scholarly journals Understanding Fluid Dynamics from Langevin and Fokker–Planck Equations

Fluids ◽  
2020 ◽  
Vol 5 (1) ◽  
pp. 40 ◽  
Author(s):  
Andrei Medved ◽  
Riley Davis ◽  
Paula A. Vasquez
Keyword(s):  
Fluid Dynamics ◽  
Undergraduate Students ◽  
Microstructure Evolution ◽  
Physical Interpretation ◽  
Particle Motion ◽  
Coarse Grained ◽  
Langevin Equations ◽  
Fluid Behavior ◽  
Fokker Planck Equations ◽  
Fokker Planck

The Langevin equations (LE) and the Fokker–Planck (FP) equations are widely used to describe fluid behavior based on coarse-grained approximations of microstructure evolution. In this manuscript, we describe the relation between LE and FP as related to particle motion within a fluid. The manuscript introduces undergraduate students to two LEs, their corresponding FP equations, and their solutions and physical interpretation.

scholarly journals Fokker–Planck representations of non-Markov Langevin equations: application to delayed systems

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180131 ◽  
Author(s):  
Luca Giuggioli ◽  
Zohar Neu
Keyword(s):  
Langevin Equation ◽  
Complex Dynamics ◽  
Probability Distributions ◽  
Delay Systems ◽  
Langevin Equations ◽  
Delayed Systems ◽  
Temporal Features ◽  
Fokker Planck Equations ◽  
Random Fluctuations ◽  
Fokker Planck

Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker–Planck equations for the n -time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n  = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker–Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

scholarly journals Langevin Equations, Fokker–Planck Equations and Entropic Analysis of a Model-Independent Classical Plasma

2021 ◽  
Author(s):  
Kendra Jean Jacques ◽  
Preet Sharma
Keyword(s):  
Electric Field ◽  
Small Deviation ◽  
Important Change ◽  
Langevin Equations ◽  
Plasma Dynamics ◽  
Classical Plasma ◽  
Model Independent ◽  
Fokker Planck Equations ◽  
Sizeable Difference ◽  
Fokker Planck

Plasma dynamics have been studied extensively and there is a fair amount of understanding where the scientific community has reached at. However, there is still a very big gap in completely explaining plasma physics at the classical as well as the quantum level. The dynamics of plasma from an entropic approach are not very well understood or explained. There is too much chaos to account for and even a small deviation in terms of perturbations of any kind makes a sizeable difference. This study is based on the entropic approach where we take a model independent classical plasma. Then we apply Langevin equations and Fokker–Planck equations to explain the entropy generated and entropy produced. Then we study various conditions in which we apply an electric field and a magnetic field and understand the various trends in entropy changes. When we apply the electric field and the magnetic fields independently of each other and together in the plasma model, we see that there is a very important change in the increase in entropy. There are also changes in the plasma flow, but the overall flow does not drastically change since we have considered a model independent plasma. Finally, we show that there are indeed changes to the entropy in a model-independent classical plasma in the various cases as mentioned in this study.

NUMERICAL APPROACH TO FOKKER–PLANCK EQUATIONS FOR BROWNIAN MOTORS

2002 ◽  
Vol 13 (09) ◽  
pp. 1157-1176 ◽  
Author(s):  
MARCIN KOSTUR
Keyword(s):  
Monte Carlo ◽  
Numerical Approach ◽  
Langevin Equations ◽  
Two Dimensional ◽  
Finite Element Scheme ◽  
Brownian Motors ◽  
Element Scheme ◽  
Mc Simulation ◽  
Fokker Planck Equations ◽  
Fokker Planck

The numerical approach to a large class of one- and two-dimensional Fokker–Planck equations (FPE) often encountered in modeling Brownian Motors is presented. The method is based on Finite Element scheme with additional modifications for specific problems. We compare results from discretization of FPE with those obtained from Monte Carlo (MC) simulation of the corresponding Langevin equations. Accuracy, efficiency and applicability are also discussed.

scholarly journals Efficient Stabilization of Advection Terms Involved in Separated Representations of Boltzmann and Fokker-Planck Equations

2015 ◽  
Vol 17 (4) ◽  
pp. 975-1006 ◽  
Author(s):  
F. Chinesta ◽  
E. Abisset-Chavanne ◽  
A. Ammar ◽  
E. Cueto
Keyword(s):  
Distribution Function ◽  
Dimensional Space ◽  
Computing Time ◽  
Planck Equation ◽  
Coarse Grained ◽  
High Dimensional ◽  
Separated Representations ◽  
Time Required ◽  
Fokker Planck Equations ◽  
Fokker Planck

AbstractThe fine description of complex fluids can be carried out by describing the evolution of each individual constituent (e.g. each particle, each macromolecule, etc.). This procedure, despite its conceptual simplicity, involves many numerical issues, the most challenging one being that related to the computing time required to update the system configuration by describing all the interactions between the different individuals. Coarse grained approaches allow alleviating the just referred issue: the system is described by a distribution function providing the fraction of entities that at certain time and position have a particular conformation. Thus, mesoscale models involve many different coordinates, standard space and time, and different conformational coordinates whose number and nature depend on the particular system considered. Balance equation describing the evolution of such distribution function consists of an advection-diffusion partial differential equation defined in a high dimensional space. Standard mesh-based discretization techniques fail at solving high-dimensional models because of the curse of dimensionality. Recently the authors proposed an alternative route based on the use of separated representations. However, until now these approaches were unable to address the case of advection dominated models due to stabilization issues. In this paper this issue is revisited and efficient procedures for stabilizing the advection operators involved in the Boltzmann and Fokker-Planck equation within the PGD framework are proposed.

Modified Fokker-Planck equations

1978 ◽  
Vol 36 (1) ◽  
pp. 65-78 ◽  
Author(s):  
Glenn T. Evans
Keyword(s):  
Fokker Planck Equations ◽  
Fokker Planck
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