Einstein's theory of Brownian motion is revisited in order to formulategeneralized kinetic theory of anomalous diffusion. It is shown that if theassumptions of analyticity and the existence of the second moment of thedisplacement distribution are relaxed, the fractional derivative naturallyappears in the diffusion equation. This is the first demonstration of thephysical origin of the fractional derivative, in marked contrast to the usualphenomenological introduction of it. Furthermore, Einstein's approach isgeneralized to nonlinear kinetic theory to derive the porous-medium-typeequation by the appropriate use of the escort distribution.
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