Self-diffusion in granular gases: Green-Kubo versus Chapman-Enskog

TL;DR

This paper compares Green-Kubo and Chapman-Enskog methods for calculating self-diffusion in granular gases, finding Chapman-Enskog more practical for realistic impact-velocity dependent restitution coefficients.

Contribution

It demonstrates the equivalence of both methods for constant restitution and advocates for Chapman-Enskog in realistic scenarios with velocity-dependent restitution.

Findings

Both methods agree for constant restitution coefficient.

Chapman-Enskog is more practical for impact-velocity dependent restitution.

Diffusion coefficients are computed for realistic viscoelastic particles.

Abstract

We study the diffusion of tracers (self-diffusion) in a homogeneously cooling gas of dissipative particles, using the Green-Kubo relation and the Chapman-Enskog approach. The dissipative particle collisions are described by the coefficient of restitution $\epsilon$ which for realistic material properties depends on the impact velocity. First, we consider self-diffusion using a constant coefficient of restitution, $\epsilon=$const, as frequently used to simplify the analysis. Second, self-diffusion is studied for a simplified (stepwise) dependence of $\epsilon$ on the impact velocity. Finally, diffusion is considered for gases of realistic viscoelastic particles. We find that for $\epsilon=$const both methods lead to the same result for the self-diffusion coefficient. For the case of impact-velocity dependent coefficients of restitution, the Green-Kubo method is, however, either restrictive or too complicated for practical application, therefore we compute the diffusion coefficient using the Chapman-Enskog method. We conclude that in application to granular gases, the Chapman-Enskog approach is preferable for deriving kinetic coefficients.