On Fractional Diffusion and its Relation with Continuous Time Random Walks

Time evolutions whose infinitesimal generator is a fractional time derivative arise generally in the long time limit. Such fractional time evolutions are considered here for random walks. An exact relationship is established between the fractional master equation and a separable continuous time random walk of the Montroll-Weiss type. The waiting time density can be expressed using a generalized Mittag-Leffler function. The first moment of the waiting density does not exist. in: Anomalous Diffusion From Basics to Applications, R. Kutner, A. Pekalski and K. Sznaij-Weron (eds.), Lecture Notes in Physics, vol. 519, Springer, Berlin 1999, pages 77-82 1 Fractional Time Evolution A series of recent investigations [1, 2, 3, 4, 5] has found that, in a suitable long time limit, the macroscopic time evolution T t of a physical observable X(t) is given as a convolution of the form