Mobility and diffusion in granular fluids

A recent analysis of Brownian motion in granular media undergoing homogeneous cooling implies a violation of the Einstein relation between the diffusion and mobility coefficients. It is shown here that this violation occurs more generally. The usual method of linear response to an external force is generalized for inelastic collisions, where the reference state at zero applied force is the homogeneous cooling state. Formally exact Green-Kubo type expressions are compared for the mobility and diffusion coefficients of an impurity particle in a granular fluid. The results show that the absence of the Gibbs state, the cooling of the reference state, and the occurrence of different kinetic temperatures for the particle and surrounding fluid are responsible for a violation of the Einstein relation. A quantitative description of the effect over a wide range of densities and restitution coefficients is provided by an approximate evaluation of the associated response functions.