Linear Response and Hydrodynamics for Granular Fluids

TL;DR

This paper derives linear hydrodynamics for granular fluids using statistical mechanics, identifying transport coefficients and susceptibilities, and discusses differences from normal fluids and conditions for idealized models.

Contribution

It provides a formal derivation of granular fluid hydrodynamics, generalizing classical relations and analyzing the effects of inelastic collisions.

Findings

Transport coefficients are expressed via time correlation functions.

Differences between granular and normal fluids are identified.

Scaling limits for inelastic hard sphere models are described.

Abstract

A formal derivation of linear hydrodynamics for a granular fluid is given. The linear response to small spatial perturbations of the homogeneous reference state is studied in detail using methods of non-equilibrium statistical mechanics. A transport matrix for macroscopic excitations in the fluid is defined in terms of the response functions. An expansion in the wavevector to second order allows identification of all phenomenological susceptibilities and transport coefficients through Navier-Stokes order, in terms of appropriate time correlation functions. The transport coefficients in this representation are the generalization to granular fluids of the familiar Helfand and Green-Kubo relations for normal fluids. The analysis applies to a wide range of collision rules. Important differences in both the analysis and results from those for normal fluids are identified and discussed. A scaling limit is described corresponding to the conditions under which idealized inelastic hard sphere models can apply. Further details and interpretation are provided in the paper following this one, by specialization to the case of smooth, inelastic hard spheres with constant coefficient of restitution.