Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental...

The fractional diffusion equation that is constructed replacing the time derivative with a fractional derivative, (0)D(alpha)(t) f = C(alpha) theta(2) f/theta x(2), where (0)D(alpha)(t) is the Riemann-Liouville derivative operator, is characterized by a probability density that decays with time as t(alpha -1) (alpha < 1) and an initial condition that diverges as t -->0 [R. Hilfer, J. Phys. Chem. B 104, 3914 (2000)]. These seemingly unphysical features have obstructed the application of the fractional diffusion equation. The paper clarifies the meaning of these properties adopting concrete physical interpretations of experimentally verified models: the decay of free-carrier density in a semiconductor with an exponential distribution of traps, and the decay of ion-recombination isothermal luminescence. We conclude that the fractional diffusion equation is a suitable representation of diffusion in disordered media with dissipative processes such as trapping or recombination involving an initial exponential distribution either in the energy or spatial axis. The fractional decay does not consider explicitly the starting excitation and ultrashort time-scale relaxation that forms the initial exponential distribution, and therefore it cannot be extrapolated to t = 0.