Dynamics of a hard sphere granular impurity

TL;DR

This paper derives an exact mapping for an impurity particle in a granular fluid, showing how its behavior relates to elastic collisions and revealing its long-term dynamics and temperature differences.

Contribution

It introduces an exact mapping of the impurity's distribution function to that of an elastic collision case with an effective mass, advancing understanding of granular impurity dynamics.

Findings

Impurity solutions in granular and elastic cases are equivalent classes.

Impurity approaches a different equilibrium temperature in a granular host.

Impurity reaches a scaling solution in a granular host's scaling state.

Abstract

An impurity particle coupling to its host fluid via inelastic hard sphere collisions is considered. It is shown that the exact equation for its distribution function can be mapped onto that for an impurity with elastic collisions and an effective mass. Application of this result to the Enskog--Lorentz kinetic equation leads to several conclusions: 1) every solution in the elastic case is equivalent to a class of solutions in the granular case; 2) for an equilibrium host fluid the granular impurity approaches equilibrium at a different temperature, with a dominant diffusive mode at long times; 3) for a granular host fluid in its scaling state, the granular impurity approaches the corresponding scaling solution.