Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
975881 | Physica A: Statistical Mechanics and its Applications | 2013 | 12 Pages |
Abstract
We analyze a specific class of random systems that, while being driven by a symmetric Lévy stable noise, asymptotically set down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) Ïâ(x)â¼exp[âΦ(x)]. This behavior needs to be contrasted with the standard Langevin representation of Lévy jump-type processes. It is known that the choice of the drift function in the Newtonian form â¼ââΦ excludes the existence of the Boltzmannian pdf â¼exp[âΦ(x)] (Eliazar-Klafter no go theorem). In view of this incompatibility statement, our main goal here is to establish the appropriate path-wise description of the equilibrating jump-type process. A priori given inputs are (i) jump transition rates entering the master equation for Ï(x,t) and (ii) the target (invariant) pdf Ïâ(x) of that equation, in the Boltzmannian form. We resort to numerical methods and construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. The generated sample trajectories show up a qualitative typicality, e.g. they display structural features of jumping paths (predominance of small vs large jumps) specific to particular stability indices μâ(0,2). The obtained random paths statistical data allow us to infer an associated pdf Ï(x,t) dynamics which stands for a validity check of our procedure. The considered exemplary Boltzmannian equilibria â¼exp[âΦ(x)] refer to (i) harmonic potential Φâ¼x2, (ii) logarithmic potential Φâ¼nln(1+x2) with n=1,2 and (iii) locally periodic confining potential Φâ¼sin2(2Ïx),|x|â¤2, Φâ¼(x2â4),|x|>2.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematical Physics
Authors
Mariusz Żaba, Piotr Garbaczewski, Vladimir Stephanovich,