Kinetic Theory of Response Functions for the Hard Sphere Granular Fluid

TL;DR

This paper develops a kinetic theory framework for response functions in granular fluids, deriving a linear kinetic equation that generalizes existing theories and applies to various densities and inelasticities, with validation through transport coefficient calculations.

Contribution

It introduces a new kinetic equation for granular fluids based on response functions, extending the linearized Enskog theory without restrictions on density or inelasticity.

Findings

Derived a kinetic equation for granular response functions.

Validated the theory by calculating transport coefficients.

Results agree with Chapman-Enskog solutions.

Abstract

The response functions for small spatial perturbations of a homogeneous granular fluid have been described recently. In appropriate dimensionless variables, they have the form of stationary state time correlation functions. Here, these functions are expressed in terms of reduced single particle functions that are expected to obey a linear kinetic equation. The functional assumption required for such a kinetic equation, and a Markov approximation for its implementation are discussed. If, in addition, static velocity correlations are neglected, a granular fluid version of the linearized Enskog kinetic theory is obtained. The derivation makes no a priori limitation on the density, space and time scale, nor degree of inelasticity. As an illustration, recently derived Helfand and Green-Kubo expressions for the Navier-Stokes order transport coefficients are evaluated with this kinetic theory. The results are in agreement with those obtained from the Chapman-Enskog solution to the nonlinear Enskog kinetic equation.