Revising and Extending the Linear Response Theory for Statistical Mechanical Systems: Evaluating Observables as Predictors and Predictands

Valerio Lucarini

Published 2018-06-11Version 1

Linear response theory has developed into a formidable set of tools for studying the response of a large variety of systems - including out of equilibrium ones - to perturbations. Mathematically rigorous derivations of linear response theory have been provided for systems obeying stochastic dynamics as well as for deterministic chaotic systems. In this paper we provide a new angle on the problem, by studying under which conditions it is possible to perform predictions on the response of a given observable to perturbations, using one or more other observables as predictors. By taking an approach that breaks the separation between forcing and response, which is key in linear response theory, the surrogate Green functions one constructs have support that is not necessarily limited to the nonnegative time axis. In other terms, we show that not all observables are equally good as predictands when a given forcing is applied, as result of the properties of their corresponding susceptibility. In particular problems emerge from the presence of complex zeros. We propose a new general method for reconstructing the response of an observable to the perturbation due to the application of N independent forcings by using as predictors N other observables by defining a new class of surrogate susceptibilities. We provide a thorough test of the theory and of the possible pathologies by using numerical simulations of the paradigmatic Lorenz '96 model. Our results are potentially relevant for problems like the reconstruction of climatic data from proxy data and provide a possible mathematical basis for the theory of the so-called emergent constraints in climate, as well as, possibly for the study of linear feedbacks in complex systems. It might also serve for reconstructing the response to forcings of a spatially extended system in a given location looking at the response in a separate location.