A simple model kinetic equation for inelastic Maxwell particles

TL;DR

This paper introduces a simplified kinetic model for inelastic Maxwell particles that retains key properties of the Boltzmann equation, accurately predicts transport coefficients, and can be exactly solved for the homogeneous cooling state.

Contribution

A new kinetic model that simplifies the Boltzmann equation for inelastic Maxwell particles while preserving essential physical properties and enabling exact solutions.

Findings

Reproduces true shear viscosity.

Predicts accurate heat flux transport coefficients.

Exhibits algebraic high-energy tail in homogeneous cooling state.

Abstract

The model of inelastic Maxwell particles (IMP) allows one to derive some exact results which show the strong influence of inelasticity on the nonequilibrium properties of a granular gas. The aim of this work is to propose a simple model kinetic equation that preserves the most relevant properties of the Boltzmann equation (BE) for IMP and reduces to the BGK kinetic model in the elastic limit. In the proposed kinetic model the collision operator is replaced by a relaxation-time term toward a reference Maxwellian distribution plus a term representing the action of a friction force. It contains three parameters (the relaxation rate, the effective temperature of the reference Maxwellian, and the friction coefficient) which are determined by imposing consistency with basic exact properties of the BE for IMP. As a consequence, the kinetic model reproduces the true shear viscosity and predicts accurate expressions for the transport coefficients associated with the heat flux. The model can be exactly solved for the homogeneous cooling state, the solution exhibiting an algebraic high-energy tail with an exponent in fair agreement with the correct one.