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

Linear response theory has developed into a formidable set of tools for studying the forced behaviour of a large variety of systems - including out of equilibrium ones. 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 of a system to perturbations, using one or more other observables of the same system as predictors, and thus bypassing the need to know all the details of the acting forcing. Thus, we break the rigid separation between forcing and response, which is key in linear response theory, and revisit the concept of causality. As a result, the surrogate Green functions one constructs for predicting the response of the observable of interest may have support that is not necessarily limited to the nonnegative time axis. This implies 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 derive general explicit formulas that, in absence of such pathologies, allow one to reconstruct the response of an observable of interest to N independent forcings by using as predictors N other observables. 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.

💡 Research Summary

The paper revisits linear response theory (LRT) and proposes a novel framework that allows the prediction of a system’s response to external perturbations without explicit knowledge of the forcing. Traditionally, LRT treats forcing and response as separate entities; the Green function that maps a forcing to an observable’s response is constructed under the assumption that the time‑profile of the forcing is known. In many real‑world contexts—paleoclimate reconstruction, neuroscience, or spatially extended climate models—the forcing is either poorly known or completely unknown, while multiple observables are available. The authors therefore introduce the concepts of “predictor observables” and “predictand observables”. By exploiting the linearity of Ruelle’s response formula, they show that the response of a target observable can be expressed as a linear combination of the responses of other observables, each convolved with a surrogate Green function that depends only on the cross‑susceptibilities of the observables themselves.

The core technical development proceeds as follows. Starting from the Ruelle response expression, the authors write the causal Green function Γτ for an observable Ψ under a forcing G. Fourier transforming yields the susceptibility χ(ω). They demonstrate that the existence of complex zeros (zeros of χ(ω) in the complex frequency plane) is the decisive factor governing whether the inverse susceptibility χ⁻¹(ω) can be used to construct a well‑behaved surrogate Green function. If χ(ω) has no zeros, χ⁻¹ exists everywhere and the surrogate Green function obtained by inverse Fourier transform is causal (support only for τ≥0). In this case, for N independent forcings and N predictor observables, a set of linear equations can be solved explicitly: the response of the target observable is a sum over convolutions of each predictor’s time series with a kernel K_{Bi}(τ) = ℱ⁻¹