We discuss the problem of the equivalence between continuous-time random walk ͑CTRW͒ and generalized master equation ͑GME͒. The walker, making instantaneous jumps from one site of the lattice to another, resides in each site for extended times. The sojourn times have a distribution density (t) that is assumed to be an inverse power law with the power index . We assume that the Onsager principle is fulfilled, and we use this assumption to establish a complete equivalence between GME and the Montroll-Weiss CTRW. We prove that this equivalence is confined to the case where (t) is an exponential. We argue that is so because the Montroll-Weiss CTRW, as recently proved by Barkai ͓E. Barkai, Phys. Rev. Lett. 90, 104101 ͑2003͔͒, is nonstationary, thereby implying aging, while the Onsager principle is valid only in the case of fully aged systems. The case of a Poisson distribution of sojourn times is the only one with no aging associated to it, and consequently with no need to establish special initial conditions to fulfill the Onsager principle. We consider the case of a dichotomous fluctuation, and we prove that the Onsager principle is fulfilled for any form of regression to equilibrium provided that the stationary condition holds true. We set the stationary condition on both the CTRW and the GME, thereby creating a condition of total equivalence, regardless of the nature of the waiting-time distribution. As a consequence of this procedure we create a GME that is a bona fide master equation, in spite of being non-Markov. We note that the memory kernel of the GME affords information on the interaction between system of interest and its bath. The Poisson case yields a bath with infinitely fast fluctuations. We argue that departing from the Poisson form has the effect of creating a condition of infinite memory and that these results might be useful to shed light on the problem of how to unravel non-Markov quantum master equations.

We study the transport of information between two complex systems with similar properties. Both systems generate non-Poisson renewal fluctuations with a power-law spectrum 1/f 3−μ , the case μ = 2 corresponding to ideal 1/f noise. We denote by μ S and μ P the power-law indexes of the system of interest S and the perturbing system P , respectively. By adopting a generalized fluctuation-dissipation theorem (FDT) we show that the ideal condition of 1/f noise for both systems corresponds to maximal information transport. We prove that to make the system S respond when μ S < 2 we have to set the condition μ P < 2. In the latter case, if μ P < μ S , the system S inherits the relaxation properties of the perturbing system. In the case where μ P > 2, no response and no information transmission occurs in the long-time limit. We consider two possible generalizations of the fluctuation dissipation theorem and show that both lead to maximal information transport in the condition of 1/f noise.

Nonergodic renewal processes have recently been shown by several authors to be insensitive to periodic perturbations, thereby apparently sanctioning the death of linear response, a building block of nonequilibrium statistical physics. We show that it is possible to go beyond the "death of linear response" and establish a permanent correlation between an external stimulus and the response of a complex network generating nonergodic renewal processes, by taking as stimulus a similar nonergodic process. The ideal condition of 1/f noise corresponds to a singularity that is expected to be relevant in several experimental conditions.

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