On the Einstein relation in a heated granular gas

TL;DR

This paper analytically investigates the Einstein relation in heated granular gases, showing deviations due to non-Maxwellian distributions and comparing effects of different thermostats, aligning well with simulation results.

Contribution

It provides an analytical solution to the Boltzmann-Lorentz equation for heated granular gases, clarifying the origin of Einstein relation violations under different thermostats.

Findings

Deviation from Einstein relation is due to non-Maxwellian velocity distributions.

Gaussian thermostats cause larger deviations than stochastic thermostats.

Results agree with previous computer simulation findings.

Abstract

Recent computer simulation results [Barrat {\em et al.}, Physica A 334 (2004) 513] for granular mixtures subject to stochastic driving have shown the validity of the Einstein relation $\epsilon\equiv D/(T_0\lambda)=1$ between the diffusion $D$ and mobility $\lambda$ coefficients when the temperature of the gas $T$ is replaced by the temperature of the impurity $T_0$ in the usual Einstein relation. This problem is analyzed in this paper by solving analytically the Boltzmann-Lorentz equation from the Chapman-Enskog method. The gas is heated by the action of an external driving force (thermostat) which does work to compensate for the collisional loss of energy. Two types of thermostats are considered: (a) a deterministic force proportional to the particle velocity (Gaussian thermostat), and (b) a white noise external force (stochastic thermostat). The diffusion and mobility coefficients are given in terms of the solutions of two linear integral equations, which are approximately solved up to the second order in a Sonine polynomial expansion. The results show that the violation of the Einstein relation ($\epsilon\neq 1$) is only due to the non-Maxwellian behavior of the impurity velocity distribution function (absence of the Gibbs state). At a quantitative level, the kinetic theory results also show that the deviation of $\epsilon$ from 1 is more significant in the case of the Gaussian thermostat than in the case of the stochastic one, in which case the deviation of the Einstein relation is in general smaller than 1%. This conclusion agrees quite well with the results found in computer simulations.