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

Johan Anderson,Tariq Rafiq,Sara Moradi

doi:10.3390/e20100760

Johan Anderson, Tariq Rafiq + Show 1 more

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https://doi.org/10.3390/e20100760

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Abstract

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.

Highlights

In magnetically confined (MC) plasma devices transport driven by turbulent fluctuations often severely limit the confinement time and impede the development of fusion as an alternative for electricity production
It is commonly accepted that turbulence is the primary cause of anomalous transport [1,2]
It has been recognized that the nature of the anomalous transport processes is dominated by a significant ballistic or non-local component where a diffusive description is improper

Summary

Introduction

In magnetically confined (MC) plasma devices transport driven by turbulent fluctuations often severely limit the confinement time and impede the development of fusion as an alternative for electricity production. The super-diffusive properties are often ubiquitously found in plasmas, such as the thermal and particle fluxes in the gradient region or in the Scrape-Off Layer (SOL) where the transport is dominated by the coherent structures (blobs) [3,4,5,6,7,8,9] and inherently possess a non-local character [10,11,12,13,14,15,16]. There is a large quantity of experimental evidence that density and potential fluctuations measured by Langmuir probes at different fusion devices support the idea that these fluctuations are distributed according to Lévy statistics. This was illustrated for example in [4], Entropy 2018, 20, 760; doi:10.3390/e20100760 www.mdpi.com/journal/entropy

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