Energy distribution of inelastic gas in a box is dominated by a power law -- a derivation in the framework of sample space reducing processes

TL;DR

This paper derives the energy distribution of an inelastic gas in a box using sample space reducing processes, revealing power-law behaviors influenced by driving rate and restitution, confirmed by simulations.

Contribution

It introduces SSR processes as an alternative to Boltzmann approaches for deriving energy distributions in dissipative systems, highlighting power-law solutions.

Findings

Energy distribution follows power laws over entire energy range.

Power-law exponents decrease with increased driving rate.

Results are validated through molecular dynamics simulations.

Abstract

We use the framework of sample space reducing processes (SSR) as an alternative to Boltzmann equation based approaches, to derive the energy and velocity distribution functions of an inelastic gas in a box as an example for a dissipative, driven system. SSR processes do not assume molecular chaos and are characterized by a specific type of eigenvalue equation whose solutions represent stationary distribution functions. The equations incorporate the geometry of inelastic collisions and a driving mechanism in a transparent way. Energy is injected by boosting particles that hit the walls of the container to high energies. The numerical solution of the resulting equations yields power laws over the entire energy region. The exponents decrease with the driving rate from about $2$ to below $1.5$ and depend on the coefficient of restitution. Results are confirmed with a molecular dynamics simulation in 3D with the same driving mechanism. We believe that these distribution functions are observable in experimental situations; the abundance of power laws in driven dissipative systems might be a sign of the generality of the result.