## The Minerva Center for Nonlinear Physics of Complex Systems

Itamar Procaccia, DirectorThe Barbara and Morris L. Levinson Professorial Chair of Chemical PhysicsThe Center was established jointly by the Technion and the Weizmann Institute. It maintains strong links with the Max Planck Institute for Physics of Complex Systems in Dresden. The Center supports the activities of three groups at the Weizmann Institute, in addition to a variety of exchanges, workshops, schools and seminars.

The main subjects covered by the Center in the last few years were:

Hydrodynamics, Turbulence and Pattern Formation-Theory

Turbulence. Our research in the last few years concentrated on understanding the universality of turbulence, with a focus on the "anomalous" exponents that characterize the scaling properties of correlation functions and structure functions. We had three major lines of progress, in each of which we have achieved significant results. First, anomalous scaling was related to the existence of "Statistically Preserved Structures", which remain invariant (on the average) on the background of the turbulent flow. Such invariant functions are responsible for the observed anomalous exponents that were mysterious for decades to turbulence researchers. Now we can identify them as eigenfunctions of eigenvalue 1 of appropriate operators. Second, we have presented a systematic and accurate approach to peeling off anisotoropic contributions from turbulent statistical objects. We showed how each isotropic sector exhibits its own set of anomalous exponents which are universal. This way we have settled long standing issues related to the decay of aniostropy as a function of length scale and Reynolds number. Finally we have addressed the problem of drag reduction in turbulence by polymeric additives, and proposed a rather complete theory of this mysterious phenomenon. In particular we explained the universal "Maximum Drag Reduction" (MDR) asymptote, and offered explanation to the non-universal cross over back to Newtonian behavior. The theory provides an explanation to the common aspects and to the differences in drag reduction by flexible and rodlike polymers.

Our main focus on a fundamental side is in symmetries. We studiy anomalies i.e. breakdown of symmetries that do not disappear when symmetry-breaking factor goes to zero. In particular, we derived recently new exact flux relations for compressible and incompressible turbulence, they shed some unexpeceted light on how properties of turbulence may depend on th espace dimensionality. A different set of phenomena is related to symmetries emerging as a result of turbulence cascade. With collaborators in France and Italy, we discovered new emerging symmetries like conformal invariance in inverse turbulent cascades and now try to incorporate this in turbulence theory. We also work on exploring the whole class of systems (in fluid mechanics, plasma physics and geophysics) which allow for conformal invariance in turbulence. Another direction of work is on turbulence interacting with a coherent mode. Here we studued ultra-long fiber lasers and discovered an unusual process of appearance of optical turbulence very much similar to turbulence appearance in pipe flows. We also study spectral condensates that appear in inverse cascades and condensate-turbulence interaction with applications to atmospheric physics.

Fractal Grourth. We considered Laplacian growth and Diffusion Limited Aggregates (DLA). By constructing conformal maps from the unit circle to the fractal patterns we obtained dynamical equations for the conformal maps, allowing us to solve for the interesting patterns that evolve in these systems. We understood completely Laplacian dynamics and developed a theory of DLA which is able to predict all the multifractal properties from first principles. In particular we have offered a convergent calculation of the fractal dimension of DLA, settling a long standing question whether it is fractal or not (it is).

In the last year we have developed the dynamics of conformal maps to problems of fracture of brittle materials. We have solved the quasi-static fracture problem and are making progress in dynamic fracture. We have offered novel methods to analyze the roughening of fractured interfaces, and begun to explore the physics of plasticity and its implications on fracture.

Hydrodynamics, Turbulence, and Pattern Formation-Experimental

During the past years the main progress has been made on the following projects:

