Tag: Anomalous Diffusion

Experimental Implications of Bochner-Levy-Riesz Diffusion

Post author By R Hilfer
Post date August 18, 2014

R. Hilfer

Fractional Calculus and Applied Analysis 18, 333-341 (2015)
https://doi.org/10.1515/fca-2015-0022

submitted on
Monday, August 18, 2014

Fractional Bochner-Levy-Riesz diffusion arises from ordinary diffusion by replacing the Laplacean with a noninteger power of itself. Bochner- Levy-Riesz diffusion as a mathematical model leads to nonlocal boundary value problems. As a model for physical transport processes it seems to predict phenomena that have yet to be observed in experiment.

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Tags Anomalous Diffusion, fractional Laplacian

diffusion Fractional Calculus

Fractional Diffusion based on Riemann-Liouville Fractional Derivatives

Post author By R Hilfer
Post date October 12, 1999

R. Hilfer

The Journal of Physical Chemistry B 104, 3914-3917 (2000)
DOI: 10.1021/jp9936289

submitted on
Tuesday, October 12, 1999

A fractional diffusion equation based on Riemann−Liouville fractional derivatives is solved exactly. The initial values are given as fractional integrals. The solution is obtained in terms of H-functions. It differs from the known solution of fractional diffusion equations based on fractional integrals. The solution of fractional diffusion based on a Riemann−Liouville fractional time derivative does not admit a probabilistic interpretation in contrast with fractional diffusion based on fractional integrals. While the fractional initial value problem is well defined and the solution finite at all times, its values for t → 0 are divergent.

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Tags Anomalous Diffusion

Fractals Lattice Models Renormalisation Stochastic Processes

On Finitely Ramified Fractals and Their Extensions

Post author By R Hilfer
Post date July 11, 1985

R. Hilfer, A. Blumen

in: Fractals in Physics
edited by: L. Pietronero and E. Tosatti
Elsevier Publishing Co., Amsterdam, 33 (1986)

submitted on
Thursday, July 11, 1985

We construct deterministic fractal lattices using generators with tetrahedral symmetry. From the corresponding master equation we determine the spectral dimension d and prove that d<2. Furthermore we extend our set of fractals (with d dense in [1,2]) by direct multiplication, thus obtaining fractals whose d are dense for all real numbers larger than or equal to 1 .

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Tags Anomalous Diffusion, finitely ramified, Markov chain, post-critically finite, real space renormalization, Sierpinski gasket

Critical phenomena Disordered Systems Fractals Lattice Models Random Walks Renormalisation Stochastic Processes

Renormalisation on Symmetric Fractals

Post author By R Hilfer
Post date July 9, 1984

R. Hilfer, A. Blumen

J.Phys.A: Math. Gen. 17, L783 (1984)
https://doi.org/10.1088/0305-4470/17/14/011

submitted on
Monday, July 9, 1984

We introduce and investigate new classes of Sierpinski-type fractals. We determine their fractal and spectral dimensions using renormalisation procedures and, for particular classes, we give these dimensions in closed form. The spectral dimensions densely fill the interval [1,2], allowing us to choose flexibly models for applications.

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Tags Anomalous Diffusion, finitely ramified, Markov chain, post-critically finite, real space renormalization

Disordered Systems Fractals Lattice Models Renormalisation Stochastic Processes

Renormalisation on Sierpinski-type Fractals

Post author By R Hilfer
Post date April 13, 1984

R. Hilfer, A. Blumen

Journal of Physics A: Mathematical and General 17, L573-L545 (1984)
10.1088/0305-4470/17/10/004

submitted on
Friday, April 13, 1984

We present a family of deterministic fractals which generalise the d-dimensional Sierpinski gaskets and we establish their order of ramification and their fractal and spectral dimensions. Random walks on these fractals are renormalisable and lead to rational, not necessarily polynomial, mappings.

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Tags Anomalous Diffusion, finitely ramified, Markov chain, post-critically finite, real space renormalization, Sierpinski gasket, spectral dimension