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

Abstract

We consider the steady states of a driven inelastic Maxwell gas consisting of two types of particles with scalar velocities. Motivated by experiments on bilayers where only one layer is driven, we focus on the case when only one of the two types of particles are driven externally, with the other species receiving energy only through inter-particle collision. The velocity v of a particle that is driven is modified to −r w v + η, where r w parameterises the dissipation upon the driving and the noise η is taken from a fixed distribution. We characterize the statistics for small velocities by computing exactly the mean energies of the two species, based on the simplifying feature that the correlation functions are seen to form a closed set of equations. The asymptotic behaviour of the velocity distribution for large speeds is determined for both components through a combination of exact analysis for a range of parameters or obtained numerically to a high degree of accuracy from an analysis of the large moments of velocity. We show that the tails of the velocity distribution for both types of particles have similar behaviour, even though they are driven differently. For dissipative driving (r w < 1), the tails of the steady state velocity distribution show nonuniversal features and depend strongly on the noise distribution. On the other hand, the tails of the velocity distribution are exponential for diffusive driving (r w = 1) when the noise distribution decays faster than exponential. arXiv:1910.07745v2 [cond-mat.stat-mech] 3 Jan 2020

Mpemba effect in driven granular Maxwell gases

“…While this makes the Maxwell gas more unrealistic, it renders it more amenable to exact analysis, at the same time retaining the qualitative features. This advantageous feature has been exploited in obtaining more rigorous results in both freely cooling granular gas [32][33][34][35] as well as in the velocity distributions of driven granular gases [36][37][38][39][40][41][42]. The equations for the time evolution of the relevant two point velocity correlations for the Maxwell gas form a closed set of equations [43].…”

Mpemba effect in driven granular gases: role of distance measures

“…( 42) requires to have information regarding the time evolution of the velocity distribution function at any time t which is not analytically feasible in the case of granular gases. The previous theoretical studies on velocity distribution of driven granular gases attempts to find the velocity distribution of the steady state for different contexts [43,44,[50][51][52]. But in all the cases, it was not possible to derive the form of instantaneous velocity distribution at an instant of time t even for the simplest model of inelastic Maxwell gas.…”