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 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 Nonlinear stochastic modelling with Langevin regression

2021 ◽  
Vol 477 (2250) ◽  
pp. 20210092
Author(s):  
J. L. Callaham ◽  
J.-C. Loiseau ◽  
G. Rigas ◽  
S. L. Brunton
Keyword(s):  
Langevin Equation ◽  
Degrees Of Freedom ◽  
Bluff Body ◽  
Gaussian White Noise ◽  
Stochastic Modelling ◽  
State Dependent ◽  
Nonlinear Langevin Equation ◽  
Statistical Consistency ◽  
Fokker Planck Equations ◽  
Random Fluctuations

Many physical systems characterized by nonlinear multiscale interactions can be modelled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behaviour are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like Gaussian white noise, introducing non-Markovian behaviour to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using forward and adjoint Fokker–Planck equations to enforce statistical consistency. If the form of the Langevin equation is unknown, a simple sparsifying procedure can provide an appropriate functional form. We demonstrate that this method can learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by coloured noise and approximating the second-order dynamics of a particle in a double-well potential with the corresponding first-order bifurcation normal form. Finally, we apply Langevin regression to experimental measurements of a turbulent bluff body wake and show that the statistical behaviour of the centre of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise.

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 Deterministic and Stochastic Research of Cubic Population Model with Harvesting

2021 ◽  
pp. 2150023
Author(s):  
Özgür Gültekin ◽  
Çağatay Eskin ◽  
Esra Yazicioğlu
Keyword(s):  
Langevin Equation ◽  
Allee Effect ◽  
Deterministic Model ◽  
Probability Distributions ◽  
Gaussian White Noise ◽  
Quadratic Function ◽  
Planck Equation ◽  
Detailed Examination ◽  
Fokker Planck Equation ◽  
Fokker Planck

A detailed examination of the effect of harvesting on a population has been carried out by extending the standard cubic deterministic model by considering a population under Allee effect with a quadratic function representing harvesting. Weak and strong Allee effect transitions, carrying capacity, and Allee threshold change according to harvesting are first discussed in the deterministic model. A Fokker–Planck equation has been obtained starting from a Langevin equation subject to correlated Gaussian white noise with zero mean, and an Approximate Fokker–Planck Equation has been obtained from a Langevin equation subject to correlated Gaussian colored noise with zero mean. This allowed to calculate the stationary probability distributions of populations, and thus to discuss the effects of linear and nonlinear (Holling type-II) harvesting for populations under Allee effect and subject to white and colored noises, respectively.

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