Superposition principle for non-local Fokker–Planck–Kolmogorov operators

2020 ◽  
Vol 178 (3-4) ◽  
pp. 699-733
Author(s):  
Michael Röckner ◽  
Longjie Xie ◽  
Xicheng Zhang
Keyword(s):  
Superposition Principle ◽  
Kolmogorov Operators ◽  
Non Local ◽  
Fokker Planck

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.

scholarly journals A non-local scalar conservation law describing navigation processes

2020 ◽  
Vol 17 (04) ◽  
pp. 809-841
Author(s):  
Paulo Amorim ◽  
Florent Berthelin ◽  
Thierry Goudon
Keyword(s):  
Conservation Law ◽  
Stability Result ◽  
Planck Equation ◽  
Small Region ◽  
Effective Velocity ◽  
Fokker Planck Equation ◽  
Scalar Conservation Law ◽  
Non Local ◽  
Scalar Conservation ◽  
Fokker Planck

We consider a non-local scalar conservation law in two space dimensions which arises as the formal hydrodynamic limit of a Fokker–Planck equation. This Fokker–Planck equation is, in turn, the kinetic description of an individual-based model describing the navigation of self-propelled particles in a pheromone landscape. The pheromone may be linked to the agent distribution itself, leading to a nonlinear, non-local scalar conservation law where the effective velocity vector depends on the pheromone field in a small region around each point, and thus, on the solution itself. After presenting and motivating the problem, we present some numerical simulations of a closely related problem, and then prove a well-posedness and stability result for the conservation law.

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.

On the Superposition Principle for Fokker–Planck–Kolmogorov Equations

2019 ◽  
Vol 100 (1) ◽  
pp. 363-366
Author(s):  
V. I. Bogachev ◽  
M. Röckner ◽  
S. V. Shaposhnikov
Keyword(s):  
Superposition Principle ◽  
Kolmogorov Equations ◽  
Fokker Planck

scholarly journals Quantum Networks: Dynamics of Open Nanostructures

VLSI Design ◽  
1998 ◽  
Vol 8 (1-4) ◽  
pp. 191-196 ◽  
Author(s):  
Günter Mahler ◽  
Rainer Wawer
Keyword(s):  
Quantum Noise ◽  
Superposition Principle ◽  
Quantum Trajectories ◽  
Quantum Networks ◽  
Numerical Examples ◽  
Composite Systems ◽  
Local Coherence ◽  
Non Local ◽  
Quantum Parallelism

The superposition principle makes quantum networks behave very differently from their classical counterparts: We discuss how local and non-local coherence are generated and how these may affect the function of composite systems. Numerical examples concern quantum trajectories, quantum noise and quantum parallelism.

scholarly journals Fokker-Planck simulation of non-local thermal smoothing due to non-uniform laser heating

2008 ◽  
Vol 112 (2) ◽  
pp. 022044
Author(s):  
M Chen ◽  
Y Kishimoto ◽  
J Q Li ◽  
W B Pei
Keyword(s):  
Laser Heating ◽  
Non Local ◽  
Fokker Planck
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