic turbulence. The physical reason for this difference is the length dependent buoyancy forcing in this case, in contrast to forcing on the largest scale only in Kolmogorovs picture. The calculation appears to be in agreement with preliminary experimental findings in convective turbulence with helium as the working fluid.

(1990) Physical Review A. 41, 12, p. 66026614 Abstract
Chaotic dynamical systems can be organized around an underlying strange set, which is comprised of all the unstable periodic orbits. In this paper, we quantify the complexity of such an organization; this complexity addresses the difficulty of predicting the structure of the strange set from loworder data and is independent of the entropy and the algorithmic complexity. We refer to the new measure as the grammatical complexity. The notion is introduced, discussed, and illustrated in the context of simple dynamical systems. In addition, the grammatical complexity is generalized to include metric properties arising due to the nonuniform distribution of the invariant measure on the strange set.

(1990) Physical review letters. 65, 1, p. 5255 Abstract
On the basis of a renormalization group in the space of shapes, conformal mapping techniques, and numerical simulations, it is argued that noisereduced and regular diffusionlimited aggregates on a lattice are in the same universality class. These models lead to the same asymptotic shape of fractal aggregates. This shape is a fixed point of a functional equation, realized by a conformal transformation.
1989


(1989) Physical review letters. 62, 18, p. 21282131 Abstract
A dynamical theory for the structure functions in convective turbulence at high Rayleigh numbers is presented. A range of scales is identified in which buoyancy forces dominate and the scaling exponents differ significantly from isotropic turbulence. A crossover to Kolmogorovtype scaling is predicted as a function of the Rayleigh number.

(1989) Physical review letters. 63, 12, p. 12371240 Abstract
The dynamics of a single topological defect and the interaction between pairs, including the process of annihilation of defects of opposite topological charge, are studied experimentally and theoretically near the onset of weak turbulence in Williams domains of electroconvecting nematics. The existence of topological defects requires a gauge field theoretical treatment, enriching the commonly used amplitude equations. The gauge field carries the interaction which is found to be of finite range in agreement with detailed observations.

(1989) Physical Review A. 39, 10, p. 53595372 Abstract
The calculation of the scaling properties of multifractal sets is presented as an eigenvalue problem. The eigenvalue equation unifies the treatment of sets that can be organized on regular treesbe they complete or incomplete (pruned) trees. In particular, this approach unifies the multifractal analyses of sets at the borderline of chaos with those of chaotic sets. Phase transitions in the thermodynamic formalism of multifractals are identified as a crossing of the largest discrete eigenvalue with the continuous part of the spectrum of the relevant operator. An analysis of the eigenfunctions is presented and several examples are solved in detail. Of particular interest is the analysis of intermittent maps, which shows the existence of an infiniteorder phase transitions, and of (Smale) incomplete maps of the interval with finite and infinite rules of pruning of the multifractal trees.

1988


(1988) Nonlinearity. 1, 1, p. 157180 Abstract
Strange attractors in dynamical systems that go to chaos via quasiperiodicity are considered. It is shown that there exists an infinite number of points in parameter space where the topology of the strange attractors is universal. At such points the periodic points belonging to unstable periodic orbits can be organised on ternary trees which are pruned by local rules. The grammar is universal, and thus the topological entropy is universal at each of these points in parameter space. The complete understanding of the topology is used to calculate systematically the metric properties of the attractors. The spectrum of scaling indices f(α) is computed. It is found that there is no metric universality, although some aspects of the metric properties are universal. Experiments to test some of the predictions of this theory are suggested.


(1988) Physical review letters. 60, 24, p. 25112514 Abstract
Fractal, dendritic patterns with invariant distributions which are harmonic measures are represented by Julia sets of polynomial mappings. The equipotential lines around such patterns are obtained, leading to analytic estimates of the minimal and maximal scaling exponents of the f() function of the harmonic measure. These and physical stability arguments rationalize the shapes of diffusionlimited aggregates.

(1988) Physical Review A. 37, 6, p. 22342236 Abstract
The spectrum of scaling exponents of multifractal, hyperbolic strange attractors is quantitatively understood by realizing that each unstable periodic point contributes one scaling exponent to the spectrum. The calculation of the f(±) function can thus be reduced to counting periodic orbits.

