Jump to ContentJump to Main Navigation
Quantum Field Theory and Critical Phenomena$
Users without a subscription are not able to see the full content.

Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 24 January 2021

Stochastic Differential Equations: Langevin, Fokker–Planck Equations

Stochastic Differential Equations: Langevin, Fokker–Planck Equations

Quantum Field Theory and Critical Phenomena


Oxford University Press

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 (FP) equation for the probability distribution of the stochastic variables is derived. The FP equation has a form analogous to the equation for the statistical operator in a magnetic field studied in Section 3.2. The chapter shows that averaged observables can also be calculated from path integrals, whose integrands define automatically positive measures. In some cases, like brownian motion on Riemannian manifolds, difficulties appear in the precise definition of stochastic equations, quite similar to the quantization problem encountered in quantum mechanics. Time discretization provides a solution to the problem. This chapter is also meant to serve as an introduction to Chapters 17 and 36 in which stochastic quantization and critical dynamics are discussed.

Keywords:   Langevin equation, Fokker–Planck equations, path integrals, brownian motion, Riemannian manifolds

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .