Thermalization of Levy flights: Path-wise picture in 2D

Abstract

We analyze two-dimensional (2D) random systems driven by a symmetric Lévy stable noise which, under the sole influence of external (force) potentials $Φ(x) $, asymptotically set down at Boltzmann-type thermal equilibria. Such behavior is excluded within standard ramifications of the Langevin approach to Lévy flights. In the present paper we address the response of Lévy noise not to an external conservative force field, but directly to its potential $Φ(x)$. We prescribe a priori the target pdf $ρ_*$ in the Boltzmann form $\sim \exp[- Φ(x)]$ and next select the Lévy noise of interest. Given suitable initial data, this allows to infer a reliable path-wise approximation to a true (albeit analytically beyond the reach) solution of the pertinent master equation, with the property $ρ(x,t)\rightarrow ρ_*(x)$ as time $t$ goes to infinity. We create a suitably modified version of the time honored Gillespie's algorithm, originally invented in the chemical kinetics context. A statistical analysis of generated sample trajectories allows us to infer a surrogate pdf dynamics which consistently sets down at a pre-defined target pdf. We pay special attention to the response of the 2D Cauchy noise to an exemplary locally periodic "potential landscape" $Φ(x), x\in R^2$.