Entropy for relaxation dynamics in granular media

TL;DR

This paper explores how entropic concepts influence relaxation dynamics in granular media, highlighting hierarchical relaxation, phase space modifications, and a mean-field model with a free-energy-like functional.

Contribution

It introduces a mean-field model incorporating a free-energy-like functional to describe relaxation and segregation in granular systems, including a diffusive limit analysis.

Findings

Hierarchical relaxation phenomena with multiple time scales

Existence of a free-energy-like functional guiding asymptotic behavior

Derivation of a diffusive limit model for granular relaxation

Abstract

We investigate the role of entropic concepts for the relaxation dynamics in granular systems. In these systems the existence of a geometrical frustration induces a drastic modification of the allowed phase space, which in its turn induces a dynamic behavior characterized by hierarchical relaxation phenomena with several time scales associated. In particular we show how, in the framework of a mean-field model introduced for the compaction phenomenon, there exists a free-energy-like functional which decreases along the trajectories of the dynamics and which allows to account for the asymptotic behavior: e.g. density profile, segregation phenomena. Also we are able to perform the continuous limit of the above mentioned model which turns out to be a diffusive limit. In this framework one can single out two separate physical ingredients: the free-energy-like functional that defines the phase-space and the asymptotic states and a diffusion coefficient $D(\rho)$ accounting for the velocity of approach to the asymptotic stationary states.