Thermodynamics of Superdiffusion Generated by Lévy–Wiener Fluctuating Forces

Abstract

Scale free Lévy motion is a generalized analogue of the Wiener process. Its time derivative extends the notion of “white noise” to non-Gaussian noise sources, and as such, it has been widely used to model natural signal variations described by an overdamped Langevin stochastic differential equation. Here, we consider the dynamics of an archetypal model: a Brownian-like particle is driven by external forces, and noise is represented by uncorrelated Lévy fluctuations. An unperturbed system of that form eventually attains a steady state which is uniquely determined by the set of parameter values. We show that the analyzed Markov process with the stability index α < 2 violates the detailed balance, i.e., its stationary state is quantified by a stationary probability density and nonvanishing current. We discuss consequences of the non-Gibbsian character of the stationary state of the system and its impact on the general form of the fluctuation–dissipation theorem derived for weak external forcing.

Peculiarities of escape kinetics in the presence of athermal noises

“…Divergent moments of Lévy statistics and Lévy motion seem to stay in conflict with energetic and thermodynamics of the stochastic differential equation of the Langevin type 23,44,57,58 . Yet, accumulating evidence shows that Markovian Lévy flights (LFs) with distribution of jumps emerging from the generalized version of the central limit theorem are well suited representations of complex phenomena, to name just a few recent applications of LFs in description of mental searches 61 , analysis of free neutron output in a fusion experiment with a deuteron plasma 56 , investigations of generegulatory networks 62 or examination of self-regulatory motion of insects 63 .…”

“…The central part of the jump length distribution control short jumps which are responsible for penetration of the potential barrier 4 . Therefore, in the current subsection, we assume that the particle is driven by two stochastic forces [55][56][57][58] , so that the Langevin equation assumes the form…”

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

“…There are many formalisms that describe anomalous diffusion, ranging from thermodynamics [78][79][80], and fractional derivatives [4,6] to generalized Langevin equations (GLE) [81][82][83]. The goal of this short review is to call attention to relevant research within the GLE framework and to some fundamental theorems in statistical mechanics.…”

Density fluctuations and random walks in an overdamped and supercooled simple liquid

“…The properties of such oscillations can be highlighted in the most direct way by the exploration of the mean square velocity displacement and connected with the thermodynamic density fluctuations. Note also that such behavior can be associated with the recent topic of non-stationary transient modes in model Ornstein-Uhlenbeck and related stochastic processes [40][41][42]. In addition, the approach considered in this work is not limited by the particular L-J potential only, and may be applied to systems with other potentials, e.g.…”