## Properties of Langevin and Fokker–Planck equations

*Helmut Hofmann*

### in The Physics of Warm Nuclei: with Analogies to Mesoscopic Systems

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198504016
- eISBN:
- 9780191708480
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504016.003.0025
- Subject:
- Physics, Nuclear and Plasma Physics

This chapter introduces Langevin- and Fokker-Planck equations by way of a heuristic approach to the physics of Brownian particles. Variances of physical quantities are defined and related to ... More

## SINGLE CHAINS IN EXTERNAL FIELDS

*Glenn H. Fredrickson*

### in The Equilibrium Theory of Inhomogeneous Polymers

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198567295
- eISBN:
- 9780191717956
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567295.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter examines the subject of polymers experiencing external potential fields that vary in space. The statistical mechanical problem is reduced to the solution of Fokker-Planck equations, and ... More

## The master equation

*J. C. Garrison and R. Y. Chiao*

### in Quantum Optics

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198508861
- eISBN:
- 9780191708640
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508861.003.0019
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter examines the evolution of an open system — the sample — with the quantum Liouville equation for the world density operator. The fundamental approximation is that the action of the sample ... More

## Generalized Fokker–Planck equation

*Melvin Lax, Wei Cai, and Min Xu*

### in Random Processes in Physics and Finance

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Thermal noise and Gaussian noise sources combine to create a category of Markovian processes known as Fokker–Planck processes. This chapter discusses Fokker–Planck processes such as ... More

## Large-scale collective motion at finite thermal excitations

*Helmut Hofmann*

### in The Physics of Warm Nuclei: with Analogies to Mesoscopic Systems

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198504016
- eISBN:
- 9780191708480
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504016.003.0011
- Subject:
- Physics, Nuclear and Plasma Physics

This chapter uses a Kramers-Moyal expansion to derive from short time propagators transport equations of Fokker-Planck type for global motion. Because of the inherent non-linearities, the transport ... More

## Stochastic Modeling and Analysis

*Richard M. Murray*

### in Biomolecular Feedback Systems

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161532
- eISBN:
- 9781400850501
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161532.003.0004
- Subject:
- Biology, Biochemistry / Molecular Biology

This chapter explores stochastic behavior in biomolecular systems. It does so by first building on the preliminary discussion of stochastic modeling laid out in Chapter 2. The chapter reviews methods ... More

## Fokker–Planck Equations

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0008
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter deals with deriving Fokker–Planck equations (FPEs) that govern the behaviour of phase space distribution functions (normalised and unnormalised) for boson and fermion systems due to ... More

## STOCHASTIC EQUATIONS FROM A STATISTICAL MECHANICAL VIEWPOINT

*Robert M. Mazo*

### in Brownian Motion: Fluctuations, Dynamics, and Applications

- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.003.0011
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter begins with a heuristic derivation of the Langevin equation using the Mori projection operator as the main tool. The random force and the friction constant are identified. A similar ... More

## IDEAL CHAIN MODELS

*Glenn H. Fredrickson*

### in The Equilibrium Theory of Inhomogeneous Polymers

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198567295
- eISBN:
- 9780191717956
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567295.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter discusses elementary statistical mechanical models of ideal chains, i.e., polymers in isolation and not experiencing interactions with other polymers. Freely jointed and bead-spring ... More

## Stochastic Differential Equations: Langevin, Fokker–Planck Equations

*JEAN ZINN-JUSTIN*

### in Quantum Field Theory and Critical Phenomena

- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses Langevin equations, that is, stochastic differential equations related to diffusion processes, brownian motion, or random walk. From the Langevin equation, the Fokker–Planck ... More

## Classical Probability Theory and Stochastic Processes

*Heinz-Peter Breuer and Francesco Petruccione*

### in The Theory of Open Quantum Systems

- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.01
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental concepts of probability space and Kolmogorov axioms. These ... More

## FLUCTUATIONS

*HARRY SUHL*

### in Relaxation Processes in Micromagnetics

- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780198528029
- eISBN:
- 9780191713545
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528029.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter examines how the magnetization senses random fluctuations. Mathematical descriptions of stochastic processes of physical origin involve one or other of various incarnations of the ... More

## Langevin Equations

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0009
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

In this chapter, Langevin equations (or Ito stochastic differential equations, SDEs) are derived that are equivalent to Fokker–Planck equations for bosons and fermions. The approach involves ... More

## Langevin Field Equations

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0014
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

In this chapter, Langevin field equations (or Ito stochastic field equations, SFEs) are derived that are equivalent to functional Fokker–Planck equations (FFPEs) for bosons and fermions. Phase space ... More

## Phase Space Methods for Degenerate Quantum Gases

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

Recent experimental progress has enabled cold atomic gases to be studied at nanokelvin temperatures, creating new states of matter where quantum degeneracy occurs—Bose–Einstein condensates and ... More

## THE SMOLUCHOWSKI EQUATION

*Robert M. Mazo*

### in Brownian Motion: Fluctuations, Dynamics, and Applications

- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.003.0008
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter discusses the Smoluchowski equation, a configuration space description of a Brownian particle in a field of force. It begins with the Kramers–Klein equation, essentially the ... More

## PATH INTEGRALS AND QUANTIZATION

*Jean Zinn-Justin*

### in Path Integrals in Quantum Mechanics

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0005
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

This chapter constructs path integrals for general Hamiltonians with potentials linear in the velocities, like Hamiltonians of particles in a magnetic field. Two examples are considered: a quantum ... More

## INTERACTING BROWNIAN PARTICLES

*Robert M. Mazo*

### in Brownian Motion: Fluctuations, Dynamics, and Applications

- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.003.0017
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter studies the effect of the concentration of Brownian particles on Brownian motion. It starts from the multi-particle Fokker–Planck equation and the multi-particle Smoluchowski equation. ... More

## Functional Fokker–Planck Equations

*Bryan J. Dalton, John Jeffers, and Stephen M. Barnett*

### in Phase Space Methods for Degenerate Quantum Gases

- Published in print:
- 2014
- Published Online:
- April 2015
- ISBN:
- 9780199562749
- eISBN:
- 9780191747311
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562749.003.0013
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter deals with deriving functional Fokker–Planck equations (FFPEs) that govern the behaviour of phase space distribution functionals (normalised and unnormalised) for boson and fermion ... More

## Fluctuations, Dissipation, and Noise

*Peter W. Milonni*

### in An Introduction to Quantum Optics and Quantum Fluctuations

- Published in print:
- 2019
- Published Online:
- April 2019
- ISBN:
- 9780199215614
- eISBN:
- 9780191868689
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199215614.003.0006
- Subject:
- Physics, Atomic, Laser, and Optical Physics, Particle Physics / Astrophysics / Cosmology

General concepts in the theory of fluctuations and dissipation are reviewed and applied to examples in quantum optics. Brownian motion, Fokker-Planck and Langevin equations, and the Wiener-Khintchine ... More