The grazing collision limit of the inelastic Kac model around a L\'evy-type equilibrium

TL;DR

This paper investigates the grazing collision limit of the inelastic Kac model with heavy-tailed Lévy-type equilibrium distributions, showing convergence to a fractional Fokker-Planck equation.

Contribution

It establishes the convergence of solutions to a fractional Fokker-Planck equation in the heavy-tailed equilibrium setting of the inelastic Kac model.

Findings

Solutions converge to a fractional Fokker-Planck equation.

The equilibrium distribution is a heavy-tailed Lévy-type distribution.

The analysis extends the grazing collision limit to infinite variance cases.

Abstract

This paper is devoted to the grazing collision limit of the inelastic Kac model introduced in [A. Pulvirenti and G. Toscani. J. Statist. Phys., 114(5-6):1453--1480, 2004], when the equilibrium distribution function is a heavy-tailed L\'evy-type distribution with infinite variance. We prove that solutions in an appropriate domain of attraction of the equilibrium distribution converge to solutions of a Fokker-Planck equation with a fractional diffusion operator.