Abstract/Summary

Linear response theory, originally formulated for studying how near-equilibrium statistical mechanical systems respond to small perturbations, has developed into a formidable set of tools for investigating the forced behaviour of a large variety of systems, including non-equilibrium ones. 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. We study under which conditions it is possible to perform predictions of the response of a given observable of a forced system, using, as predictors, the response of one or more different observables of the same system. This allows us to bypass the need to know all the details of the acting perturbation. Thus, we break the rigid separation between forcing and response, which is key in linear response theory, and revisit the concept of causality. We find that that not all observables are equally good as predictors when a given forcing is applied. In fact, the surrogate Green function one constructs for predicting the response of an observable of interest using a “bad” observable as predictor has support that is not limited to the nonnegative time axis. We explain the mathematical reasons behind the fact that an observable is an inefficient predictor. We derive general explicit formulas that, in absence of such pathologies, allow one to reconstruct the response of an observable of interest to N independent external forcings by using as predictors N other observables, with N≥1 . 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 data from proxy signals, like in the case of paleoclimate, and, in general, the analysis of signals and feedbacks in complex systems where our knowledge on the system is limited, as in neurosciences. Our technique might also be useful for reconstructing the response to forcings of a spatially extended system in a given location by looking at the response in a separate location.