(1988) Physical Review A. 38, 3, p. 15031520 Abstract
We use the set of all periodic points of Hénontype mappings to develop a theory of the topological and metric properties of their attractors. The topology of a Hénontype attractor is conveniently represented by a twodimensional symbol plane, with the allowed and disallowed orbits cleanly separated by the pruning front. The pruning front is a function discontinuous on every binary rational number, but for maps with finite dissipation b
1987

(1987) Nuclear and Particle Physics Proceedings. 2, p. 517520 Abstract
Nonanalyticities in the generalized dimensions of multifractal sets of physical interest are interpreted as phase transitions. The problem is mapped onto thermodynamics of onedimensional spin models. The spin Hamiltonians are explicitly constructed and their phase transitions discussed. This mapping can provide insight in both directions. PACS numbers: 05.70.a.

(1987) Nuclear and Particle Physics Proceedings. 2, C, p. 527538 Abstract
The application of standard dimension algorithms is not always successful in deciding whether a given data set represents deterministic chaotic motion. We propose to supplement such algorithms with a search of the underlying unstable periodic orbits. These appear to give important clues to the organization of strange attractors and to their scaling properties. It is shown how to extract these orbits from experimental data sets, and how to use to obtain information on dimensions, entropies and the spectrum of scaling exponents of the ergodic invariant measure. Finally they are used to obtain approximants for the invariant measure itself. These approximants can be used to calculate a variety of interesting invariants that characterize the chaotic motion.

(1987) Nuclear and Particle Physics Proceedings. 2, p. 513516 Abstract
We study an experimental orbit on a twotorus with a goldenmean winding number obtained from a forced RayleighBénard system at the onset of chaos. This experimental orbit is compared with the orbit generated by a simple theoretical model, the circle map, at its goldenmean winding number at the onset of chaos. The "spectrum of singularities" of the two orbits are compared. Within error, these are identical. Since the spectrum characterizes the metric properties of the entire orbit, this result confirms theoretical speculations that these orbits, taken as a whole, enjoy a kind of universality. PACS numbers: 47.20. + m, 05.45. + b, 47.25. c.

(1987) Physical review letters. 59, 13, p. 13771380 Abstract
The organization, encoding, and hierarchical construction of a generic chaotic attractor is presented. A systematic calculation of the multifractal properties is accomplished, and an understanding of the spectrum of singularities is reported. In general we expect nontrivial phase transitions in the thermodynamic formalism of strange attractors.

(1987) Physical review letters. 58, 12, p. 11691172 Abstract
Nonanalyticities in the generalized dimensions of multifractal sets of physical interest are interpreted as phase transitions. The problem is mapped onto thermodynamics of onedimensional spin models. The spin Hamiltonians are explicitly constructed and their phase transitions discussed. This mapping can provide insight in both directions.

(1987) Physical review letters. 58, 23, p. 23872389 Abstract
The fractal invariant measure of chaotic strange attractors can be approximated systematically by the set of unstable nperiodic orbits of increasing n. Algorithms for extracting the periodic orbits from a chaotic time series and for calculating their stabilities are presented. With this information alone, important properties like the topological entropy and the Hausdorff dimension can be calculated.

(1987) Physical Review A. 35, 9, p. 40084011 Abstract
It is shown that the system of parametrically excited surface waves falls into the class of critical flows whose dynamics and transition to chaos can be understood from firstreturn maps derived in the vicinity of one saddle point in phase space. The onset of chaos is via gluing bifurcations, which are also common to Lorenzlike flows, but these are intermingled here with usual perioddoubling bifurcations. Similar parametrically excited systems might show the full array of routes to chaos which appear in critical flows.

(1987) Physical Review A. 35, 4, p. 18841900 Abstract
We propose to look at firstreturn maps into a specified region of phase space as a basis for a unified renormalization scheme for dynamical systems. The choice of the region for first return is dictated by the symbolic dynamics (e.g., kneading sequence) of the relevant trajectories. The renormalization group can be formulated on the symbolic level, but once translated to maps it yields the said renormalization scheme. We show how the wellstudied examples of the onset of chaos via period doubling and quasiperiodicity fit into this approach, and argue that these problems get in fact unified. The unification leads also to a generalization that allows us to study the onset of chaos in maps that belong to larger spaces of functions than those usually considered. In these maps we discover a host of new scenarios for the onset of chaos. These scenarios are physically relevant since the maps considered are reductions of simple flows. We present a theoretical analysis of some of these new scenarios, and report universal results. Finally we show that all the available renormalization groups can be found using symbolic manipulations only.
1986

(1986) Physical review letters. 57, 13, p. 15031506 Abstract
From the spectrum of dimensions of a fractal invariant measure of a dynamical system one can extract information about the dynamical process that gave rise to the measure. This is equivalent to finding the class of Hamiltonians of an Ising model with a given thermodynamics.

