Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models

TL;DR

This paper analyzes the spectral properties and convergence rates of inelastic Boltzmann models, deriving explicit estimates for diffusivity and demonstrating convergence to heat equations in the diffusive limit.

Contribution

It provides a complete spectral analysis for Maxwell molecules and establishes explicit entropy-based convergence rates for hard spheres, including diffusivity estimates in Fick's law.

Findings

Explicit exponential convergence rates to equilibrium.

Derivation of the diffusive limit and heat equation convergence.

First explicit estimates of diffusivity in Fick's law for inelastic gases.

Abstract

We consider the linear dissipative Boltzmann equation describing inelastic interactions of particles with a fixed background. For the simplified model of Maxwell molecules first, we give a complete spectral analysis, and deduce from it the optimal rate of exponential convergence to equilibrium. Moreover we show the convergence to the heat equation in the diffusive limit and compute explicitely the diffusivity. Then for the physical model of hard spheres we use a suitable entropy functional for which we prove explicit inequality between the relative entropy and the production of entropy to get exponential convergence to equilibrium with explicit rate. The proof is based on inequalities between the entropy production functional for hard spheres and Maxwell molecules. Mathematical proof of the convergence to some heat equation in the diffusive limit is also given. From the last two points we deduce the first explicit estimates on the diffusive coefficient in the Fick's law for (inelastic hard-spheres) dissipative gases.