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Entropy 2018, 20(9), 658; https://doi.org/10.3390/e20090658

Thermodynamics of Superdiffusion Generated by Lévy–Wiener Fluctuating Forces

1
Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
2
Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagiellonian University, ul. S. Łojasiewicza 11, 30-348 Kraków, Poland
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Received: 5 April 2018 / Revised: 16 August 2018 / Accepted: 29 August 2018 / Published: 31 August 2018
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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. View Full-Text
Keywords: nonequilibrium and irreversible thermodynamics; fluctuation phenomena; random walks and Lévy flights nonequilibrium and irreversible thermodynamics; fluctuation phenomena; random walks and Lévy flights
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Kuśmierz, Ł.; Dybiec, B.; Gudowska-Nowak, E. Thermodynamics of Superdiffusion Generated by Lévy–Wiener Fluctuating Forces. Entropy 2018, 20, 658.

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