Phenomenological approach to non-linear Langevin equations

TL;DR

This paper develops a phenomenological framework for constructing non-linear Langevin equations that accurately describe fluctuations in complex systems, aligning with microscopic models under certain conditions.

Contribution

It introduces a method to determine the properties of the random force in non-linear Langevin equations based on weak coupling and symmetry considerations.

Findings

The phenomenological approach agrees with microscopic models for Brownian motion and diode systems.

Weak coupling allows complete specification of the random force's statistical properties.

Symmetry relations are derived for odd variables under time-reversal.

Abstract

In this paper we address the problem of consistently construct Langevin equations to describe fluctuations in non-linear systems. Detailed balance severely restricts the choice of the random force, but we prove that this property together with the macroscopic knowledge of the system is not enough to determine all the properties of the random force. If the cause of the fluctuations is weakly coupled to the fluctuating variable, then the statistical properties of the random force can be completely specified. For variables odd under time-reversal, microscopic reversibility and weak coupling impose symmetry relations on the variable-dependent Onsager coefficients. We then analyze the fluctuations in two cases: Brownian motion in position space and an asymmetric diode, for which the analysis based in the master equation approach is known. We find that, to the order of validity of the Langevin equation proposed here, the phenomenological theory is in agreement with the results predicted by more microscopic models.