Velocity Distributions in Homogeneously Cooling and Heated Granular Fluids

TL;DR

This paper analyzes the velocity distribution of granular fluids under cooling and heating, finding small deviations from Gaussian behavior and a high-energy tail in the steady state, using the Enskog-Boltzmann equation.

Contribution

It provides explicit calculations of the velocity distribution and fourth cumulant for inelastic hard spheres, comparing theoretical results with simulations, and characterizes the high-energy tail in the heated state.

Findings

Good agreement with simulations for undriven cooling.

Small non-Gaussian corrections at all inelasticities.

High-energy tail decays as exp(-A c^{3/2}) in heated steady state.

Abstract

We study the single particle velocity distribution for a granular fluid of inelastic hard spheres or disks, using the Enskog-Boltzmann equation, both for the homogeneous cooling of a freely evolving system and for the stationary state of a uniformly heated system, and explicitly calculate the fourth cumulant of the distribution. For the undriven case, our result agrees well with computer simulations of Brey et al. \cite{brey}. Corrections due to non-Gaussian behavior on cooling rate and stationary temperature are found to be small at all inelasticities. The velocity distribution in the uniformly heated steady state exhibits a high energy tail $\sim \exp(-A c^{3/2})$, where $c$ is the velocity scaled by the thermal velocity and $A\sim 1/\sqrt{\eps}$ with $\eps$ the inelasticity.