Anomalous velocity distributions in inelastic Maxwell gases

TL;DR

This review investigates how inelastic collisions affect the velocity distribution in Maxwell gases, revealing overpopulated high-energy tails with algebraic or stretched Gaussian forms in different cooling regimes.

Contribution

It demonstrates the existence of scaling solutions for velocity distributions in inelastic Maxwell gases and characterizes their high-energy tail behaviors.

Findings

High-energy tails are overpopulated compared to Maxwellian distributions.

Algebraic tails in freely cooling systems depend on inelasticity.

Stretched Gaussian tails appear in forced systems.

Abstract

This review is a kinetic theory study investigating the effects of inelasticity on the structure of the non-equilibrium states, in particular on the behavior of the velocity distribution in the high energy tails. Starting point is the nonlinear Boltzmann equation for spatially homogeneous systems, which supposedly describes the behavior of the velocity distribution function in dissipative systems as long as the system remains in the homogeneous cooling state, i.e. on relatively short time scales before the clustering and similar instabilities start to create spatial inhomogeneities. This is done for the two most common models for dissipative systems, i.e. inelastic hard spheres and inelastic Maxwell particles. In systems of Maxwell particles the collision frequency is independent of the relative velocity of the colliding particles, and in hard sphere systems it is linear. We then demonstrate the existence of scaling solutions for the velocity distribution function, $F(v,t) \sim v_0(t)^{-d} f((v/v_0(t))$, where $v_0$ is the r.m.s. velocity. The scaling form $f(c)$ shows overpopulation in the high energy tails. In the case of freely cooling systems the tails are of algebraic form, $ f(c)\sim c^{-d-a}$, where the exponent $a$ may or may not depend on the degree of inelasticity, and in the case of forced systems the tails are of stretched Gaussian type $f(v)\sim\exp[-\beta (v/v_0)^b]$ with $b <2$.