On the propagation of a perturbation in an anharmonic system

We give a not trivial upper bound on the velocity of disturbances in an infinitely extended anharmonic system at thermal equilibrium. The proof is achieved by combining a control on the non equilibrium dynamics with an explicit use of the state invariance with respect to the time evolution.Comment: 14 page

2-D constrained Navier-Stokes equation and intermediate asymptotics

We introduce a modified version of the two-dimensional Navier-Stokes equation, preserving energy and momentum of inertia, which is motivated by the occurrence of different dissipation time scales and related to the gradient flow structure of the 2-D Navier-Stokes equation. The hope is to understand intermediate asymptotics. The analysis we present here is purely formal. A rigorous study of this equation will be done in a forthcoming paper

Recent Results on the Periodic Lorentz Gas

The Drude-Lorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption -- known as the Boltzmann-Grad scaling by analogy with the kinetic theory of rarefied gases -- this system can be described in some limit by a linear Boltzmann equation, assuming that the configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) vol. 185 (1969), 308]). The case of a periodic configuration of obstacles (like atoms in a crystal) leads to a completely different limiting dynamics. These lecture notes review several results on this problem obtained in the past decade as joint work with J. Bourgain, E. Caglioti and B. Wennberg.Comment: 62 pages. Course at the conference "Topics in PDEs and applications 2008" held in Granada, April 7-11 2008; figure 13 and a misprint in Theorem 4.6 corrected in the new versio

Response properties in a model for granular matter

We investigate the response properties of granular media in the framework of the so-called {\em Random Tetris Model}. We monitor, for different driving procedures, several quantities: the evolution of the density and of the density profiles, the ageing properties through the two-times correlation functions and the two-times mean-square distance between the potential energies, the response function defined in terms of the difference in the potential energies of two replica driven in two slightly different ways. We focus in particular on the role played by the spatial inhomogeneities (structures) spontaneously emerging during the compaction process, the history of the sample and the driving procedure. It turns out that none of these ingredients can be neglected for the correct interpretation of the experimental or numerical data. We discuss the problem of the optimization of the compaction process and we comment on the validity of our results for the description of granular materials in a thermodynamic framework.Comment: 22 pages, 35 eps files (21 figures

Relative entropy via non-sequential recursive pair substitutions

The entropy of an ergodic source is the limit of properly rescaled 1-block entropies of sources obtained applying successive non-sequential recursive pairs substitutions (see P. Grassberger 2002 ArXiv:physics/0207023 and D. Benedetto, E. Caglioti and D. Gabrielli 2006 Jour. Stat. Mech. Theo. Exp. 09 doi:10.1088/1742.-5468/2006/09/P09011). In this paper we prove that the cross entropy and the Kullback-Leibler divergence can be obtained in a similar way.Comment: 13 pages , 2 figure

Coarsening and Slow-Dynamics in Granular Compaction

We address the problem of the microscopic reorganization of a granular medium under a compaction process in the framework of Tetris-like models. We point out the existence of regions of spatial organization which we call domains, and study their time evolution. It turns out that after an initial transient, most of the activity of the system is concentrated on the boundaries between domains. One can then describe the compaction phenomenon as a coarsening process for the domains, and a progressive reduction of domain boundaries. We discuss the link between the coarsening process and the slow dynamics in the framework of a model of active walkers on active substrates.Comment: Revtex 4 pages, 4 figures, in press in PRL. More info http://axtnt3.phys.uniroma1.it/Tetri

Spherical averages in the space of marked lattices

A marked lattice is a $d$-dimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on ${\mathbb Z}^d$. We prove that, if the field is strongly mixing with a faster-than-logarithmic rate, then for every given lattice and almost every marking, large spheres become equidistributed in the space of marked lattices. A key aspect of our study is that the space of marked lattices is not a homogeneous space, but rather a non-trivial fiber bundle over such a space. As an application, we prove that the free path length in a crystal with random defects has a limiting distribution in the Boltzmann-Grad limit

Free-volume kinetic models of granular matter

We show that the main dynamical features of granular media can be understood by means of simple models of fragile-glass forming liquid provided that gravity alone is taken into account. In such lattice-gas models of cohesionless and frictionless particles, the compaction and segregation phenomena appear as purely non-equilibrium effects unrelated to the Boltzmann-Gibbs measure which in this case is trivial. They provide a natural framework in which slow relaxation phenomena in granular and glassy systems can be explained in terms of a common microscopic mechanism given by a free-volume kinetic constraint.Comment: 4 pages, 6 figure

Memory in aged granular media

Stimulated by recent experimental results, we simulate ``temperature''-cycling experiments in a model for the compaction of granular media. We report on the existence of two types of memory effects: short-term dependence on the history of the sample, and long-term memory for highly compact (aged) systems. A natural interpretation of these results is provided by the analysis of the density heterogeneities.Comment: 5 eps figures, uses euromacr.tex and europhys.sty (included

Algebraic Correlation Function and Anomalous Diffusion in the HMF model

In the quasi-stationary states of the Hamiltonian Mean-Field model, we numerically compute correlation functions of momenta and diffusion of angles with homogeneous initial conditions. This is an example, in a N-body Hamiltonian system, of anomalous transport properties characterized by non exponential relaxations and long-range temporal correlations. Kinetic theory predicts a striking transition between weak anomalous diffusion and strong anomalous diffusion. The numerical results are in excellent agreement with the quantitative predictions of the anomalous transport exponents. Noteworthy, also at statistical equilibrium, the system exhibits long-range temporal correlations: the correlation function is inversely proportional to time with a logarithmic correction instead of the usually expected exponential decay, leading to weak anomalous transport properties