Category: Renormalisation

Fluctuation-Dissipation on Fractals: A Probabilistic Approach

Post author By R Hilfer
Post date March 31, 1987

R. Hilfer, A. Blumen

in: Time Dependent Effects in Disordered Materials
edited by: R.Pynn and T. Riste
Plenum Press, New York, 217 (1987)

submitted on
Tuesday, March 31, 1987

The analogies between the diffusion problem and the resistor network problem as witnessed by the Einstein relation have been very important for analytical and numerical investigations of linear problems in disordered geometries (e.g. percolating clusters). This raises the question whether the resistor problem can be identified in a purely probabilistic context. An affirmative answer has recently been given and it was shown that the Einstein relation follows from a simple probabilistic argument. Here we present the results of a more general treatment.

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Tags finitely ramified, Markov chain

Disordered Systems Fractals Lattice Models Renormalisation Stochastic Processes

Renormalisation Group Approach in the Theory of Disordered Systems

Post author By R Hilfer
Post date July 23, 1986

R. Hilfer

Verlag Harri Deutsch, Frankfurt, 1986
ISBN-10: 3871449792, ISBN-13: 978-3871449796

submitted on
Wednesday, July 23, 1986

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Tags order parameter, real space renormalization

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