Fractional diffusion as the limit of a short range potential Rayleigh gas

TL;DR

This paper rigorously derives a fractional diffusion equation as a hydrodynamic limit of a Rayleigh gas with short-range interactions, using probabilistic and semigroup methods to connect microscopic dynamics to macroscopic fractional diffusion.

Contribution

It establishes the fractional diffusion as a limit of a Rayleigh gas with short-range potentials, extending the understanding of hydrodynamic limits in kinetic theory.

Findings

Fractional diffusion equation derived as a limit of Rayleigh gas.

Intermediate linear Boltzmann equation obtained in the Boltzmann-Grad limit.

Convergence of particle dynamics to Boltzmann-type dynamics proven.

Abstract

The fractional diffusion equation is rigorously derived as a scaling limit from a deterministic Rayleigh gas, where particles interact via short range potentials with support of size $\varepsilon$ and the background is distributed in space $\mathbb{R}^3$ according to a Poisson process with intensity $N$ and in velocity according to some fat-tailed distribution. As an intermediate step a linear Boltzmann equation is obtained in the Boltzmann-Grad limit as $\varepsilon$ tends to zero and $N$ tends to infinity with $N \varepsilon^2 =c$. The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. The fractional diffusion equation is a hydrodynamic limit for times $t \in [0,T]$, where $T$ and inverse mean free path $c$ can both be chosen as some negative rational power $\varepsilon^{-k}$.