Hard Sphere Dynamics for Normal and Granular Fluids

TL;DR

This paper reviews the dynamics of hard sphere fluids, including elastic and inelastic collisions, and introduces exact solutions to the Liouville equation related to granular fluid hydrodynamics.

Contribution

It provides a novel application of generators for Liouville dynamics to granular fluids, identifying eigenvalues and eigenfunctions for exact solutions.

Findings

Eigenvalues and eigenfunctions of the Liouville generator are identified.

Exact solutions related to granular fluid hydrodynamics are derived.

The approach bridges microscopic dynamics and macroscopic hydrodynamics.

Abstract

A fluid of N smooth, hard spheres is considered as a model for normal (elastic collisions) and granular (inelastic collisions) fluids. The potential energy is discontinuous for hard spheres so the pairwise forces are singular and the usual forms of Newtonian and Hamiltonian mechanics do not apply. Nevertheless, particle trajectories in the N particle phase space are well defined and the generators for these trajectories can be identified. The first part of this presentation is a review of the generators for the dynamics of observables and probability densities. The new results presented in the second part refer to applications of these generators to the Liouville dynamics for granular fluids. A set of eigenvalues and eigenfunctions of the generator for this Liouville dynamics is identified in a special "stationary representation". This provides a class of exact solutions to the Liouville equation that are closely related to hydrodynamics for granular fluids.