Evaluation of the probability current in the stochastic path integral formalism

TL;DR

This paper reviews and derives a comprehensive expression for the probability current in stochastic path integral formalism, clarifying its role in non-equilibrium systems with complex drift and diffusion characteristics.

Contribution

It provides a self-contained derivation of the probability current within the Onsager-Machlup framework for non-equilibrium processes, including explicit evaluation for complex systems.

Findings

Derived formulas for probability current with non-constant coefficients

Explicit evaluation in an Ornstein-Uhlenbeck process with shear flow

Clarified the concept of probability current in path integral approach

Abstract

The probability current is a vital quantity in the Fokker-Planck description of stochastic processes. It characterizes non-equilibrium stationary states and appears in linear response calculations. We recover and review the probability current in the Onsager-Machlup functional approach to Markov processes by deriving a self-contained expression in general non-equilibrium fluctuation-dissipation relations using field theoretical methods. The derived formulas hold for non-constant drift and diffusion tensors and are explicitly evaluated in an Ornstein-Uhlenbeck process with non-reciprocal interactions specified as a harmonically bound particle in shear flow. Our work clarifies the concept of the probability current -- familiar from the Fokker-Planck equation -- in the path integral approach.