Relevance of sampling schemes in light of Ruelle's linear response theory

TL;DR

This paper explores how different sampling schemes affect the application of Ruelle's linear response theory to non-equilibrium steady states, emphasizing the importance of choosing the correct time horizon for accurate response calculation.

Contribution

It introduces a functional decomposition approach for space-time forcings and clarifies the impact of sampling schemes on applying Ruelle's response formula, including periodic perturbations.

Findings

Only specific sampling schemes yield Ruelle's formula.

Periodic perturbations allow flexible sampling schemes.

Critical review of Reick's proposed response formula.

Abstract

We reconsider the theory of the linear response of non-equilibrium steady states to perturbations. We first show that by using a general functional decomposition for space-time dependent forcings, we can define elementary susceptibilities that allow to construct the response of the system to general perturbations. Starting from the definition of SRB measure, we then study the consequence of taking different sampling schemes for analysing the response of the system. We show that only a specific choice of the time horizon for evaluating the response of the system to a general time-dependent perturbation allows to obtain the formula first presented by Ruelle. We also discuss the special case of periodic perturbations, showing that when they are taken into consideration the sampling can be fine-tuned to make the definition of the correct time horizon immaterial. Finally, we discuss the implications of our results in terms of strategies for analyzing the outputs of numerical experiments by providing a critical review of a formula proposed by Reick.