Third and fourth degree collisional moments for inelastic Maxwell models

TL;DR

This paper provides exact evaluations of third and fourth degree collisional moments for inelastic Maxwell models in multiple dimensions, revealing how anisotropy affects relaxation times and moment behaviors in cooling states.

Contribution

It introduces explicit formulas for collisional moments and eigenvalues in inelastic Maxwell models, extending understanding to higher dimensions and anisotropic effects.

Findings

Relaxation time decreases with increasing anisotropy.

All anisotropic moments up to degree four vanish in the homogeneous cooling state for dimensions two and higher.

Explicit expressions for eigenvalues and cross coefficients are provided.

Abstract

The third and fourth degree collisional moments for $d$-dimensional inelastic Maxwell models are exactly evaluated in terms of the velocity moments, with explicit expressions for the associated eigenvalues and cross coefficients as functions of the coefficient of normal restitution. The results are applied to the analysis of the time evolution of the moments (scaled with the thermal speed) in the free cooling problem. It is observed that the characteristic relaxation time toward the homogeneous cooling state decreases as the anisotropy of the corresponding moment increases. In particular, in contrast to what happens in the one-dimensional case, all the anisotropic moments of degree equal to or less than four vanish in the homogeneous cooling state for $d\geq 2$.