Thermalization of Lévy Flights: Path-Wise Picture in 2D. (3rd October 2013) - Żaba, Mariusz; Garbaczewski, Piotr
Abstract : We analyze two-dimensional (2D) random systems driven by a symmetric Lévy 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 Lévy 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 ρ ∗ in the Boltzmann form exp[− Φ ( x ) ] and next select the Lévy noise (e.g., its Lévy 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 pdfρ ( x, t ) dynamics which sets down at a predefined target, in consistency with the associated kinetic (master) equation.