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Quantum Field Theory and Critical Phenomena$
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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

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Stochastic Differential Equations: Langevin, Fokker–Planck Equations

Stochastic Differential Equations: Langevin, Fokker–Planck Equations

Chapter:
(p.60) 4 STOCHASTIC DIFFERENTIAL EQUATIONS: LANGEVIN, FOKKER–PLANCK EQUATIONS
Source:
Quantum Field Theory and Critical Phenomena
Author(s):

JEAN ZINN-JUSTIN

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198509233.003.0004

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

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