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

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.