Anomalous Diffusion in View of Einstein's 1905 Theory of Brownian Motion
arXiv:cond-mat/0411645 · doi:10.1016/j.physa.2005.03.035
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.
arXiv comment7 pages, no figures
References in corpus (6)
- Anomalous diffusion in the presence of external forces: exact time-dependent solutions and entropy
- On Generalized Fractional Kinetic Equations
- Geometry of escort distributions
- Fractional Calculus as a Macroscopic Manifestation of Randomness
- Kinetical Foundations of Non Conventional Statistics
- Anomalous diffusion on dynamical networks: A model for interacting epithelial cell migration
Cited by in corpus (9)
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- Variational principle for fractional kinetics and the Lévy Ansatz
- Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type
- Occupation times of long-range exclusion and connections to KPZ class exponents