Stable Equilibrium Based on L\'evy Statistics: Stochastic Collision Models Approach

TL;DR

This paper explores how two stochastic collision models lead to stable Le9vy distributions as equilibrium states, generalizing Maxwell's distribution through fractional kinetic equations.

Contribution

It demonstrates that stable power-law equilibria arise naturally in different stochastic collision models, extending classical Maxwellian results via fractional kinetic equations.

Findings

Both models yield Le9vy distributions as equilibrium states.

The Maxwell distribution is a special case within this framework.

Equilibrium distributions are stable and independent of model specifics.

Abstract

We investigate equilibrium properties of two very different stochastic collision models: (i) the Rayleigh particle and (ii) the driven Maxwell gas. For both models the equilibrium velocity distribution is a L\'evy distribution, the Maxwell distribution being a special case. We show how these models are related to fractional kinetic equations. Our work demonstrates that a stable power-law equilibrium, which is independent of details of the underlying models, is a natural generalization of Maxwell's velocity distribution.