The generalized Langevin equation with Gaussian fluctuations

@article{Fox1977TheGL,
  title={The generalized Langevin equation with Gaussian fluctuations},
  author={Ronald Forrest Fox},
  journal={Journal of Mathematical Physics},
  year={1977},
  volume={18},
  pages={2331-2335}
}
  • R. Fox
  • Published 1 December 1977
  • Physics
  • Journal of Mathematical Physics
It is shown that all statistical properties of the generalized Langevin equation with Gaussian fluctuations are determined by a single, two‐point correlation function. The resulting description corresponds with a stationary, Gaussian, non‐Markovian process. Fokker–Planck‐like equations are discussed, and it is explained how they can lead one to the erroneous conclusion that the process is nonstationary, Gaussian, and Markovian. 

Generalized Fokker–Planck equation for non‐Markovian processes

This paper deals with a system of linear non‐Markovian–Langevin equations with memory functions that are not constant in time and a nonzero initial instant of time. A set of statistical means, based

A generalized Langevin equation for dealing with nonadditive fluctuations

A suitable extension of the Mori memory-function formalism to the non-Hermitian case allows a “multiplicative” process to be described by a Langevin equation of non-Markoffian nature. This

Fluctuation theories and Gaussian stochastic processes

The theory of fluctuations for systems near equilibrium has given rise to two developments which generalize the theory in two distinct ways. One of these developments is focused on the theory of

The long-time fluctuations of a Brownian sphere

Statistical-mechanical problems of principle associated with long-time correlation are discussed using the example of retarded Brownian motion. The well known 'stochastic' solutions are quoted and

A colored-noise approach to Brownian motion in position space. Corrections to the Smoluchowski equation

The contraction of the description of Brownian motion from phase space to position space is discussed by means of non-Markovian Langevin equations in position space. A Fokker-Planck equation valid

The generalized Smoluchowski equation and non‐Markovian dynamics

Starting from the generalized Langevin equation, we derive a generalized Smoluchowski equation. This equation provides a general framework for problems of non‐Markovian barrier crossing dynamics. We

Non-Markovian effects on the Brownian motion of a free particle

Dynamical properties of non‐Markovian stochastic differential equations

We study nonstationary non‐Markovian processes defined by Langevin‐type stochastic differential equations with an Ornstein–Uhlenbeck driving force. We concentrate on the long time limit of the
...

References

SHOWING 1-5 OF 5 REFERENCES

Analysis of nonstationary, Gaussian and non-Gaussian, generalized Langevin equations using methods of multiplicative stochastic processes

  • R. Fox
  • Mathematics, Computer Science
  • 1977
TLDR
Using the methods of multiplicative stochastic processes, a thorough analysis of “non-Markovian,” generalized Langevin equations is presented and shows that the methods already used in the Gaussian case lead directly to results for the non-Gaussian case.

Fokker-Planck equations for simple non-Markovian systems

Exact generalized Fokker–Planck equations are derived from the linear Mori–Kubo generalized Langevin equation for the case of Gaussian but non‐Markovian noise. Fokker–Planck equations which generate

The Brownian Movement and Stochastic Equations

The irregular movements of small particles immersed in a liquid, caused by the impacts of the molecules of the liquid, were described by Brown in 1828.1 Since 1905 the Brownian movement has been

On the Theory of the Brownian Motion

With a method first indicated by Ornstein the mean values of all the powers of the velocity $u$ and the displacement $s$ of a free particle in Brownian motion are calculated. It is shown that