Trajectory statistics of confined Lévy flights and Boltzmann-type equilibria

Abstract

We analyze a specific class of random systems that are driven by a symmetric Lévy stable noise, where Langevin representation is absent. In view of the Lévy noise sensitivity to environmental inhomogeneities, the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) ρ * (x) ∼ exp [−Φ(x)]. Here, we infer pdf ρ(x, t) based on numerical path-wise simulation of the underlying jump-type process. A priori given data are jump transition rates entering the master equation for ρ(x, t) and its target pdf ρ * (x). To simulate the above processes, we construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. We exemplified our algorithm simulating different jump-type processes and discuss the dynamics of real physical systems where it can be useful. I. INTRODUCTIONDespite many attempts to pin down the essential features of dynamics and relaxation in random systems, the problem is still far from its complete solution. It turns out that Lévy flight models are adequate for the description of different random systems ranging from the motion of defects in disordered solids to the dynamics of assets in stock markets, see, e.g. [1]. They are especially useful to model the random systems on the semi-phenomenological, mesoscopic level, when the (often unknown) details of their microscopic random behavior are substituted by a properly tailored (e.g. based on experimental data) (Gaussian or Lévy) noise. Paradoxically, in disordered solids, the noise can promote order and organization, switching them between different equilibrium states. The latter situation emerges in disordered ferroelectrics, where the fluctuations of order parameter (spontaneous polarization) give rise to selflocalization of charge carrier, generating a fluctuon, an analog of well-known polaron in disordered substance [2]. These fluctuons make a substantial contribution into conductivity and optical properties of disordered dielectrics.Many random processes admit a description based on stochastic differential equations. In such case there is a routine passage procedure from microscopic random variables to macroscopic (statistical ensemble) data. The latter are encoded in the time evolution of an associated probability density function (pdf) which is a solution of a deterministic transport equation. A paradigmatic example is the so-called Langevin modeling of diffusion-type and jump-type processes. The presumed microscopic model of the dynamics in external force fields is provided by the Langevin (stochastic) equation whose direct consequence is the Fokker-Planck equation, [3] and [4]. We note that in case of jump-type processes the familiar Laplacian (Wiener noise generator) needs to be replaced by a suitable pseudo-differential operator (fractional Laplacian, in case of a symmetric Lévy-stable noise).We pay a particular attention to jump-type processes which are omnipresent in Nature (see [5] and references therein). Their chara...