2003/01/01 by M. H. Ernst, Ricardo Brito, R. Brito · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Boltzmann equation #Classical mechanics #Dissipative system #Exponent #Gas Dynamics and Kinetic Theory #Kinetic energy #Mathematical physics #Physics #Power law #Thermodynamics #Thermoelastic and Magnetoelastic Phenomena #Thermostat #cond-mat.stat-mech
paper · pdf · doi:10.1007/978-3-540-39843-1_1
Submitted to: "Granular Gas Dynamics", T. Poeschel, N. Brilliantov (eds.), Lecture Notes in Physics, Vol. LNP 624, Springer-Verlag, Berlin-Heidelberg-New York, 2003
openalex publication_date 2003/01/01 · arxiv created 2003/04/27 · arxiv updated 2015/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Analytic solutions F(v,t) of the nonlinear Boltzmann equation in d-dimensions are studied for a new class of dissipative models, called inelastic repulsive scatterers, interacting through pseudo-power law repulsions, characterized by a strength parameter ν, and embedding inelastic hard spheres (ν=1) and inelastic Maxwell models (ν=0). The systems are either freely cooling without energy input or driven by thermostats, e.g. white noise, and approach stable nonequilibrium steady states, or marginally stable homogeneous cooling states, where the data, vd0(t) F(v,t) plotted versus c=v/v0(t), collapse on a scaling or similarity solution f(c), where v0(t) is the r.m.s. velocity. The dissipative interactions generate overpopulated high energy tails, described generically by stretched Gaussians, f(c) ∼ exp[-βcb] with 0 < b < 2, where b=ν with ν>0 in free cooling, and b=1+1/2 ν with ν≥ 0 when driven by white noise. Power law tails, f(c) ∼ 1/ca+d, are only found in marginal cases, where the exponent a is the root of a transcendental equation. The stability threshold depend on the type of thermostat, and is for the case of free cooling located at ν=0. Moreover we analyze an inelastic BGK-type kinetic equation with an energy dependent collision frequency coupled to a thermostat, that captures all qualitative properties of the velocity distribution function in Maxwell models, as predicted by the full nonlinear Boltzmann equation, but fails for harder interactions with ν>0.