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Open AccessArticle

Non-Linear Langevin and Fractional Fokker–Planck Equations for Anomalous Diffusion by Lévy Stable Processes

1
Department of Space, Earth and Environment, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
2
Laboratory for Plasma Physics—LPP-ERM/KMS, Royal Military Academy, 1000 Brussels, Belgium
3
Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA
*
Author to whom correspondence should be addressed.
Entropy 2018, 20(10), 760; https://doi.org/10.3390/e20100760
Received: 28 July 2018 / Revised: 17 August 2018 / Accepted: 29 September 2018 / Published: 3 October 2018
The numerical solutions to a non-linear Fractional Fokker–Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where Lévy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable Lévy distribution as solutions to the FFP equation. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data. View Full-Text
Keywords: non-local theory; Lévy noise; Tsallis entropy; fractional Fokker–Plank equation; anomalous diffusion non-local theory; Lévy noise; Tsallis entropy; fractional Fokker–Plank equation; anomalous diffusion
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Anderson, J.; Moradi, S.; Rafiq, T. Non-Linear Langevin and Fractional Fokker–Planck Equations for Anomalous Diffusion by Lévy Stable Processes. Entropy 2018, 20, 760.

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