%0 Journal Article
%T Thermalization of L¨¦vy Flights: Path-Wise Picture in 2D
%A Mariusz £¿aba
%A Piotr Garbaczewski
%J International Journal of Statistical Mechanics
%D 2013
%R 10.1155/2013/738345
%X We analyze two-dimensional (2D) random systems driven by a symmetric L¨¦vy stable noise which in the presence of confining potentials may asymptotically set down at Boltzmann-type thermal equilibria. In view of the Eliazar-Klafter no-go statement, such dynamical behavior is plainly incompatible with the standard Langevin modeling of L¨¦vy flights. No explicit path-wise description has been so far devised for the thermally equilibrating random motion we address, and its formulation is the principal goal of the present work. To this end we prescribe a priori the target pdf ¦Ñ£¿ in the Boltzmann form ~exp[ ] and next select the L¨¦vy noise (e.g., its L¨¦vy measure) of interest. To reconstruct random paths of the underlying stochastic process we resort to numerical methods. We create a suitably modified version of the time honored Gillespie algorithm, originally invented in the chemical kinetics context. A statistical analysis of generated sample trajectories allows us to infer a surrogate pdf dynamics which sets down at a predefined target, in consistency with the associated kinetic (master) equation. 1. Introduction Various random processes in real physical systems admit a simplified description based on stochastic differential equations. Then, there is a routine passage procedure from microscopic random variables to macroscopic (statistical ensemble, mean field) data, like, for example, the time evolution of an associated probability density function (pdf) which is a solution of a deterministic transport equation. A paradigm example is so-called the Langevin modeling of diffusion-type and jump-type processes. The presumed microscopic model of random dynamics is provided by the Langevin (stochastic) equation, which additively decomposes into a (Newtonian by origin) drift and purely random (perturbing noise) term. Its direct consequence is the Fokker-Planck equation for an associated probability density function (pdf); confer [1] for a discussion of the Brownian motion and [2, 3] for that of L¨¦vy flights in external forces. We note that the L¨¦vy-Langevin formulation results in the space-fractional Fokker-Planck equation. The subject of our further discussion is two-dimensional (2D) random systems driven by a symmetric L¨¦vy stable noise which, under the sole influence of external (force) potentials , asymptotically set down at Boltzmann-type thermal equilibria. Such behavior is excluded within standard ramifications of the Langevin approach to L¨¦vy flights, where the action of a conservative force field stands for an explicit reason for the emergence of an
%U http://www.hindawi.com/journals/ijsm/2013/738345/