Formal Fluctuation-Response Relations for Non-Stationary Systems: The Dynamic Conjugate Variable

Abstract

Fluctuation-response relations (FRRs) connect the linear response of a system to an external perturbation with properties of spontaneous fluctuations in the unperturbed system. We provide a simple derivation of FRRs for non-stationary dynamics following an almost standard way based on introducing a variable conjugated to perturbation (a dynamic conjugate). The derivation relies solely on the Markovianity of the underlying dynamics, with the only additional assumption that the transition PDF of the process is differentiable with respect to the strength of a constant, time- independent perturbation. The structure of this conjugate variable is however unusual: it is a two-time one, and reduces to a usual single-time conjugate in stationary situations. We show how several known results follow from this approach.