Complete characterization of convergence to equilibrium for an inelastic Kac model

TL;DR

This paper characterizes the convergence to equilibrium for an inelastic Kac model, showing it occurs if and only if initial distributions are in the domain of attraction of a symmetric stable law with an index related to inelasticity.

Contribution

It provides a complete characterization of convergence to equilibrium for the inelastic Kac model, linking it to stable laws and determining stationary solutions.

Findings

Convergence to equilibrium occurs iff initial distribution is in the domain of attraction of a symmetric stable law.

Stationary solutions are exactly the symmetric stable laws with a specific index.

The solution to a key stochastic functional equation is characterized without additional assumptions.

Abstract

Pulvirenti and Toscani introduced an equation which extends the Kac caricature of a Maxwellian gas to inelastic particles. We show that the probability distribution, solution of the relative Cauchy problem, converges weakly to a probability distribution if and only if the symmetrized initial distribution belongs to the standard domain of attraction of a symmetric stable law, whose index $\alpha$ is determined by the so-called degree of inelasticity, $p>0$, of the particles: $\alpha=\frac{2}{1+p}$. This result is then used: (1) To state that the class of all stationary solutions coincides with that of all symmetric stable laws with index $\alpha$. (2) To determine the solution of a well-known stochastic functional equation in the absence of extra-conditions usually adopted.