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

TL;DR

This paper resolves the ambiguities in discretizing Langevin equations with multiplicative noise for systems with known equilibrium distributions, ensuring consistent physical interpretation especially under reversibility conditions.

Contribution

It demonstrates that ambiguities in multiplicative noise Langevin equations are uniquely resolved using equilibrium and reversibility principles, beyond standard discretization conventions.

Findings

Ambiguities depend on discretization conventions but are resolved by equilibrium conditions.

Reversibility at a fundamental level ensures unique interpretation of multiplicative noise equations.

Application to effective theories of hot non-Abelian plasmas illustrates practical relevance.

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 d^2q/dt^2 + e^2(q) dq/dt = - grad V(q) + e^{-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 Stratanovich or Ito. [One application of Langevin equations with multiplicative noise is to certain effective theories for hot, non-Abelian plasmas.]