On the validity of linear response theory in high-dimensional deterministic dynamical systems

TL;DR

This paper demonstrates that linear response theory can be valid in high-dimensional deterministic systems with many unresolved chaotic degrees of freedom, supported by a theoretical proof and numerical simulations.

Contribution

It provides a proof of concept showing linear response theory's validity in high-dimensional systems where individual components violate the theory.

Findings

Linear response theory holds in high-dimensional systems with many unresolved chaotic degrees of freedom.

Numerical simulations support the theoretical conditions for linear response validity.

The study introduces a model with weakly coupled resolved and unresolved degrees of freedom.

Abstract

This theoretical work considers the following conundrum: linear response theory is successfully used by scientists in numerous fields, but mathematicians have shown that typical low-dimensional dynamical systems violate the theory's assumptions. Here we provide a proof of concept for the validity of linear response theory in high-dimensional deterministic systems for large-scale observables. We introduce an exemplary model in which observables of resolved degrees of freedom are weakly coupled to a large, inhomogeneous collection of unresolved chaotic degrees of freedom. By employing statistical limit laws we give conditions under which such systems obey linear response theory even if all the degrees of freedom individually violate linear response. We corroborate our result with numerical simulations.