Fluctuation-dissipation relations in driven granular gases

We study the dynamics of a two-dimensional driven inelastic gas, by means of direct simulation Monte Carlo techniques, i.e., under the assumption of molecular chaos. Under the effect of a uniform stochastic driving in the form of a white noise plus a friction term, the gas is kept in a nonequilibrium steady state characterized by fractal density correlations and non-Gaussian distributions of velocities; the mean-squared velocity, that is the so-called granular temperature, is lower than the bath temperature. We observe that a modified form of the Kubo relation, which relates the autocorrelation and the linear response for the dynamics of a system at equilibrium, still holds for the off equilibrium, though stationary, dynamics of the systems under investigation. Interestingly, the only needed modification to the equilibrium Kubo relation is the replacement of the equilibrium temperature with an effective temperature, which results equal to the global granular temperature. We present two independent numerical experiments, where two different observables are studied: (a) the staggered density current, whose response to an impulsive shear is proportional to its autocorrelation in the unperturbed system and (b) the response of a tracer to a small constant force, switched on at time $t_w,$ which is proportional to the mean-square displacement in the unperturbed system. Both measures confirm the validity of Kubo’s formula, provided that the granular temperature is used as the proportionality factor between response and autocorrelation, at least for not too large inelasticities.