Fractional diffusion limit for collisional kinetic equations

TL;DR

This paper investigates the diffusion limits of linear Boltzmann equations with heavy-tailed equilibrium distributions, demonstrating that the small mean free path limit leads to fractional diffusion equations instead of classical ones.

Contribution

It extends the understanding of kinetic equations by deriving fractional diffusion limits for heavy-tailed equilibrium distributions, a case not covered in classical diffusion limit results.

Findings

Heavy-tailed distributions lead to fractional diffusion limits

The derived fractional diffusion equation differs from classical diffusion equations

The results apply to kinetic models with infinite variance equilibria

Abstract

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation.