High-Energy Tail of the Velocity Distribution of Driven Inelastic Maxwell Gases

TL;DR

This paper models a driven inelastic Maxwell gas system, deriving exact velocity distribution behaviors, including Gaussian and exponential tails, depending on system parameters, and explores conditions for steady states and energy growth.

Contribution

It provides an exact analytical framework for the velocity distribution and energy dynamics in a driven inelastic Maxwell gas, including tail behaviors and steady state conditions.

Findings

Velocity distribution has Gaussian tails for |r_w|<1.

Exponential tail observed at r_w=+1.

System reaches steady state for -1<r_w≤1, with exact energy and correlation calculations.

Abstract

A model of homogeneously driven dissipative system, consisting of a collection of $N$ particles that are characterized by only their velocities, is considered. Adopting a discrete time dynamics, at each time step, a pair of velocities is randomly selected. They undergo inelastic collision with probability $p$. With probability $(1-p)$, energy of the system is changed by changing the velocities of both the particles independently according to $v\rightarrow -r_w v +\eta$, where $\eta$ is a Gaussian noise drawn independently for each particle as well as at each time steps. For the case $r_w=- 1$, although the energy of the system seems to saturate (indicating a steady state) after time steps of $O(N)$, it grows linearly with time after time steps of $O(N^2)$, indicating the absence of a eventual steady state. For $ -1 <r_w \leq 1$, the system reaches a steady state, where the average energy per particle and the correlation of velocities are obtained exactly. In the thermodynamic limit of large $N$, an exact equation is obtained for the moment generating function. In the limit of nearly elastic collisions and weak energy injection, the velocity distribution is shown to be a Gaussian. Otherwise, for $|r_w| < 1$, the high-energy tail of the velocity distribution is Gaussian, with a different variance, while for $r_w=+1$ the velocity distribution has an exponential tail.