Anomalous diffusion in view of Einstein's 1905 theory of Brownian motion
By S. Abe, Stefan Thurner 4. 2004
140 citations recorded by the source corpus.
Einstein's 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 Einstein's 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.
Subfields: Modeling and Simulation, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/47c112355e8802795e4aa5e4bd84169511e4e30d
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In Statistical and Nonlinear Physics
Citations in the corpus
- A classification of structural dissipations for evolution operators (2016, background): “Anomalous diffusion problems have been deeply investigated in literature (see, for instance, [2]).”
- The Porous Medium Equation: Mathematical Theory (2006, unknown): “Such a deviation is not just a particular case, since we will show in Chapter 18 that the ZKB solutions represent the standard asymptotic behaviour of finite mass solutions both in size and spread rate (see Chapter 18); we conclude that the PME is an example of anomalous diffusion in the sense descr”