(1986) Physical Review A. 34, 4, p. 32213237 Abstract
A detailed theory of the appearance of lowdimensional chaos in a hydrodynamic system is presented. The system chosen has been subjected to a careful experimental study; it involves dynamics of surface waves in a cylinder of fluid which is oscillated vertically. All the major experimental findings are rationalized by the theory. It should be stressed that in addition to lowdimensional nonlinear evolution equations the theory results also in an approximate solution of the original partial differential equations.

(1986) Physical review letters. 57, 8, p. 925928 Abstract
It is shown that in simple threedimensional flows there are uncountably many different codimension2 scenarios for the onset of chaos. We analyze these scenarios via onedimensional Poincaré maps, and show how to construct renormalization groups to obtain new universal results. Comparisons with numerical studies are given.

(1986) Physical review letters. 56, 13, p. 13231326 Abstract
The experiment on chaos in surface waves reported by Ciliberto and Gollub is used as a case model for an understanding of the appearance of fewdimensional chaos in systems with an infinite number of degrees of freedom. Centermanifold and normalform theories are utilized to derive the pertinent dynamical system and to provide an approximate solution of the partial differential equations. Comparisons of theory and experiment are discussed.

(1986) Physical Review A. 34, 1, p. 659661 Abstract
Due to fluctuations around the metric entropy of chaotic dynamical systems there exists a spectrum of invariant dynamical scaling indices, complementary to the range of invariant static scaling indices which have been discovered and applied recently. The basic scaling behavior in both cases is shown to be rooted in the thermodynamic formalism of dynamical systems. An explicit simple example of these ideas is given, stressing their use in characterizing complex behavior.
1985

(1985) Physical Review A. 32, 2, p. 12251228 Abstract
The chaotic regime of systems which go to chaos via quasiperiodicity is studied, with the aim of establishing universal scaling laws for the invariants like the Lyapunov exponents, fractal dimensions, etc. Despite the complicated interventions of periodic windows, the envelopes of these quantities appear to agree extremely well with the theoretical predictions. This agreement is tied to the smooth development of the underlying invariant manifold.

(1985) Physical Review A. 32, 5, p. 30733083 Abstract
The issue of diffusion on fractals is addressed with stress on the question of how to generalize the diffusion equation for Euclidean lattices to the case of fractal lattices. Such an equation is proposed on the basis of scaling arguments. The solutions of this equation are interpreted as analytic envelopes of the exact probability distribution P(r,t) which is far from smooth. In fact it is shown that P(r,t) has discontinuities which appear selfsimilarly on all length scales. The exactly soluble examples of Sierpiński gaskets embedded in general dimension d are analyzed. The predstudied by assuming that the sticking probability of the particles depends on the local orientation of the interface. Directional solidification is simulated by the deposition of particles undergoing biased random walks. Linearly stable patterns are generated if the basic features of the directional solidification experiments are taken into account. The resulting patterns are very similar to those observed experimentally.

(1985) Physical review letters. 55, 25, p. 27982801 Abstract
We study an experimental orbit on a twotorus with a goldenmean winding number obtained from a forced RayleighBénard system at the onset of chaos. This experimental orbit is compared with the orbit generated by a simple theoretical model, the circle map, at its goldenmean winding number at the onset of chaos. The "spectrum of singularities" of the two orbits are compared. Within error, these are identical. Since the spectrum characterizes the metric properties of the entire orbit, this result confirms theoretical speculations that these orbits, taken as a whole, enjoy a kind of universality.

(1985) Physica Scripta. 1985, T9, p. 4046 Abstract
A short review of the theory and experimental measurement of the static and dynamical invariants that characterize chaotic motions is pesented. There is an infinity of such quantities which remain invariant under a smooth change of coordinates, but only the low order ones are relevant for experimental applications. From the theoretical point of view we stress the many interesting relations which exist between the invariants.

