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

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 760 ◽  
Author(s):  
Johan Anderson ◽  
Sara Moradi ◽  
Tariq Rafiq
Keyword(s):  
Anomalous Diffusion ◽  
Transport Coefficient ◽  
Numerical Solutions ◽  
Distribution Functions ◽  
Space Distribution ◽  
Non Linear ◽  
Non Local ◽  
Local Transport ◽  
Fokker Planck Equations ◽  
Fokker Planck

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.

Dynamics of normal and anomalous diffusion in nonlinear Fokker-Planck equations

2009 ◽  
Vol 70 (1) ◽  
pp. 107-116 ◽  
Author(s):  
V. Schwämmle ◽  
E. M.F. Curado ◽  
F. D. Nobre
Keyword(s):  
Anomalous Diffusion ◽  
Fokker Planck Equations ◽  
Fokker Planck

scholarly journals Numerical Solutions of Fractional Fokker-Planck Equations Using Iterative Laplace Transform Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Limei Yan
Keyword(s):  
Laplace Transform ◽  
Numerical Solutions ◽  
Convergent Series ◽  
Space Time ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Promising Tool ◽  
The Laplace Transform ◽  
Fokker Planck Equations ◽  
Fokker Planck

A relatively new iterative Laplace transform method, which combines two methods; the iterative method and the Laplace transform method, is applied to obtain the numerical solutions of fractional Fokker-Planck equations. The method gives numerical solutions in the form of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and straightforward when applied to space-time fractional Fokker-Planck equations. The method provides a promising tool for solving space-time fractional partial differential equations.

scholarly journals Multipoint Reconstruction of Wind Speeds

2020 ◽  
Author(s):  
Christian Behnken ◽  
Matthias Wächter ◽  
Joachim Peinke
Keyword(s):  
Wind Speed ◽  
Time Scales ◽  
Numerical Solutions ◽  
Wind Data ◽  
Conditional Probability Density ◽  
Wind Speeds ◽  
Fokker Planck Equations ◽  
Short Time ◽  
Speed Fluctuations ◽  
Fokker Planck

Abstract. The most intermittent behavior of atmospheric turbulence is found for very short time scales. Based on a concatenation of conditional probability density functions (cpdfs) of nested wind speeds increments, inspired by a Markov process in scale, we derive a short-time predictor for wind speed fluctuations around a non-stationary mean value and with a corresponding non-stationary variance. As a new quality this short time predictor enables a multipoint reconstruction of wind data. The used cpdfs are (1) directly estimated from historical data from the offshore research platform FINO1 and (2) obtained from numerical solutions of a family of Fokker-Planck equations in the scale domain. The explicit forms of the Fokker-Planck equations are estimated from the given wind data. A good agreement between the statistics of the generated synthetic wind speed fluctuations and the measured is found even on time scales below 1 s. This shows that our approach captures the short-time dynamics of real wind speed fluctuations very well. Our method is extended by taking the non-stationarity of the mean wind speed and its non-stationary variance into account.

The lattice Fokker–Planck equation for models of wealth distribution

2020 ◽  
Vol 378 (2175) ◽  
pp. 20190401
Author(s):  
Shaurya Kaushal ◽  
Santosh Ansumali ◽  
Bruce Boghosian ◽  
Merek Johnson
Keyword(s):  
Steady State ◽  
Soft Matter ◽  
Wealth Distribution ◽  
Planck Equation ◽  
Agent Based ◽  
Non Local ◽  
Free Parameters ◽  
Fokker Planck Equations ◽  
Steady State Solutions ◽  
Fokker Planck

Recent work on agent-based models of wealth distribution has yielded nonlinear, non-local Fokker–Planck equations whose steady-state solutions describe empirical wealth distributions with remarkable accuracy using only a few free parameters. Because these equations are often used to solve the ‘inverse problem’ of determining the free parameters given empirical wealth data, there is much impetus to find fast and accurate methods of solving the ‘forward problem’ of finding the steady state corresponding to given parameters. In this work, we derive and calibrate a lattice Boltzmann equation for this purpose. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

scholarly journals Long-time behavior of non-local in time Fokker–Planck equations via the entropy method

2019 ◽  
Vol 29 (02) ◽  
pp. 209-235 ◽  
Author(s):  
Jukka Kemppainen ◽  
Rico Zacher
Keyword(s):  
Decay Rate ◽  
Entropy Method ◽  
Decay Rates ◽  
Time Behavior ◽  
Optimal Decay ◽  
Local Case ◽  
Special Cases ◽  
Non Local ◽  
Fokker Planck Equations ◽  
Fokker Planck

We consider a rather general class of non-local in time Fokker–Planck equations and show by means of the entropy method that as [Formula: see text], the solution converges in [Formula: see text] to the unique steady state. Important special cases are the time-fractional and ultraslow diffusion case. We also prove estimates for the rate of decay. In contrast to the classical (local) case, where the usual time derivative appears in the Fokker–Planck equation, the obtained decay rate depends on the entropy, which is related to the integrability of the initial datum. It seems that higher integrability of the initial datum leads to better decay rates and that the optimal decay rate is reached, as we show, when the initial datum belongs to a certain weighted [Formula: see text] space. We also show how our estimates can be adapted to the discrete-time case thereby improving known decay rates from the literature.

Chaos-Induced Diffusion. Analogues to Nonlinear Fokker-Planck Equations

1985 ◽  
Vol 40 (9) ◽  
pp. 867-873 ◽  
Author(s):  
H. Fujisaka ◽  
S. Grossmann ◽  
S. Thomae
Keyword(s):  
Anomalous Diffusion ◽  
Chaotic Dynamics ◽  
Particle Diffusion ◽  
Jump Size ◽  
Fokker Planck Equations ◽  
Nonlinear Increase ◽  
Fokker Planck

Abstract Deterministic chaotic dynamics induced by a climbing map with variable local jump size is shown to yield anomalous diffusion, i.e. a nonlinear increase of the variance with time. The relation to Fokker-Planck equations with variable drift and diffusivity is given. Turbulent two-particle diffusion may be a possible application.

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