Collisional rates for the inelastic Maxwell model: application to the divergence of anisotropic high-order velocity moments in the homogeneous cooling state

TL;DR

This paper derives exact collisional rates for velocity moments in the inelastic Maxwell model and shows that high-order anisotropic moments diverge over time in the homogeneous cooling state, despite the distribution becoming isotropic.

Contribution

It provides exact formulas for collisional rates of isotropic and anisotropic moments and analyzes their divergence in the homogeneous cooling state.

Findings

High-order moments diverge at certain inelasticities.

Anisotropic moments do not decay but their ratios to isotropic moments tend to zero.

The distribution converges to an isotropic self-similar form despite anisotropic divergence.

Abstract

The collisional rates associated with the isotropic velocity moments $<V^{2r}>$ and the anisotropic moments $<V^{2r}V_i>$ and $<V^{2r}(V_iV_j-d^{-1}V^2\delta_{ij})>$ are exactly derived in the case of the inelastic Maxwell model as functions of the exponent $r$, the coefficient of restitution $\alpha$, and the dimensionality $d$. The results are applied to the evolution of the moments in the homogeneous free cooling state. It is found that, at a given value of $\alpha$, not only the isotropic moments of a degree higher than a certain value diverge but also the anisotropic moments do. This implies that, while the scaled distribution function has been proven in the literature to converge to the isotropic self-similar solution in well-defined mathematical terms, nonzero initial anisotropic moments do not decay with time. On the other hand, our results show that the ratio between an anisotropic moment and the isotropic moment of the same degree tends to zero.