Integrated Calendar

Brownian motion is a Gaussian process described by the central limit theorem. However, exponential decays of the positional probability density function $P(X,t)$ of packets of spreading random walkers, were observed in numerous situations that include glasses, live cells and bacteria suspensions. We show that such exponential behavior is generally valid in a large class of problems of transport in random media. By extending the Large Deviations approach for a continuous time random walk we uncover a general universal behavior for the decay of the density. It is found that fluctuations in the number of steps of the random walker, performed at finite time, lead to exponential decay (with logarithmic corrections) of $P(X,t)$. This universal behavior holds also for short times, a fact that makes experimental observations readily achievable.

Heat transfer in solids is usually described in terms of Fourier's law according to which
the thermal conductance of a material scales inversely with its length or, equivalently, thermal
conductivity is independent of sample length. Theoretical and experimental studies over the
past decade have demonstrated that Fourier's law is violated for a variety of one-dimensional
systems. Despite the large number of studies of many intriguing models, the validity criteria
for Fourier's law remain elusive, and a breakdown of Fouriers law seems to be commonplace.
In this talk, I will discus heat conduction mediated by longitudinal phonons in one dimensional disordered harmonic chains to understand the role of different parameters that may affect
the scaling of thermal conductance in these systems. Using scaling properties of the phonon
density of states and localization in disordered systems, we find non-trivial scaling of the thermal conductance with the system size. Our theoretical findings are corroborated by extensive
numerical analysis. We show that, suprisingly, the thermal conductance of a system with strong
disorder, characterized by a `heavy-tailed' probability distribution, and with large impedance
mismatch between the bath and the system scales normally with the system size, i.e., in a
manner consistent with Fourier's law. We identify a dimensionless scaling parameter, related
to the temperature scale and the localization length of the phonons, through which the thermal conductance for different models of disorder and different temperatures follows a universal
behavior.

Planarity plays a crucial role in determining the electronic and optical properties of π-conjugated backbones. Here I will discuss two examples of non-planar and planar systems: twisted acenes and planar furan-based macrocycles.
In the first part, I will demonstrate how twisting affects the electronic, optical and chiroptical properties of acenes. We have introduced a series of twisted acenes, having an anthracene backbone diagonally tethered by an n-alkyl bridge, which induces different degrees of twisting. This helically-locked system allows us to systematically monitor the effect of twisting on electronic and optical properties of anthracene. The effect of twisting on chiroptical properties, charge delocalization and π-conjugation will also be demonstrated.
In the second part, I will present bifuranimide as a stable furan containing analog, which resulted in the introduction of the first macrocyclic furans. These π-conjugated macrocycles were found to be completely planar, in contrast with thiophene macrocycles which are highly-twisted. The prospects of macrocyclic furans as π-conjugated analogs of crown-ethers and synthons for cycloarenes by multiple Diels–Alder cycloadditions will be discussed.

Many people are aware of Feigenbaum's astonishing discovery of the universality of period doubling, and the constant delta=4.66920 which carries his name.

In the last 13 years of his life Feigenbaum worked on other subjects, and he wrote the manuscript (in TeX) of a book whose title is "Reflections on a Tube".

This is closely related to his life-long interest in optics and aspects of vision. It deals with the optics of images reflected in a cylindrical mirror (usually called anamorphic pictures). He shows that the eye does not interpret ray-tracing, but caustics. But there are two caustics, and therefore, the viewer can actually see two different images. The visual system will often prefer one over the other. The question is the "which" and "why"? Starting from this discovery, Feigenbaum derived other aspects of this observation, dealing with the vision of fish, the "broken" pencil in water, or aspects of the floor of swimming pools. All these examples show two possible images. His study tells me how a simple study in classical optics can lead to interesting questions in perception and the visual system.

I will give an overview of this project. As I discussed with him, over those 13 years, many aspects of his work, I have edited his manuscript so it can be published as a book which should appear in a forseeable future.