Response Functions for a Granular Fluid

TL;DR

This paper develops a theoretical framework for understanding the linear response and transport properties of a granular fluid, extending classical kinetic theories to include inelastic collisions and correlations.

Contribution

It introduces a formalism for response functions in granular fluids, deriving hydrodynamic modes, Langevin equations, and Green-Kubo formulas, extending kinetic theories to inelastic particle systems.

Findings

Exact response functions in the long wavelength limit show hydrodynamic behavior.

Derived generalized kinetic equations including velocity correlations and recollision effects.

Extended classical theories like Boltzmann and Enskog to granular systems with inelastic collisions.

Abstract

The response of an isolated granular fluid to small perturbations of the hydrodynamic fields is considered. The corresponding linear response functions are identified in terms of a formal solution to the Liouville equation including the effects of the cooling reference state. These functions are evaluated exactly in the asymptotic long wavelength limit and shown to represent hydrodynamic modes. More generally, the linear granular Navier-Stokes equations for the response functions and related Langevin equations are obtained from an extension of Mori's identity. The resulting Green-Kubo expressions for transport coefficients are compared and contrasted with those for a molecular fluid. Next the response functions are described in terms of an effective dynamics in the single particle phase space. A closed linear kinetic equation is obtained formally in terms of a linear two particle functional. This closure is evaluated for two examples: a short time Markovian approximation, and a low density expansion on length and time scales of the mean free time and mean free path. The former is a generalization of the revised Enskog kinetic theory to include velocity correlations. The latter is an extension of the Boltzmann equation to include the effects of recollisions (rings) among the particles.