Trajectory statistics of confined L\'{e}vy flights and Boltzmann-type equilibria

TL;DR

This paper develops a numerical simulation method for analyzing the long-term behavior of symmetric Lévy flights in confined systems, revealing their convergence to Boltzmann-type equilibria without relying on Langevin equations.

Contribution

It introduces a modified Gillespie algorithm to simulate jump processes driven by Lévy noise and studies their asymptotic equilibrium distributions.

Findings

Simulated Lévy flights reach Boltzmann-type equilibrium distributions.

The modified Gillespie algorithm effectively models jump-type stochastic processes.

Application potential for physical systems with non-Gaussian noise.

Abstract

We analyze a specific class of random systems that are driven by a symmetric L\'{e}vy stable noise, where Langevin representation is absent. In view of the L\'{e}vy noise sensitivity to environmental inhomogeneities, the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) $\rho_*(x) \sim \exp [-\Phi (x)]$. Here, we infer pdf $\rho (x,t)$ based on numerical path-wise simulation of the underlying jump-type process. A priori given data are jump transition rates entering the master equation for $\rho (x,t)$ and its target pdf $\rho_*(x)$. To simulate the above processes, we construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. We exemplified our algorithm simulating different jump-type processes and discuss the dynamics of real physical systems where it can be useful.