Leading order response of statistical averages of a dynamical system to small stochastic perturbations

TL;DR

This paper develops a fluctuation-response theory to predict how statistical averages of dynamical systems respond to small stochastic perturbations, extending classical fluctuation-dissipation concepts to stochastic systems.

Contribution

It introduces a computational framework for leading order response formulas applicable to both deterministic and stochastic systems under stochastic perturbations.

Findings

Effective response formulas for stochastic perturbations validated on Lorenz 96 system.

Response formulas work in both deterministic and stochastic unperturbed regimes.

Numerical tests show good agreement with theoretical predictions.

Abstract

The classical fluctuation-dissipation theorem predicts the average response of a dynamical system to an external deterministic perturbation via time-lagged statistical correlation functions of the corresponding unperturbed system. In this work we develop a fluctuation-response theory and test a computational framework for the leading order response of statistical averages of a deterministic or stochastic dynamical system to an external stochastic perturbation. In the case of a stochastic unperturbed dynamical system, we compute the leading order fluctuation-response formulas for two different cases: when the existing stochastic term is perturbed, and when a new, statistically independent, stochastic perturbation is introduced. We numerically investigate the effectiveness of the new response formulas for an appropriately rescaled Lorenz 96 system, in both the deterministic and stochastic unperturbed dynamical regimes.