Anomalous diffusion in view of Einsteinʹs 1905 theory of Brownian motion

Einsteinʹs theory of Brownian motion is revisited in order to formulate a generalized kinetic theory of anomalous diffusion. It is shown that if the assumptions of analyticity and the existence of the second moment of the displacement distribution are relaxed, the fractional derivative naturally appears in the diffusion equation. This is the first demonstration of the physical origin of the fractional derivative, in marked contrast to the usual phenomenological introduction of it. It is also examined if Einsteinʹs approach can be generalized to nonlinear kinetic theory to derive a generalization of the porous-medium-type equation by the appropriate use of the escort distribution.