The fractional diffusion equation that was constructed replaced the time derivative with a fractional derivative. This was characterized by a probability density that decayed with time, and an initial condition that diverged as t tended to zero. These seemingly unphysical features had obstructed the application of the fractional diffusion equation. This work clarified the meaning of these properties, and adopted 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. It was concluded that the fractional diffusion equation was 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 did not consider explicitly the starting excitation and ultra-short time-scale relaxation that formed the initial exponential distribution, and therefore it could not be extrapolated to t = 0.

Interpretation of a Fractional Diffusion Equation with Non-Conserved Probability Density in Terms of Experimental Systems with Trapping or Recombination. J.Bisquert: Physical Review E, 2005, 72[1-1], 011109