COMPUTING THE KOLMOGOROVENTROPY FROM TIME SIGNALS OF DISSIPATIVE AND CONSERVATIVE DYNAMICALSYSTEMS(1985) Physical Review A. 31, 3, p. 18721882 Abstract

(1985) Physical Review A. 31, 6, p. 39903992 Abstract
The transition from a modelocked torus to a chaotic fractal attractor via the generic saddlenode bifurcation is considered. Although superficially the transition seems to be one of the known types of intermittency, it is shown to be in fact discontinuous. All the invariants that characterize chaos show a distinct jump. It is also argued that the power spectra are not sensitive to the discontinuous nature of the transition.

(1985) Physical Review A. 31, 3, p. 18301840 Abstract
We discuss apparent and real 1/f divergencies in the smallfrequency limit of power spectra of dynamical systems. Apparent 1/f2 divergencies appear in typeI intermittency and whenever there is infrequent random hopping between regions with chaotic motion. Real 1/f divergencies appear in typesII and III intermittency. The universal properties of these divergencies are discussed in the context of one and twodimensional maps.

(1985) Physical review letters. 54, 5, p. 455458 Abstract
A generalization of the Euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument, a renormalizationgroup theory for the Green's function, and numerical tests. Conjectures on the applicability to natural fractals (such as porous media) are presented.
1984

(1984) Journal of Statistical Physics. 36, 56, p. 649663 Abstract
We present a qualitative overview of our work on the issue of fractal structures in turbulence. We explain why fully developed turbulence is not space filling and describe how its fractal dimension can be estimated theoretically. The implications of the fractal nature of turbulence on transport processes like turbulent diffusion and on fluctuations in passive scalars are discussed. The latter affect wave propagation in turbulent media and these effects are examined. In addition we consider clouds in the atmosphere which are claimed to have fractal perimeters (or surfaces) and outline the physical reasons for this phenomenon. The fractal dimension of clouds is tied to the theory of turbulent diffusion and is computed theoretically. Indications of the road ahead are given.

(1984) Physics Letters A. 104, 3, p. 140142 Abstract
Turbulent diffusion is anomalously large and considered as universal, i.e. independent of molecular properties. A recent report in Science raised doubts on this, based on measurements of pesticide mixture dispersion in air. This touches the very basis of current understanding of turbulence. We offer an explanation why and how molecular properties affect dispersion despite universal eddy transport.

(1984) PHYSICA D. 13, 12, p. 3454 Abstract
It is shown that the fluctuations in the divergence of nearby trajectories on (strictly deterministic) strange attractors can be modelled by stochastic concepts. In particular, we propose KramersMoyal type equations for correlation functions between points on the attractor. The drift terms are the Lyapunov exponents, the diffusion terms depend on the above fluctuations. From this, we obtain bounds on generalized dimensions and entropies. Numerical results show that in nearly all studied cases (Hénon map, Zaslavskii map, MackeyGlass eq.) the attractors are fractal measures in the sense of Farmer (information dimension ≠ Hausdorff dimension; metric entropy ≠ topological entropy).

(1984) Physical Review A. 30, 1, p. 647 Abstract
The relative importance of critical opalescence versus absorption in chemically reactive critical systems in the light of comments by Morrison and by Wheeler and Petschek is discussed.

(1984) Physical Review A. 29, 2, p. 975977 Abstract
In experiments involving deterministic chaotic signals, contamination by random noise is unavoidable. We discuss a practical method that disentangles the deterministic chaos from the random part. The method yields a characterization of the strange attractors together with an estimate of the size of random noise.

(1984) Physical Review A. 30, 6, p. 32143220 Abstract
A modecoupling formalism is developed for nonequilibrium stationary systems which are close to thermodynamic criticality. Formulas for the calculation of dressed transport coefficients in such systems are found. Applications of the formalism are presented in the following paper.

(1984) Physical Review A. 30, 6, p. 32213234 Abstract
The modecoupling formalism, developed in the preceding paper, is applied to heatconducting fluid systems near the gasliquid critical point. Expressions for the nonequilibrium dressed thermal diffusitivity and kinematic viscosity are presented. The main findings are as follows: The dressed transport coefficients become anisotropic. For a wave vector parallel to the temperature gradient, the transport coefficients gain a new divergent contribution. For a wave vector perpendicular to the temperature gradient, the equilibrium form is retained.


