Thermalization of Lévy Flights: Path-Wise Picture in 2D

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 pdfρ∗in the Boltzmann form~exp[] 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 pdf dynamics which sets down at a predefined target, in consistency with the associated kinetic (master) equation.

Path-wise versus kinetic modeling for equilibrating non-Langevin jump-type processes

“…Even with the parallel computing on small clusters the procedure may take days, for larger values of the stability index µ ∈ (0, 2). Especially when increasing the spatial dimensionality from 1d to 2D, [17], or 3D.…”

“…On the other hand, if one can devise a path-wise picture of the underlying random dynamics, whose statistical consequence is the evolution (1) of ρ(x, t), we tell about an indirect path-wise method of solution of the master equation. The latter case was the subject of our recent paper [14], followed by a straightforward extension from 1D to 2D, [17].…”