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<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 ψt is obtained exactly as
ψt=tω-1/CEω,ω-tω/C where
Eω,ωx 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
