TY - JOUR

T1 - Stable equilibrium based on Lévy statistics: A linear Boltzmann equation approach

T2 - A linear Boltzmann equation approach

AU - Barkai, Eli

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2004/1/1

Y1 - 2004/1/1

N2 - To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, we consider stochastic collision models. The models are a generalization of the Rayleigh collision model, for a heavy one dimensional particle M interacting with ideal gas particles with a mass m ≪ M. Similar to previous approaches we assume elastic, uncorrelated, and impulsive collisions. We let the bath particle velocity distribution function to be of general form, namely we do not postulate a specific form of power-law equilibrium. We show, under certain conditions, that the velocity distribution function of the heavy particle is Lévy stable, the Maxwellian distribution being a special case. We demonstrate our results with numerical examples. The relation of the power law equilibrium obtained here to thermodynamics is discussed. In particular we compare between two models: a thermodynamic and an energy scaling approaches. These models yield insight into questions like the meaning of temperature for power law equilibrium, and into the issue of the universality of the equilibrium (i.e., is the width of the generalized Maxwellian distribution functions obtained here, independent of coupling constant to the bath).

AB - To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, we consider stochastic collision models. The models are a generalization of the Rayleigh collision model, for a heavy one dimensional particle M interacting with ideal gas particles with a mass m ≪ M. Similar to previous approaches we assume elastic, uncorrelated, and impulsive collisions. We let the bath particle velocity distribution function to be of general form, namely we do not postulate a specific form of power-law equilibrium. We show, under certain conditions, that the velocity distribution function of the heavy particle is Lévy stable, the Maxwellian distribution being a special case. We demonstrate our results with numerical examples. The relation of the power law equilibrium obtained here to thermodynamics is discussed. In particular we compare between two models: a thermodynamic and an energy scaling approaches. These models yield insight into questions like the meaning of temperature for power law equilibrium, and into the issue of the universality of the equilibrium (i.e., is the width of the generalized Maxwellian distribution functions obtained here, independent of coupling constant to the bath).

KW - Collision models

KW - Generalized equilibrium

KW - Lévy statistics

UR - http://www.scopus.com/inward/record.url?scp=3543120717&partnerID=8YFLogxK

U2 - 10.1023/b:joss.0000028068.93241.3a

DO - 10.1023/b:joss.0000028068.93241.3a

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SN - 0022-4715

VL - 115

SP - 1537

EP - 1565

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 5-6

ER -