# Fractional diffusion limit for collisional kinetic equations: A moments method

## TL;DR

This paper investigates the hydrodynamic limits of linear kinetic equations with heavy-tail equilibrium distributions, demonstrating that their long-time behavior is governed by fractional diffusion equations using a moments-based method.

## Contribution

It introduces a moments method to derive fractional diffusion limits for kinetic equations with heavy-tail equilibria, extending classical hydrodynamic limit results.

## Key findings

- Long-time behavior described by fractional diffusion equations
- Heavy-tail distributions lead to anomalous diffusion
- Method applicable to linear kinetic equations with non-Maxwellian equilibria

## Abstract

This paper is devoted to hydrodynamic limits of linear kinetic equations when the thermodynamical equilibrium is described by a heavy-tail distribution function rather than a Maxwellian distribution. We show that the long time/small mean free path behavior of the solution of the kinetic equation is described by a fractional diffusion equation. The method that we introduce is somewhat reminiscent of the so-called "moments method" which plays an important role in kinetic theory.