Relevance of sampling schemes in light of Ruelles linear response theory

We reconsider the theory of the linear response of non-equilibrium steady states to perturbations. We first show that by using a general functional decomposition for space-time dependent forcings, we can define elementary susceptibilities that allow to construct the response of the system to general perturbations. Starting from the definition of SRB measure, we then study the consequence of taking different sampling schemes for analysing the response of the system. We show that only a specific choice of the time horizon for evaluating the response of the system to a general time-dependent perturbation allows to obtain the formula first presented by Ruelle. We also discuss the special case of periodic perturbations, showing that when they are taken into consideration the sampling can be fine-tuned to make the definition of the correct time horizon immaterial. Finally, we discuss the implications of our results in terms of strategies for analyzing the outputs of numerical experiments by providing a critical review of a formula proposed by Reick.

💡 Research Summary

The paper revisits the linear response theory for non‑equilibrium steady states (NESS) originally formulated by Ruelle, with a focus on how the choice of sampling scheme influences the resulting response formulas. Starting from the definition of the Sinai‑Ruelle‑Bowen (SRB) invariant measure, the authors emphasize that physical observables are obtained as long‑time averages over trajectories whose initial conditions are absolutely continuous with respect to Lebesgue. This SRB measure underlies all subsequent derivations.

First, the authors treat the simplest case of a separable perturbation X(t,x)=φ(t)χ(x). By expanding the perturbed dynamics around the unperturbed map f, they recover Ruelle’s classic linear response expression

 δ_T ρ(A)≈∑_{j∈ℤ} G_A(j) φ(T−j)

with the causal Green function G_A(j)=θ(j)∫ρ χ D(A∘f^j). In the frequency domain this yields a susceptibility κ_A(ω)=∑_{j≥0}e^{iωj}∫ρ χ D(A∘f^j), analytic in the upper half‑plane and obeying Kramers‑Kronig relations.

To treat arbitrary space‑time forcings, the authors introduce a Schauder basis {φ_r(t)} for temporal functions and {ψ_s(x)} for spatial functions. Any perturbation can be expanded as X(t,x)=∑{r,s}a{r,s}φ_r(t)ψ_s(x). Because each basis element is separable, the linear response follows by superposition:

 δ_ω ρ(A)≈∑{r,s}a{r,s} \hat φ_r(ω) κ_{s,A}(ω)

where κ_{s,A}(ω) is the elementary susceptibility associated with the spatial pattern ψ_s. Transforming back to the time domain gives the compact formula

 δ_T ρ(A)≈∫ρ ∑_{j≥0}X(T−j,x) D(A∘f^j)(x).

The central methodological issue concerns the definition of the “time horizon” used when averaging observables. Two natural schemes are identified: (1) a backward‑looking horizon that includes all past times up to the present (the approach Ruelle implicitly adopts), and (2) a forward‑looking horizon that starts at the present and averages into the future. By performing a first‑order expansion of the perturbed trajectory and inserting it into the definition of the perturbed SRB average, the authors show that scheme (2) eliminates the explicit time dependence of the forcing after the ergodic limit is taken; the response becomes time‑independent, contradicting Ruelle’s formula. In contrast, scheme (1) preserves the convolution structure between the causal Green function and the forcing, reproducing the exact Ruelle result.

When the forcing is periodic, X(t+τ,x)=X(t,x), the situation changes. If the sampling interval matches the period τ and the sub‑sampling does not alter the invariant measure (i.e., the SRB measure is the same for each phase), the averaged forcing 1/τ∑_{n=1}^τ X(t+n,x) replaces the original X in the response expression. Under this condition the response again takes the Ruelle form, but with the periodic average. The authors point out that for a pure sinusoidal forcing the average vanishes, leading to a zero response, which illustrates how a careless choice of horizon can inadvertently filter out the signal.

Finally, the paper critically examines a formula proposed by Reick for analyzing numerical experiments. Reick’s expression is based on the forward‑looking horizon and is convenient for simulations with periodic forcings, but it fails to reproduce the correct linear response for generic, non‑periodic perturbations. The authors delineate the precise circumstances under which Reick’s formula is valid and argue that, for most climate‑model or geophysical‑model applications, the backward‑looking horizon (Ruelle’s original prescription) should be employed to avoid systematic biases.

In summary, the work clarifies that the correct linear response of a NESS to arbitrary space‑time perturbations is obtained only when the sampling scheme respects the causal structure inherent in Ruelle’s theory. It provides a rigorous bridge between abstract response theory and practical numerical experimentation, offering concrete guidance for researchers in climate science, statistical physics, and related fields who wish to compute susceptibilities and Green functions with confidence.