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

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

2004-11-25

arxiv.org/abs/cond-mat/0411645v1

Physics

Condensed Matter

Statistical Mechanics

7 pages, no figures

Scientific paper

10.1016/j.physa.2005.03.035

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.

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