Anomalous Diffusion in View of Einstein's 1905 Theory of Brownian Motion

TL;DR

This paper revisits Einstein's Brownian motion theory to derive a generalized kinetic framework for anomalous diffusion, revealing the natural emergence of fractional derivatives when certain assumptions are relaxed, and extending to nonlinear kinetics.

Contribution

It provides the first physical derivation of fractional derivatives in diffusion equations and generalizes Einstein's approach to nonlinear kinetic theory using escort distributions.

Findings

Fractional derivatives naturally arise in anomalous diffusion when analyticity assumptions are relaxed.

The approach offers a physical basis for fractional calculus in diffusion processes.

Extension to nonlinear kinetics yields porous-medium-type equations.

Abstract

Einstein's theory of Brownian motion is revisited in order to formulate 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. Furthermore, Einstein's approach is generalized to nonlinear kinetic theory to derive the porous-medium-type equation by the appropriate use of the escort distribution.