Categories
Disordered Systems Lattice Models Renormalisation Stochastic Processes

Fluctuation-Dissipation on Fractals: A Probabilistic Approach

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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Categories
Disordered Systems Fractals Lattice Models Renormalisation Stochastic Processes

Renormalisation Group Approach in the Theory of Disordered Systems

R. Hilfer

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

submitted on
Wednesday, July 23, 1986



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Categories
Fractals Lattice Models Renormalisation Stochastic Processes

On Finitely Ramified Fractals and Their Extensions

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

Renormalisation on Symmetric Fractals

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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Categories
Disordered Systems Fractals Lattice Models Renormalisation Stochastic Processes

Renormalisation on Sierpinski-type Fractals

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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