Lévy Flights in Random Environments

Abstract

We consider Lévy flights characterized by the step index f in a quenched random force field. By means of a dynamic renormalization group analysis we find that the dynamic exponent z for f < 2 locks onto f , independent of dimension and independent of the presence of weak quenched disorder. The critical dimension, however, depends on the step index f for f < 2 and is given by d c = 2f − 2. For d < d c the disorder is relevant, corresponding to a non trivial fixed point for the force correlation function.Irrespective of the spatial dimension d, ordinary Brownian motion traces out a manifold of fractal dimension d F = 2 [17]. In the presence of a quenched disordered force field in d dimensions the Brownian walk is unaffected for d > d F , i.e., for d larger than the critical dimension d c = d F the walk is transparent and the dynamic exponent z locks onto the value 2 for the pure case. Below the critical dimension d c = 2 the long time characteristics of the walk is changed to subdiffusive behaviour with z > 2 [18,19,20]. In d = 1, r 2 (t) ∝ [log t] 4 , independent of the strength of the quenched disorder [21].Lévy flights constitute an interesting generalization of ordinary Brownian walks.

Fractional generalization of Liouville equations

“…The main aim of most of these papers is to formulate fractional integro-differential equations to describe some scaling process. Modifications of equations governing physical processes such as the Langevin equation [6], diffusion equations, and Fokker-Plank equation have been suggested [7]- [13] which incorporate fractional derivatives with respect to time. It was shown in Ref.…”

Fractional Transport Equations for Lévy Stable Processes

“…In this Letter we consider the generalization of transport equations to describe Lévy Brownian motion. This question has already been addressed in the past by using various methods [5][6][7][8], in particular the CTRW formalism [9,10]. However, all these approaches were limited to coordinate space only.…”

“…However, it has become clear in recent years that many processes in nature, such as anomalous diffusion (for a review see [2][3][4]), cannot be described by ordinary (Gaussian) Brownian motion. A case in point is the so-called Lévy flight with a stochastic force distributed according to Lévy stable statistics and which has been introduced in connection with super-diffusion [5,6]. In this Letter we consider the generalization of transport equations to describe Lévy Brownian motion.…”

Aging Feynman–Kac equation

“…For example, they are found in systems including ultra-cold atoms, telomeres in the nucleus of cells 5 , bacterial motion 6 , and even the flight of an albatross 7 . Anomalous diffusion in the presence or absence of force fields has been modelled in many ways such as generalised diffusion equations 8 , fractional a) Electronic mail: dengwh@lzu.edu.cn Brownian motion 9 , continue time random walk 10,11 (CTRW), Langenvin equations 12,13 , and so on. The compound power law renewal process is a specific class of CTRW model, where both waiting time and jump length are i.i.d.…”