Fluctuations of Power Injection in Randomly Driven Granular Gases
Résumé
We investigate the large deviation function π∞(w) for the fluctuations of the power W(t) = wt, integrated over a time t, injected by a homogeneous random driving into a granular gas, in the infinite time limit. Our analytical study starts from a generalized Liouville equation and exploits a Molecular Chaos-like assumption. We obtain an equation for the generating function of the cumulants μ(λ) which appears as a generalization of the inelastic Boltzmann equation and has a clear physical interpretation. Reasonable assumptions are used to obtain μ(λ) in a closed analytical form. A Legendre transform is sufficient to get the large deviation function π∞(w). Our main result, apart from an estimate of all the cumulants of W(t) at large times t, is that π∞ has no negative branch. This immediately results in the inapplicability of the Gallavotti-Cohen Fluctuation Relation (GCFR), that in previous studies had been suggested to be valid for injected power in driven granular gases. We also present numerical results, in order to discuss the finite time behavior of the fluctuations of W (t) . We discover that their probability density function converges extremely slowly to its asymptotic scaling form: the third cumulant saturates after a characteristic time τ larger than ∼50 mean free times and the higher order cumulants evolve even slower. The asymptotic value is in good agreement with our theory. Remarkably, a numerical check of the GCFR is feasible only at small times (at most τ/10), since negative events disappear at larger times. At such small times this check leads to the misleading conclusion that GCFR is satisfied for π∞(w). We offer an explanation for this remarkable apparent verification. In the inelastic Maxwell model, where a better statistics can be achieved, we are able to numerically observe the “failure” of GCFR.
