Journal of Statistical Physics

, Volume 172, Issue 6, pp 1479–1498 | Cite as

On the Validity of Linear Response Theory in High-Dimensional Deterministic Dynamical Systems

  • Caroline L. WormellEmail author
  • Georg A. Gottwald


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.


Linear response theory Stochastic limit systems Statistical limit theorems Weak coupling limit 



GAG is partially supported by ARC, Grant DP180101385. CW is supported by an Australian Government Research Training Program (RTP) Scholarship.


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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