Elastic turbulence and Batchelor regime of mixing in dilute polymer solutions.The central subject in the laboratory during this period was the investigation of various aspects of hydrodynamics of polymer solutions. Significant progress has been made in our understanding of the role of elastic stresses in hydrodynamics of polymer solutions. Statistics of global (torque and injected power) and local (velocity and velocity gradient fields) characteristics of the elastic turbulence in a flow of a polymer solution between two disks was experimentally investigated. Analogy with a small scale fast dynamo in magneto-hydrodynamics and with a passive scalar turbulent advection in the Batchelor regime was used to explain the experimentally observed statistical properties, flow structure, and scaling of the elastic turbulence. Dependence of properties of elastic turbulence on polymer concentration was studied in detail. Next step in this project was to study turbulent mixing of very viscous fluids by adding polymers. We studied mixing in curved channels of macroscopic size (3mm side size in cross-section) and in a micro channel of 100 micron side size cross-section. It turned out that these studies have besides obvious importance for application also rather important implications in our basic understanding of chaotic mixing. Indeed, it was demonstrated that mixing due to elastic turbulence is an ideal system to study the Batchelor regime of mixing. The latter is a rare example of the model of dynamics of a passive scalar in a turbulent flow, for which the analytical solution was obtained. Mixing due to elastic turbulence regime provided quantitative verification of theoretical predictions and further initiated theoretical activity to quantitatively understand the results.

Single polymer dynamics and conformations in a random flow.Single polymer dynamics and statistics of conformation were studied in shear and random flows. Coil-stretch transition in polymer conformation in a random flow was identified and characterized. Dependence of the coil-stretch transition on polymer concentration and molecular weight are studied, and also the degree of polymer stretching in a random flow on its closeness to a wall. New fluorescent labeling technique with quantum dots is developing in the lab in order to conduct experiments on a single polymer in various flows to measure end-to-end vector that is used in a theoretical model. Further experiments on synthetic polymer molecules are in progress.Hydrodynamics of complex fluids in micro-channels.Dynamics of vesicles and rheology of vesicle solutions are realistic models for blood flow. Different regimes in dynamics of a single vesicle in shear flow that appears between two disks and in micro-channels were quantitatively studied. It was demonstrated that in tank-treading regime a vesicle dynamics in a wide range of vesicle deformations is described quantitatively well by the recently developed theoretical model. On the other hand, a transition from tank-treading to tumbling regime occurred rather differently from what expected. First, a new regime of vesicle trembling at lower shear rates was identified when both vesicle inclination angle and shape deformation were oscillated. Second, these shape deformations persisted also in the tumbling regime. Recently, hydrodynamics of concentrated solution of vesicles was studied by measurements of its global (pressure drop as a function of discharge in a micro-channel flow) and local (dynamics of a single vesicle) properties. Strong fluctuations in vesicle inclination angle due to vesicle interaction via flow (up to two orders of magnitude larger than thermal noise) were observed and studied specifically in a case of two vesicles interaction. Investigation of dynamics in other flows such as elongation and random flows is currently on the way. Finally, we are going to identify rheology of vesicle solution to compare it with empirical constitutive equations used for blood rheology. Similar research we are planning to perform for solution of worm-like micelles and any other fluids that show visco-elastic properties.Further development of new acoustic detection technique of vorticity distribution in turbulent flows and its application for turbulent drag reduction.During the last several years we developed a new sound scattering technique for measurements of velocity and vorticity fields in a turbulent flow. We use this technique together with Laser Doppler and particle image velocimetry methods, hot-wire anemometry, precise measurement of torque, and pressure fluctuations to study turbulent drag reduction. The latter problem we study in von Karman swirling flow between two counter-rotating disks of water or water-sugar solutions with different concentration of PAAm 18M molecular weight.Convective turbulence in SF6 near its gas-liquid critical point.Turbulent convection was studied in a gas SF6 near the gas-liquid critical point. This unique system provides us an opportunity to reach extremely large Raleigh numbers (up to 10^{15}) and to study the Pr dependence over an extremely wide range (up to 500) in the same system. The existence of the critical fluctuations provided us the possibility to perform laser Doppler velocimetry (LDV) measurements of the velocity field in a rather wide range of the closeness to the critical point. Using this novel technique developed in our laboratory, we studied statistical properties of the velocity field in a wide range of Re and Pr numbers. Together with the local temperature and global heat transfer and temperature and velocity profile measurements it provided us complete information about convective turbulence. A surprising outcome of theses studies was a very weak influence of strong non-Boussinesq effect on global and local scaling properties of convective turbulence.Quantum Chaos - Theory

