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Predicting the response of a system to perturbations is a key challenge inmathematical and natural sciences. Under suitable conditions on the nature ofthe system, of the perturbation, and of the observables of interest, responsetheories allow to construct operators describing the smooth change of theinvariant measure of the system of interest as a function of the smallparameter controlling the intensity of the perturbation. In particular,response theories can be developed both for stochastic and chaoticdeterministic dynamical systems, where in the latter case stricter conditionsimposing some degree of structural stability are required. In this paper weextend previous findings and derive general response formulae describing hown-point correlations are affected by perturbations to the vector flow. We alsoshow how to compute the response of the spectral properties of the system toperturbations. We then apply our results to the seemingly unrelated problem ofcoarse graining in multiscale systems: we find explicit formulae describing thechange in the terms describing parameterisation of the neglected degrees offreedom resulting from applying perturbations to the full system. All the termsenvisioned by the Mori-Zwanzig theory - the deterministic, stochastic, andnon-Markovian terms - are affected at 1st order in the perturbation. Theobtained results provide a more comprehesive understanding of the response ofstatistical mechanical systems to perturbations and contribute to the goal ofconstructing accurate and robust parameterisations and are of potentialrelevance for fields like molecular dynamics, condensed matter, and geophysicalfluid dynamics. We envision possible applications of our general results to thestudy of the response of climate variability to anthropogenic and naturalforcing and to the study of the equivalence of thermostatted statisticalmechanical systems.

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