Stable Equilibrium Based on Lévy Statistics: A Linear Boltzmann Equation Approach

Abstract

To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, a stochastic collision model is investigated. We consider the dynamics of a tracer particle of mass $M$, undergoing elastic collisions with ideal gas particles of mass $m$, in the Rayleigh limit $m<<M$. The probability density function (PDF) of the gas particle velocity is $f({\tilde{v}}_m)$. Assuming a uniform collision rate and molecular chaos, we obtain the equilibrium distribution for the velocity of the tracer particle $W_{eq}(V_M)$. Depending on asymptotic properties of $f({\tilde{v}}_m)$ we find that $W_{eq}(V_M)$ is either the Maxwell velocity distribution or a Lévy distribution. In particular our results yield a generalized Maxwell distribution based on Lévy statistics using two approaches. In the first a thermodynamic argument is used, imposing on the dynamics the condition that equilibrium properties of the heavy tracer particle be independent of the coupling $\epsilon =m/M$ to the gas particles, similar to what is found for a Brownian particle in a fluid. This approach leads to a generalized temperature concept. In the second approach it is assumed that bath particles velocity PDF scales with an energy scale, i.e. the (nearly) ordinary temperature, as found in standard statistical mechanics. The two approaches yield different types of Lévy equilibrium which merge into a unique solution only for the Maxwell--Boltzmann case. Thus, relation between thermodynamics and statistical mechanics becomes non-trivial for the power law case. Finally, the relation of the kinetic model to fractional Fokker--Planck equations is discussed.

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Cited by in corpus (11)

• Levy Flight Superdiffusion: An Introduction
• Stationary states in Langevin dynamics under asymmetric Lévy noises
• The Boltzmann equation for driven systems of inelastic soft spheres
• The rich behavior of the Boltzmann equation for dissipative gases
• Generalized Fokker-Planck equation, Brownian motion, and ergodicity
• On the higher order corrections to the Fokker-Planck equation
• Bimodality and hysteresis in systems driven by confined Lévy flights
• Boltzmann equation for dissipative gases in homogeneous states with nonlinear friction
• Anomalous Transport in Velocity Space, from Fokker-Planck to General Equation
• On anomalous diffusion in a plasma in velocity space
• Relativistic Lévy processes