2004/03/19 by Gerardo Aquino, Mauro Bologna, Paolo Grigolini +1 · 1 citation
Mathematics · Physics and Astronomy · #Fractional Differential Equations Solutions #Statistical Mechanics and Entropy #cond-mat.stat-mech #stochastic dynamics and bifurcation
paper · pdf · doi:10.1103/physreve.70.036105
published as Phys. Rev. E 70, 036105 (2004) · 11 pages, 4 figures
arxiv created 2004/03/19 · openalex publication_date 2004/09/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 mu in the interval 2<mu<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-mu . A brand new system is characterized by the degree o=mu-2 . If the system is prepared at time - t(a) <0 and the observation begins at time t=0 , we derive the following scenario. For times 0<t<< t(a) the system is satisfactorily described by the fractional derivative with o=3-mu . Upon time increase the system undergoes a rejuvenation process that in the time limit t>> t(a) yields o=mu-2 . The intermediate time regime is probably incompatible with a picture based on fractional derivatives, or, at least, with a mono-order fractional derivative.