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Fractional master equations and fractal time random walks

R. Hilfer and L. Anton
International School for Advanced Studies, Via Beirut 2-4, 34013 Triest, Italy (R. Hilfer, L. Anton)
Institut für Physik, Universität Mainz, 55099 Mainz, Germany (R. Hilfer)
 
Current Address:
Institute of Physics, University of Oslo, P.O.Box 1048, 0316 Oslo, Norway (R. Hilfer)
Abstract.

Fractional master equations containing fractional time derivatives of order 0<\omega\leq 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 \psi(t) is obtained exactly as \psi(t)=(t^{{\omega-1}}/C)E_{{\omega,\omega}}(-t^{\omega}/C) where E_{{\omega,\omega}}(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.

Key words and phrases:
continuous time random walks, fractional master equation
 PACS: 05.40.+j,05.60.+w,02.50+s