A path integral approach to the Langevin equation

TL;DR

This paper develops a path integral framework for the generalized Langevin equation with white and colored noise, clarifying the role of auxiliary fields and deriving correlation functions and the Fokker-Planck equation from first principles.

Contribution

It introduces a first-principles path integral formulation for the generalized Langevin equation, connecting it to correlation functions and the Fokker-Planck equation.

Findings

Derived the path integral and Hamiltonian for the generalized Langevin equation.

Explicitly calculated correlation functions for Markovian and non-Markovian processes.

Provided a simple derivation of the Fokker-Planck equation from the path integral formalism.

Abstract

We study the Langevin equation with both a white noise and a colored noise. We construct the Lagrangian as well as the Hamiltonian for the generalized Langevin equation which leads naturally to a path integral description from first principles. This derivation clarifies the meaning of the additional fields introduced by Martin, Siggia and Rose in their functional formalism. We show that the transition amplitude, in this case, is the generating functional for correlation functions. We work out explicitly the correlation functions for the Markovian process of the Brownian motion of a free particle as well as for that of the non-Markovian process of the Brownian motion of a harmonic oscillator (Uhlenbeck-Ornstein model). The path integral description also leads to a simple derivation of the Fokker-Planck equation for the generalized Langevin equation.