Exact solution of a one-dimensional Boltzmann equation for a granular tracer particle

TL;DR

This paper provides an exact analytical solution for a one-dimensional granular tracer particle system when the mass ratio equals the restitution coefficient, and explores its dynamics through simulations for other cases.

Contribution

It derives an exact solution for the Boltzmann equation in a specific one-dimensional granular system and extends understanding through numerical simulations.

Findings

Exact solution for the Boltzmann equation when M/m = α

Expressions for velocity autocorrelation and diffusion coefficient

Qualitative similarity in dynamics for M/m ≠ α

Abstract

We consider a one-dimensional system consisting of a granular tracer particle of mass $M$ in a bath of thermalized particles each of mass $m$. When the mass ratio, $M/m$, is equal to the coefficient of restitution, $\alpha$, the system maps to a a one-dimensional elastic gas. In this case, Boltzmann equation can be solved exactly. We also obtain expressions for the velocity autocorrelation function and the diffusion coefficient. Numerical simulations of the Boltzmann equation are performed for $M/m\neq \alpha$ where no analytical solution is available. It appears that the dynamical features remain qualitatively similar to those found in the exactly solvable case.