FRACTIONAL MASTER-EQUATIONS AND FRACTAL TIME RANDOM-WALKS

Fractional master equations containing fractional time derivatives of order 0<ω≤1 are introduced on the basis of a recent classification of time generators in ergodic theory. It is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks. The corresponding waiting time density ψ(t) is obtained exactly as ψ(t)=(tω-1/C)Eω,ω(-tω/C), where Eω,ω(x) is the generalized Mittag-Leffler function. This waiting time distribution is singular both in the long time as well as in the short time limit. © 1995 The American Physical Society.