Derivation of the Langevin Equation

2014 ◽  
pp. 267-282
Author(s):  
Grigorios A. Pavliotis
Keyword(s):  
Langevin Equation

scholarly journals Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

2021 ◽  
Author(s):  
Adrien Laurent ◽  
Gilles Vilmart
Keyword(s):  
Invariant Measure ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Group ◽  
Langevin Equation ◽  
Numerical Experiments ◽  
Special Linear Group ◽  
Order Conditions ◽  
High Order ◽  
Differential Equations On Manifolds

AbstractWe derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

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 The study of dynamic singularities of seismic signals by the generalized Langevin equation

2009 ◽  
Vol 388 (17) ◽  
pp. 3629-3635 ◽  
Author(s):  
Renat Yulmetyev ◽  
Ramil Khusnutdinoff ◽  
Timur Tezel ◽  
Yildiz Iravul ◽  
Bekir Tuzel ◽  
...  
Keyword(s):  
Langevin Equation ◽  
Generalized Langevin Equation ◽  
Seismic Signals

Turbulent dispersion in a non-homogeneous field

1999 ◽  
Vol 392 ◽  
pp. 45-71 ◽  
Author(s):  
ILIAS ILIOPOULOS ◽  
THOMAS J. HANRATTY
Keyword(s):  
Numerical Simulation ◽  
Direct Numerical Simulation ◽  
Stochastic Model ◽  
Point Source ◽  
Time Scales ◽  
Langevin Equation ◽  
Turbulent Dispersion ◽  
Homogeneous Field ◽  
Fully Developed Flow ◽  
Good Agreement

Dispersion of fluid particles in non-homogeneous turbulence was studied for fully developed flow in a channel. A point source at a distance of 40 wall units from the wall is considered. Data obtained by carrying out experiments in a direct numerical simulation (DNS) are used to test a stochastic model which utilized a modified Langevin equation. All of the parameters, with the exception of the time scales, are obtained from Eulerian statistics. Good agreement is obtained by making simple assumptions about the spatial variation of the time scales.

An approach towards a simple quantum Langevin equation

2011 ◽  
Vol 511 (4-6) ◽  
pp. 471-481 ◽  
Author(s):  
Joshua M. Jackson ◽  
Pietrina L. Brucia ◽  
Michael Messina
Keyword(s):  
Langevin Equation ◽  
Quantum Langevin Equation ◽  
Simple Quantum
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