Velocity distribution of driven granular gases

Abstract

The granular gas is a paradigm for understanding the effects of inelastic interactions in granular materials. Kinetic theory provides a general theoretical framework for describing the granular gas. Its central result is that the tail of the velocity distribution of a driven granular gas is a stretched exponential that, counterintuitively, decays slower than that of the corresponding elastic gas in equilibrium. However, a derivation of this result starting from a microscopic model is lacking. Here, we obtain analytical results for a microscopic model for a granular gas where particles with two-dimensional velocities are driven homogeneously and isotropically by reducing the velocities by a factor and adding a stochastic noise. We find two universal regimes. For generic physically relevant driving, we find that the tail of the velocity distribution is a Gaussian with additional logarithmic corrections. Thus, the velocity distribution decays faster than the corresponding equilibrium gas. The second universal regime is less generic and corresponds to the scenario described by kinetic theory. Here, the velocity distribution is shown to decay as an exponential with additional logarithmic corrections, in contradiction to the predictions of the phenomenological kinetic theory, necessitating a re-examination of its basic assumptions. arXiv:1804.02558v2 [cond-mat.stat-mech]

Asymptotic velocity distribution of a driven one dimensional binary granular Maxwell gas

“…For ballistic transport, the collision rate is proportional to the relative velocity. For mono dispersed gases, an analysis with this more realistic kernel shows that β remains the same, though for r w = 1, there are additional logarithmic corrections to the exponential decay [45,46]. We expect these results to generalise to the driven binary gas also, such that β, as obtained in this paper, is not modified.…”

“…Thus, the truncation of the driving term in the Boltzmann equation to lowest order in η gives the correct result only in restricted regimes. However, even this restricted equivalence between microscopic models for driving and Boltzmann equation with diffusive driving may not hold for more realistic collision kernels where the collision rates are proportional to the relative velocity [45,46].…”

“…Third, in the limit r w = 1, the diffusive term that is usually used to model driving in kinetic theory result can be realised [46]. This may be argued as follows.…”

“…where the subleading correction Ψ k (|v|) satisfies |v| −β k Ψ k (|v|) → 0 for large v. For such a stretched exponential distribution, the 2n th moment can be obtained using saddle point approximation for large n, and has the form [46] M k 2n ∼…”

Asymptotic Behavior of the Velocity Distribution of Driven Inelastic Gas with Scalar Velocities: Analytical Results

“…In this paper, we choose to study a system with scalar velocities as the analysis there is much simpler (Study of an inelastic gas with two-dimensional velocities may be found in Ref. [72]). For the inelastic gas with scalar velocities, driven homogeneously through dissipative driving, we carry out an exact analysis of the equations satisfied by the moments of the velocity for a general collision kernel, corresponding to arbitrary δ .…”