Driven inelastic Maxwell models with high energy tails

Abstract

The solutions of the homogeneous nonlinear Boltzmann equation for inelastic Maxwell models, when driven by different types of thermostats, show, in general, overpopulated high energy tails of the form ϳexp(Ϫac), with power law tails and Gaussian tails as border line cases. The results are compared with those for inelastic hard spheres, and a comprehensive picture of the long time behavior in freely cooling and driven inelastic systems is presented.

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“…These results were recently extended [25,26] to driven inelastic Maxwell models, where the high energy tail is of exponential form exp[−A|c|]. Apparently, the type of overpopulation depends sensitively on the microscopic model, on the degree of inelasticity and on the possible mode of energy supply to the dissipative system.…”

“…As discussed more extensively in Ref. [25] the differences in shape are frequently related to non-uniformities in the limits of long times, large velocities and vanishing inelasticity, which lead to different results when taking the limits in different order or when taking coupled limits such as the scaling limit (e.g. the differences between bulk and tail behavior), or performing an expansion in powers of the inelasticity, and then studying large times (typically Gaussian tails are observed [14]) or studying large times at fixed inelasticity and taking large time limits afterwards (typically overpopulated tails are observed [5,6]) with a whole wealth of coupled limits in between.…”

Boltzmann equation for dissipative gases in homogeneous states with nonlinear friction

“…Although the measured velocity distribution F (v) generically deviates from the Maxwellian, its functional form depends on material property and on the forcing mechanism used to compensate for collisional loss of energy [6,7,8,9]. A similar picture emerges from numerical simulations and analytical studies [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33], where in addition to the more common stretched exponential behavior, power-law distributions have also been reported [13,20,21,22,31].…”

“…The parameters in f (c) can be obtained from the full integral equation (23) by substituting the ansatz (22), applying the derivative, equating leading and sub-leading powers of c, and recalling the relation b > b ′ > 0. To leading order we have bβc b+θ−1 = c ν B σ , yielding…”

“…Generically these tails are stretched exponentials [15,16,23]. The more spectacular power law tails [13,20,21,22,31] are exceptional.…”