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

TL;DR

This paper examines the validity of the generalized Langevin equation with non-Gaussian noise, revealing it is only accurate for linear or quadratic noise properties and is otherwise superfluous.

Contribution

It demonstrates that the Langevin equation with non-Gaussian noise is only valid for certain properties and is superfluous for others, challenging common assumptions.

Findings

Jarzynski equality holds up to seventh order in pulse duration.

Langevin equation is only accurate for linear or quadratic noise properties.

Non-Gaussian noise makes the Langevin equation superfluous for higher-order properties.

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 $\tau$. We find that the Jarzynski equality is satisfied unconditionally only up to the seventh order in $\tau$; 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.