On the equivalence of the freely cooling granular gas to the sticky gas

TL;DR

This study compares a one-dimensional freely cooling granular gas with a velocity-dependent restitution coefficient to a sticky gas, revealing equivalence in behavior during early inhomogeneous cooling but divergence at larger times due to a finite velocity scale.

Contribution

It demonstrates the conditions under which a granular gas behaves like a sticky gas and clarifies the limitations of the inviscid Burgers equation in describing real granular gases.

Findings

Granular gas behaves like sticky gas for times less than t1 ~ δ^{-1}.

For δ approaching zero, the granular gas aligns with the inviscid Burgers equation.

Finite δ introduces a crossover time beyond which the behaviors diverge.

Abstract

A freely cooling granular gas with velocity dependent restitution coefficient is studied in one dimension. The restitution coefficient becomes near elastic when the relative velocity of the colliding particles is less than a velocity scale $\delta$. Different statistical quantities namely density distribution, occupied and empty cluster length distributions, and spatial density and velocity correlation functions, are obtained using event driven molecular dynamic simulations. We compare these with the corresponding quantities of the sticky gas (inelastic gas with zero coefficient of restitution). We find that in the inhomogeneous cooling regime, for times smaller than a crossover time $t_1$ where $t_1 \sim \delta^{-1}$, the behaviour of the granular gas is equivalent to that of the sticky gas. When $\delta \rt 0$, then $t_1 \rt \infty$ and hence, the results support an earlier claim that the freely cooling inelastic gas is described by the inviscid Burgers equation. For a real granular gas with finite $\delta$, the existence of the time scale $t_1$ shows that, for large times, the granular gas is not described by the inviscid Burgers equation.