Life and Death of Stationary Linear Response in Anomalous Continuous Time Random Walk Dynamics

Abstract

Linear theory of stationary response in systems at thermal equilibrium requires to find equilibrium correlation function of unperturbed responding system. Studies of the response of the systems exhibiting anomalously slow dynamics are often based on the continuous time random walk description (CTRW) with divergent mean waiting times. The bulk of the literature on anomalous response contains linear response functions like one by Cole-Cole calculated from such a CTRW theory and applied to systems at thermal equilibrium. Here we show within a fairly simple and general model that for the systems with divergent mean waiting times the stationary response at thermal equilibrium is absent, in accordance with some recent studies. The absence of such stationary response (or dying to zero non-stationary response in aging experiments) would confirm CTRW with divergent mean waiting times as underlying physical relaxation mechanism, but reject it otherwise. We show that the absence of stationary response is closely related to the breaking of ergodicity of the corresponding dynamical variable. As an important new result, we derive a generalized Cole-Cole response within ergodic CTRW dynamics with finite waiting time. Moreover, we provide a physically reasonable explanation of the origin and wide presence of 1/f noise in condensed matter for ergodic dynamics close to normal, rather than strongly deviating.