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[] 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.
Highlights
Various random processes in real physical systems admit a simplified description based on stochastic differential equations
There is a routine passage procedure from microscopic random variables to macroscopic data, like, for example, the time evolution of an associated probability density function which is a solution of a deterministic transport equation
Its direct consequence is the Fokker-Planck equation for an associated probability density function; confer [1] for a discussion of the Brownian motion and [2, 3] for that of Levy flights in external forces
Summary
Various random processes in real physical systems admit a simplified description based on stochastic differential equations. The subject of our further discussion is two-dimensional (2D) random systems driven by a symmetric Levy 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 Levy flights, where the action of a conservative force field ∼ −∇Φ(x) stands for an explicit reason for the emergence of an asymptotic invariant probability density function (pdf). A novel fractional generalization of the Fokker-Planck equation governing the time evolution of ρ(x, t) has been introduced in [8, 9, 16] (see [7, 10, 11]) to handle systems that are randomized by symmetric Levy-stable drivers and may asymptotically set down at Boltzmann-type equilibria under the influence of external potentials ( not Newtonian forces anymore). We note that exp(−Φ(x, y)) becomes a genuine stationary solution of (7) once we let ε1x,y → 0 and ε2x,y → ∞
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