Langevin equations with non-Gaussian thermal noise: Valid but superfluous

TL;DR

The paper investigates whether a generalized Langevin equation (GLE) with non-Gaussian thermal noise can consistently reproduce fluctuation relations, specifically the Jarzynski equality (JE), for finite-time processes without assuming ergodicity. Focusing on a Brownian oscillator with a rectangular stiffness protocol, it solves the GLE with arbitrary noise statistics constrained by the fluctuation-dissipation relation and analyzes the average exponential work $\langle e^{-W/T}\rangle$. It shows that JE holds for all noise statistics up to order $\tau^7$, while at order $\tau^8$ JE demands Gaussian noise, revealing that non-Gaussian noise affects only higher-order noise statistics and related nonlinear observables. Consequently, the GLE with non-Gaussian noise is informative mainly for linear or quadratic in-noise properties and becomes redundant for quantities sensitive to higher-order noise cumulants, unless a more complete (higher-order) model of dissipation is employed. These results deepen the understanding of when non-Gaussian noise can be meaningfully incorporated into non-Markovian Langevin descriptions and fluctuation theorem analyses.

Abstract

We discuss the statistics of additive thermal (internal) noise in systems governed by the generalized Langevin equation with linear dissipation. To assess the equation's validity, it is common to assume that the system is ergodic and to verify that solutions approach correct equilibrium values at asymptotically long times. In this paper, we instead consider the consistency of the generalized Langevin equation with the Jarzynski equality at finite times and do not assume the system's ergodicity. Specifically, we consider a classical Brownian oscillator whose initial stiffness, or frequency, is perturbed by a rectangular pulse of duration $τ$. We find that the Jarzynski equality is satisfied unconditionally only up to the seventh order in $τ$; in higher orders, the Jarzynski equality holds if and only if the noise is Gaussian. These results imply that, unless it is exact, the Langevin equation can only be used to evaluate properties that are linear or quadratic in noise and its derivatives. Such properties are insensitive to the noise statistics, so the Langevin equation with linear dissipation and non-Gaussian noise (though not inconsistent by itself) is superfluous.