Fractional diffusion equation for transport phenomena in random media

A differential equation for diffusion in isotropic and homogeneous fractal structures is derived within the context of fractional calculus. It generalizes the fractional diffusion equation valid in Euclidean systems. The asymptotic behavior of the probability density function is obtained exactly and coincides with the accepted asymptotic form obtained using scaling argument and exact enumeration calculations on large percolation clusters at criticality. The asymptotic frequency dependence of the scattering function is derived exactly from the present approach, which can be studied by X-ray and neutron scattering experiments on fractals.