349 research outputs found
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 -dimensional Euclidean lattice, where each lattice
point is assigned a mark via a given random field on . 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
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