Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model

TL;DR

This paper analyzes the convergence rate to equilibrium for an inelastic Kac model of granular materials, providing bounds on distances between solutions and the stable distribution limit, with conditions affecting the bounds' sharpness.

Contribution

It offers new bounds on convergence rates for an inelastic Kac model, including exponential bounds under specific initial data assumptions.

Findings

Solutions converge to a symmetric stable distribution.

Bounds depend on initial data properties.

Necessary conditions for relaxation are established.

Abstract

This paper deals with a one--dimensional model for granular materials, which boils down to an inelastic version of the Kac kinetic equation, with inelasticity parameter $p>0$. In particular, the paper provides bounds for certain distances -- such as specific weighted $\chi$--distances and the Kolmogorov distance -- between the solution of that equation and the limit. It is assumed that the even part of the initial datum (which determines the asymptotic properties of the solution) belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent $\a=2/(1+p)$. With such initial data, it turns out that the limit exists and is just the aforementioned stable distribution. A necessary condition for the relaxation to equilibrium is also proved. Some bounds are obtained without introducing any extra--condition. Sharper bounds, of an exponential type, are exhibited in the presence of additional assumptions concerning either the behaviour, near to the origin, of the initial characteristic function, or the behaviour, at infinity, of the initial probability distribution function.