Fluctuation-dissipation relations in driven granular gases

Abstract

We study the dynamics of a 2d driven inelastic gas, by means of Direct Simulation Monte Carlo (DSMC) 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 non-equilibrium 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 experiment, i.e. 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 tw, 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.

Temperature probes in binary granular gases

“…[14] has shown the validity of this relation in the context of a monodisperse granular media heated by a thermal bath with temperature T b , reaching in this way a stationary state with granular temperature T g < T b . In this case the FD relation holds by replacing T with T g .…”

“…Another totally independent way of checking FD relations in granular gases has also been used in [14]: once a steady-state has been reached, the system is perturbed impulsively at a given time t 0 by a non-conservative force applied (non-uniformly) on every particle (we will take t 0 = 0 without loss of generality). The response is then monitored at later times.…”

“…A recent investigation has shown that, for non-equilibrium monodisperse driven granular gases, this relation is still obeyed if the equilibrium temperature is replaced by the granular temperature [14]. A natural question then arises: what happens to the relation (1) in a mixture which displays more than one granular temperature ?…”

“…We now turn to the measure of R i (t) and C i (t). Technical details of the numerical procedure to perform this measure are given in [14]. From the definitions of ξ, C i and R i , it is clear that R i (0) = 1/(2m i ), and…”

“…Suppose to perform a global measurement of the Fluctuation-Dissipation ratio on a system of unknown composition. A global measurement on a monodisperse system would yield a well defined FD ratio and a well defined temperature equal to the granular temperature, independently of the observable used (see [14]). On the other hand a polydisperse system would feature a typically non-constant FD ratio, function also of the observable (unless the system is completely elastic).…”