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 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 the 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 first order in the perturbation. The obtained results provide a more comprehensive understanding of the response of statistical mechanical systems to perturbations. They also 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.
Highlights
A fundamental step in the direction of developing a comprehensive response theory can be found in the early work of Kubo (1957) (see Kubo et al (1988)), who studied the impact of imposing weak perturbations to a statistical mechanical system originally at the thermodynamic equilibrium described by the canonical ensemble
The Kubo response theory leads to response formulae that express the change in the expectation value of a given observable Ψ of the system as a perturbative series
The term proportional to ε is given by the sum of n terms, the first one resulting from the linear correction to the measure, which corresponds to what one would naively obtain by applying the standard response theory, and the other n − 1 terms resulting from the linear correction to each of the n − 1 Koopman operators appearing in the definition of the n-point correlation function
Summary
Understanding how a system responds to perturbations is a key challenge in mathematical and natural sciences and has long been the subject of extensive analysis through formal, exper imental, and numerical investigations. It is important to note that the Green’s function itself is constructed as an expectation value of an observable on the unperturbed measure, with the ensuing effect that the unperturbed system contains all the information needed for estimating its response to general forcings This provides the basis for the cornerstone of Kubo’s response theory, the so-called fluctuation-dissipation theorem (FDT), which links forced and free fluctuations in the linear perturbative regime. Axiom A systems featuring— on the average—a contraction in the phase space provide excellent mathematical models for nonequilibrium systems (Gallavotti 2006) In this case, the invariant measure lives on a set with a Hausdorff dimension lower than the number of degrees of freedom of the system and is singular with respect to the Lebesgue measure, as a result of the contraction taking place in the stable manifold (Eckmann and Ruelle 1985). Though, that while the presence of noise smoothens the invariant measure of the system, the weaker the noise, the harder it is for a numerical model to appreciate such smoothness given the finite length numerical simulations and the finite size of the ensemble of performed simulations
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