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

Abstract

We determine the asymptotic behavior of the tails of the steady state velocity distribution of a homogeneously driven granular gas comprising of particles having a scalar velocity. A pair of particles undergo binary inelastic collisions at a rate that is proportional to a power of their relative velocity. At constant rate, each particle is driven by multiplying its velocity by a factor −r w and adding a stochastic noise. When r w < 1, we show analytically that the tails of the velocity distribution are primarily determined by the noise statistics, and determine analytically all the parameters characterizing the velocity distribution in terms of the parameters characterizing the stochastic noise. Surprisingly, we find logarithmic corrections to the leading stretched exponential behavior. When r w = 1, we show that for a range of distributions of the noise, inter-particle collisions lead to a universal tail for the velocity distribution.

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].…”

Mpemba effect in driven granular Maxwell gases

“…The physical motivations for the form of driving may be found in Refs. [43,44], where positive r w 's can be identified as the coefficient of restitution of collisions between particle and a vibrating wall.…”

Producing suprathermal tails in the stationary velocity distribution

“…Their effects add in the above set-up (7) where the evolution equation has the test particle exposed either to an equilibrium environment at temperature T or to (effectively infinite temperature) diffusive driving. We now discuss γ(v) and B(v).…”

“…in [4,5] one finds a convincing explanation of power law tails in the velocity distribution in space plasmas by means of whistler-mode wave stochastic acceleration. For granular gases a very similar analysis is contained in [6,7]. There may even be other physical explanations, differing in various details.…”