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Title: Langevin equations with multiplicative noise: Resolution of time discretization ambiguities for equilibrium systems

Abstract

A Langevin equation with multiplicative noise is an equation schematically of the form dq/dt=-F(q)+e(q){xi}, where e(q){xi} is Gaussian white noise whose amplitude e(q) depends on q itself. Such equations are ambiguous, and depend on the details of one's convention for discretizing time when solving them. I show that these ambiguities are uniquely resolved if the system has a known equilibrium distribution exp[-V(q)/T] and if, at some more fundamental level, the physics of the system is reversible. I also discuss a simple example where this happens, which is the small frequency limit of Newton's equation qe+e{sup 2}(q)q=-{nabla}V(q)+e{sup -1}(q){xi} with noise and a q-dependent damping term. The resolution does not correspond to simply interpreting naive continuum equations in a standard convention, such as Stratonovich or Ito. (c) 2000 The American Physical Society.

Authors:
 [1]
  1. Department of Physics, University of Virginia, Charlottesville, Virginia 22901 (United States)
Publication Date:
OSTI Identifier:
20216770
Resource Type:
Journal Article
Journal Name:
Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Additional Journal Information:
Journal Volume: 61; Journal Issue: 6; Other Information: PBD: Jun 2000; Journal ID: ISSN 1063-651X
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; LANGEVIN EQUATION; STOCHASTIC PROCESSES; GAUSSIAN PROCESSES; THEORETICAL DATA

Citation Formats

Arnold, Peter. Langevin equations with multiplicative noise: Resolution of time discretization ambiguities for equilibrium systems. United States: N. p., 2000. Web. doi:10.1103/PhysRevE.61.6091.
Arnold, Peter. Langevin equations with multiplicative noise: Resolution of time discretization ambiguities for equilibrium systems. United States. doi:10.1103/PhysRevE.61.6091.
Arnold, Peter. Thu . "Langevin equations with multiplicative noise: Resolution of time discretization ambiguities for equilibrium systems". United States. doi:10.1103/PhysRevE.61.6091.
@article{osti_20216770,
title = {Langevin equations with multiplicative noise: Resolution of time discretization ambiguities for equilibrium systems},
author = {Arnold, Peter},
abstractNote = {A Langevin equation with multiplicative noise is an equation schematically of the form dq/dt=-F(q)+e(q){xi}, where e(q){xi} is Gaussian white noise whose amplitude e(q) depends on q itself. Such equations are ambiguous, and depend on the details of one's convention for discretizing time when solving them. I show that these ambiguities are uniquely resolved if the system has a known equilibrium distribution exp[-V(q)/T] and if, at some more fundamental level, the physics of the system is reversible. I also discuss a simple example where this happens, which is the small frequency limit of Newton's equation qe+e{sup 2}(q)q=-{nabla}V(q)+e{sup -1}(q){xi} with noise and a q-dependent damping term. The resolution does not correspond to simply interpreting naive continuum equations in a standard convention, such as Stratonovich or Ito. (c) 2000 The American Physical Society.},
doi = {10.1103/PhysRevE.61.6091},
journal = {Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},
issn = {1063-651X},
number = 6,
volume = 61,
place = {United States},
year = {2000},
month = {6}
}