Diffusion equations from kinetic models with non-conserved momentum

TL;DR

This paper derives coupled diffusion equations for particle and energy densities from kinetic models that include both conserved and non-conserved momentum dynamics, using Hilbert expansion and applying it to systems like hard disks.

Contribution

It introduces a method to derive macroscopic diffusion equations from kinetic models with broken momentum conservation, including systems like hard disks at intermediate densities.

Findings

Coupled diffusion equations with Onsager coefficients derived.

Inclusion of non-conserved momentum effects in macroscopic equations.

Application to Enskog equation for hard disks.

Abstract

We derive diffusive macroscopic equations for the particle and energy density of a system whose time evolution is described by a kinetic equation for the one particle position and velocity function f(r,v,t) that consists of a part that conserves energy and momentum such as the Boltzmann equation and an external randomization of the particle velocity directions that breaks the momentum conservation. Rescaling space and time by epsilon and epsilon square respectively and carrying out a Hilbert expansion in epsilon around a local equilibrium Maxwellian yields coupled diffusion equations with specified Onsager coefficients for the particle and energy density. Our analysis includes a system of hard disks at intermediate densities by using the Enskog equation for the collision kernel.