A handy fluctuation-dissipation relation to approach generic noisy systems and chaotic dynamics

TL;DR

This paper presents a generalized fluctuation-dissipation relation applicable to a wide range of stochastic systems, including non-Gaussian and chaotic dynamics, without requiring explicit stationary distributions, with potential applications in geophysics and climate modeling.

Contribution

It introduces a novel, broadly applicable FDR that works for non-equilibrium, non-Gaussian, and chaotic systems without needing stationary probability densities.

Findings

Validates the formula with numerical simulations for non-Gaussian noise

Reproduces response functions in chaotic Lorenz '63 model

Applicable to systems with multiplicative noise and non-Gaussian distributions

Abstract

We introduce a general formulation of the fluctuation-dissipation relations (FDR) holding also in far-from-equilibrium stochastic dynamics. A great advantage of this version of the FDR is that it does not require the explicit knowledge of the stationary probability density function. Our formula applies to Markov stochastic systems with generic noise distributions: when the noise is additive and Gaussian, the relation reduces to those known in the literature; for multiplicative and non-Gaussian distributions (e.g. Cauchy noise) it provides exact results in agreement with numerical simulations. Our formula allows us to reproduce, in a suitable small-noise limit, the response functions of deterministic, strongly non-linear dynamical models, even in the presence of chaotic behavior: this could have important practical applications in several contexts, including geophysics and climate. As a case of study, we consider the Lorenz '63 model, which is paradigmatic for the chaotic properties of deterministic dynamical systems.