Fractional calculus in statistical physics: The case of time fractional diffusion equation

The relationships among intermittency with fractal Waiting Time distribution, Contin- uous Time Random Walk (CTRW) and the emergence of Fractional Calculus (FC) are reviewed. The derivation, in the long-time limit, of Time Fractional Diffusion Equation (TFDE) is shown and compared with the case of normal difusion equation. Emphasis is given to the underlying connections of CTRW with concepts and results from probability theory and stochastic processes: conditional probabilities, the law of total probability, Central and (?evy) Generalized limit theorems, renewal theory. It is shown how the emergence of a well-defined scaling rigorously emerges by imposing the invariance of the probability distribution under a group of self-similarity transformations involving space and time. The physical interpretation of some crucial mathematical passages is explained. In particular, the physical meaning of self-similarity coupled with the long-time limit is explained, having in mind a experimental point of view. Finally, the emergence of FC in complexity is discussed and associated with the ubiquitous generation of short-time transition events in the dynamics of complex systems. These renewal events are associ-ated with the dynamical emergence (birth) and decay (death) of cooperative long-lived structures, thus giving rise to a intermittent birth-death process of cooperation.