Aging and rejuvenation with fractional derivatives

We discuss a dynamic procedure that makes fractional derivatives emerge in the time asymptotic limit of non-Poisson processes. We find that two-state fluctuations, with an inverse power-law distribution of waiting times, finite first moment, and divergent second moment, namely, with the power index 𝜇 in the interval 2<𝜇<3, yield a generalized master equation equivalent to the sum of an ordinary Markov contribution and a fractional derivative term. We show that the order of the fractional derivative depends on the age of the process under study. If the system is infinitely old, the order of the fractional derivative, ℴ, is given by ℴ=3−𝜇. A brand new system is characterized by the degree ℴ=𝜇−2. If the system is prepared at time −𝑡𝑎<0 and the observation begins at time 𝑡=0, we derive the following scenario. For times 0<𝑡⪡𝑡𝑎 the system is satisfactorily described by the fractional derivative with ℴ=3−𝜇. Upon time increase the system undergoes a rejuvenation process that in the time limit 𝑡⪢𝑡𝑎 yields ℴ=𝜇−2. The intermediate time regime is probably incompatible with a picture based on fractional derivatives, or, at least, with a mono-order fractional derivative.