Linear Boltzmann Equation and Fractional Diffusion

TL;DR

This paper demonstrates that under certain asymptotic conditions, the radiation pressure in a linear Boltzmann model converges to a fractional diffusion equation, illustrating a novel kinetic-to-fractional diffusion limit based on harmonic extension.

Contribution

It establishes a new fractional diffusion asymptotic limit for a kinetic model using harmonic extension, differing from previous limits based on heavy tails or scattering properties.

Findings

Radiation pressure governed by fractional diffusion in the asymptotic regime

Fractional diffusion limit derived from harmonic extension of bla

Distinct from previous kinetic limits based on equilibrium tails

Abstract

Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient $\sigma$. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient $\alpha$. Moreover, assume that there is a temperature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where $\sigma\to+\infty$ and $1-\alpha\sim C/\sigma$, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a fractional diffusion equation. This result provides an example of fractional diffusion asymptotic limit of a kinetic model which is based on the harmonic extension definition of $\sqrt{-\Delta}$. This fractional diffusion limit therefore differs from most of other such limits for kinetic models reported in the literature, which are based on specific properties of the equilibrium distributions (heavy tails) or of the scattering coefficient as in [U. Frisch-H. Frisch: Mon. Not. R. Astr. Not. 181 (1977), 273-280].