The main problem in "quantum chaos" is to reveal the quantum mechanical implications ofclassical chaos. Chaotic dynamics - a generic property in classical physics, leaveuniversalfingerprints in quantum physics, which are unraveled by the on going research in "quantum Chaos". The results are relevant and applicable in Mesoscopics, Atomic, Molecular and Nuclear physics. As a matter of fact, "quantum chaos" appears in all problems where wave propagation is studied in the short wavelength limit. Thus, "quantum chaos" is also studied in acoustics, electromagnetic propagation, cavities etc. The observation which brought "quantum chaos" to the focal point of modern theoretical physics, was the intimate connection between the distributions and statistics of many quantum observables, and the underlying classical dynamics. More precisely, it was found that the predictions of random matrix theory, a minimum-information statistical approach, accurately reproduce the properties of simple quantum systems, as long as the underlying classical dynamics is chaotic. In this way, the ergodicity which is the hallmark of classical chaos is extended into the quantum domain. In our recent research we contributed to this effort along the following lines:

Quantum graphs. In the quest for the simplest quantum systems which display spectral fluctuations which are reproduce by random matrix theory, we proposed quantum graphs, for which an exact trace formula exists, and the "classical dynamics" was shown to be mixing. An extensive test of the spectra of simple graphs have shown an excellent reproduction of various statistical measures derived from random matrix theory. Moreover, the derivation of spectral correlation functions can be reduced to the solution of combinatorial problems. With this insight, the applicability of random matrix theory for graphs was theoretically established to better degree than hitherto achieved in any other system. Various other problems of interest, such as isospectrality (which relates to the question- "Can one hear the shape of a graph?"), quantum irreversibility (dephasing) and nodal structures of wave functions on graphs are also studied. Recently we introduced a method to construct graphs which are isospectral but are not congtuent.

The statistics and structure of nodal domains.Real wave functions (in 2d for simplicity) vanish along lines which separate domains where the wave function has a constant sign. The properties of the sets of nodal lines and nodal domains are sensitive to the underlying classical dynamics. We revived the interest in this aspect of "quantum chaos" by introducing a new statistical measure for the distribution of the number of nodal domains. We derived the universal features of this distribution for quantum integrable problems, and conjectured its behavior for chaotic ones.

Can one count the shape of a drum?We study the sequences obtained by counting the number of nodal domains of wave functions ordered by increasing energies. We have shown that these sequences of integers store geometric information on the shape of the boundary of the "drum". Moreover, we conjectured and verified the conjecture numerically that these sequences resolve isospectral ambiguities. Recently we proved the validity of the conjecture for a certain class of isospectral graphs.

Fracture, Glass Physics and Plasticity - Theory

Fracture. Our research in the last couple of years concentrated on understanding the dynamics and instabilities of rapid crack propagation. The main focus was on understanding the roles played by elastic nonlinearities localized near a crack's tip in accounting for both the apparent universality of fracture dynamics and for the emerging lengthscales. One major achievement is the development of the so-called "Weakly Nonlinear Theory of Dynamic Fracture", which quantitatively accounts for pioneering experimental results and resolves various puzzles in fracture mechanics. This theoretical framework naturally gives rise to a material lengthscale that may be intimately related to several poorly understood high-speed crack instabilities. In this context, we have proposed a crack tip equation of motion that incorporates the new lengthscale and accurately predicts the onset and properties of an experimentally observed oscillatory instability. We currently further develop this theory and apply it to a broader range of fracture phenomena.

Glass Physics and Plasticity. We developed a non-equilibrium thermodynamic framework for driven disordered materials. A basic idea in our approach is that the degrees of freedom of a disordered material can be separated into two subsystems -- the slow configurational degrees of freedom, i.e. the inherent structures, and the fast kinetic-vibrational degrees of freedom which. While the latter are generally at equilibrium with the thermal reservoir, the former are characterized by an effective temperature that may depart from the reservoir temperature in externally driven situations. Within this general framework, we addressed several fundamental problems in glass physics and plasticity. We were able to explain a basic glassy memory effect – the Kovacs effect – in a continuum thermodynamic approach. We reformulated the Shear-Transformation-Zone (STZ) theory of amorphous plasticity in a self-consistent, non-equilibrium thermodynamic manner and applied it to various phenomena involving irreversible deformation.

- We currently study the linear response of glassy materials (both hard – structural, polymeric and metallic glasses – and soft – colloidal suspensions and emulsions) with the aim of understanding in a unified manner the linear viscoelastic behavior of these systems. We also started to study the irreversible, dislocation-mediated deformation of polycrystalline materials, using non-equilibrium thermodynamic ideas and concepts developed originally in the context of amorphous materials.