Transport coefficients of d-dimensional inelastic Maxwell models

TL;DR

This paper derives the transport coefficients of inelastic Maxwell models in arbitrary dimensions using the Chapman-Enskog method, comparing them with inelastic hard spheres and revealing limitations of the Maxwell model.

Contribution

It provides explicit expressions for transport coefficients of inelastic Maxwell models in multiple dimensions, including driven and unforced systems, and compares them with hard sphere results.

Findings

Transport coefficients depend on inelasticity in a qualitatively similar way to hard spheres.

Inelastic Maxwell models capture the qualitative trend but not the quantitative details of hard sphere behavior.

A simpler BGK-like model aligns more closely with hard sphere results than the Maxwell model.

Abstract

Due to the mathematical complexity of the Boltzmann equation for inelastic hard spheres, a kinetic model has recently been proposed whereby the collision rate (which is proportional to the relative velocity for hard spheres) is replaced by an average velocity-independent value. The resulting inelastic Maxwell model has received a large amount of recent interest, especially in connection with the high energy tail of homogeneous states. In this paper the transport coefficients of inelastic Maxwell models in d dimensions are derived by means of the Chapman-Enskog method for unforced systems as well as for systems driven by a Gaussian thermostat and by a white noise thermostat. Comparison with known transport coefficients of inelastic hard spheres shows that their dependence on inelasticity is captured by the inelastic Maxwell models only in a mild qualitative way. Paradoxically, a much simpler BGK-like model kinetic equation is closer to the results for inelastic hard spheres than the inelastic Maxwell model