(1984) Physical Review A. 29, 3, p. 13581365 Abstract
Anomalous relative diffusion (pair separation) in the inertial, as well as in the viscous, subranges of fully developed turbulent flow is considered. Using the Lagrangian velocitycorrelation function in lowestorder continuedfraction approximation based on the NavierStokes equations and a phenomenological closure assumption, we express the variance in terms of the static structure function. This closed form for the turbulent diffusivity unifies scaling concepts with fluid mechanics and is free of fitting parameters. We obtain the following conclusions: Various regimes of variance versus time t are identified, t2 initially, t1 and exponential (viscous subrange), t3+ (inertial subrange). Intermittency (well known to alter the scaling exponent) implies a fairly strong effect upon the magnitude of the diffusion depending on the intial separation and on the Reynolds number; diffusion is delayed with increasing intermittency. Incompressibility is expected to lead to differences in transverse versus parallel separation. Molecular diffusivity may show up even in the universal regime as a permanent enhancement of diffusion due to different incubation times.

(1984) Physical Review A. 29, 3, p. 14611470 Abstract
A recent experiment indicates that clouds in the atmosphere have fractal surfaces which are characterized by a fractal dimension D2.35. Here we present a theory of the fractal dimension of clouds. We show that the fractal dimension of clouds is calculable from the theory of relative turbulent diffusion. We claim that previous approaches to the theory of relative diffusion are in contradiction with this experiment and that a newly developed theory gives D2.35 as a natural consequence. The new theory of relative diffusion is simultaneously in agreement with the intermittencymodified "43 (power) law" [i.e., the (4+2)3 (power) law, where is the intermittency exponent].
1983

(1983) PHYSICA D. 9, 12, p. 189208 Abstract
We study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise. The exponent v is closely related to the fractal dimension and the information dimension, but its computation is considerably easier. Its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed. Algorithms for extracting v from the time series of a single variable are proposed. The relations between the various measures of strange attractors and between them and the Lyapunov exponents are discussed. It is shown that the conjecture of Kaplan and Yorke for the dimension gives an upper bound for v. Various examples of finite and infinite dimensional systems are treated, both numerically and analytically.

(1983) PHYSICA D. 8, 3, p. 435444 Abstract
We show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions D_{q}, q > 0. To this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities D_{q}. We prove that lim _{q→0} D_{q} = fractal dimension (D), lim_{q→1}D_{q} = information dimension (σ) and D_{q=2} = correlation exponent (v). D_{q} with other integer q's correspond to exponents associated with ternary, quaternary and higher correlation functions. We prove that generally Dq > D_{q} for any q > q. For homogeneous fractals D_{q} = D_{q}. A particularly interesting dimension is D_{q=∞}. For two examples (Feigenbaum attractor, generalized baker's transformation) we calculate the generalized dimensions and find that D_{∞} is a nontrivial number. All the other generalized dimensions are bounded between the fractal dimension and D_{∞}.


(1983) Physical Review A. 27, 1, p. 555557 Abstract
We show that dissociationrecombination equilibria are strongly affected by critical phenomena. The concentration of monomers can increase with infinite slope (determined by critical indices) upon cooling towards Tc, in agreement with experiments.

(1983) Physical review letters. 50, 5, p. 346349 Abstract
A new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable. The relation of this new measure to fractal dimension and informationtheoretic entropy is discussed.

(1983) Physical Review A. 27, 5, p. 25852602 Abstract
Complete hydrodynamic theory of the dynamics of equilibrium fluctuations in onecomponent fluid layers confined by rigid solid boundaries is presented. The dynamic structure factor for such a fluid layer is shown to have both diagonal and offdiagonal elements which depend on the layer height L and on the transport of energy and tangential momentum across the fluidsolid interfaces. The effects of interfacial energy transport have been analyzed in the limits of maximum or vanishing tangential momentum transport ("stick" or "slip" boundary conditions on the velocity field). In the presence of interfacial transport, two new propagating modes have been found for each distinct interface. In the limit of maximum energy and/or tangential momentum transport, the new modes differ from the bulk sound modes only by an increased attenuation coefficient. Additional dissipation is in part due to shear created by the boundaries, and in part, since the sound is not isothermal, to heat conduction between the fluid and solid. In the dynamic structure factor, the new modes appear as additional peaks in the vicinity of the Brillouin peaks of the unbounded fluid; for wave vectors typical of light scattering experiments these peaks are found to be significant for L∼100 μm. Since the positions and line shapes of these peaks are very sensitive to interfacial transport, their study may provide a useful experimental probe of transport across the fluidsolid interfaces.


