Response Formulae for $n$-point Correlations in Statistical Mechanical Systems and Application to a Problem of Coarse Graining

Abstract

Predicting the response of a system to perturbations is a key challenge in mathematical and natural sciences. Under suitable conditions on the nature of the system, of the perturbation, and of the observables of interest, response theories allow to construct operators describing the smooth change of the invariant measure of the system of interest as a function of the small parameter controlling the intensity of the perturbation. In particular, response theories can be developed both for stochastic and chaotic deterministic dynamical systems, where in the latter case stricter conditions imposing some degree of structural stability are required. In this paper we extend previous findings and derive general response formulae describing how n-point correlations are affected by perturbations to the vector flow. We also show how to compute the response of the spectral properties of the system to perturbations. We then apply our results to the seemingly unrelated problem of coarse graining in multiscale systems: we find explicit formulae describing the change in the terms describing parameterisation of the neglected degrees of freedom resulting from applying perturbations to the full system. All the terms envisioned by the Mori-Zwanzig theory - the deterministic, stochastic, and non-Markovian terms - are affected at 1st order in the perturbation. The obtained results provide a more comprehesive understanding of the response of statistical mechanical systems to perturbations and contribute to the goal of constructing accurate and robust parameterisations and are of potential relevance for fields like molecular dynamics, condensed matter, and geophysical fluid dynamics. We envision possible applications of our general results to the study of the response of climate variability to anthropogenic and natural forcing and to the study of the equivalence of thermostatted statistical mechanical systems.

17 pages; improved discussion and reference to the impact of the perturbations on the spectral properties of the system

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• Response and Sensitivity Using Markov Chains
• Quadratic response of random and deterministic dynamical systems
• Optimal linear responses for Markov chains and stochastically perturbed dynamical systems
• Analysis of a bistable climate toy model with physics-based machine learning methods
• Predictors and Predictands of Linear Response in Spatially Extended Systems
• Bridging the Gap between Koopmanism and Response Theory: Using Natural Variability to Predict Forced Response
• A Statistical Mechanical Approach for the Parametrization of the Coupling in a Fast-Slow System