Einstein relation on fractal objects
type: Item
instance of: scholarly article
title: Einstein relation on fractal objects
zbMATH Open document ID: 1242.28010
author: Uta Freiberg
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2012-03-19T00:00:00Z
review text: There exists nowadays a rich analytic machinery on fractal objects. In this survey article, the author summarizes some key features of this analytic machinery, focusing on three characteristic notions of dimension: the \textit{Hausdorff dimension} \(d_H\) (encoding \textbf{geometric} features of the object), the \textit{spectral dimension} \(d_S\) (encoding \textbf{analytic} features) and the \textit{walk dimension} \(d_W\) (encoding \textbf{stochastic} features). These three quantities are related, for a broad class of self-similar fractals, by the \textit{Einstein relation} \[ 2d_H = d_S \, d_W. \] The author reviews the basic ingredients in each of these three approaches to fractal analysis. The classical Sierpiński gasket \(SG\) is used throughout as the model example of a PCF (post critically finite) self-similar fractal. In sections 2, 3 and 4, definitions for each of the preceding notions of dimension are given. In the case of \(SG\), equipped with its standard symmetric Dirichlet form, the author provides simple heuristics for the explicit values \(d_H = \tfrac{\log 3}{\log 2}\), \(d_S = \tfrac{\log 9}{\log 5}\) and \(d_W = \tfrac{\log 5}{\log 2}\). For more general nested self-similar fractals \(F\), the author states without proof the following explicit values: \[ d_H = \frac{\log M}{\log (1/r)}, \] \[ d_S = \frac{2 \log M}{\log (M\rho)}, \] and \[ d_W = \frac{\log T}{\log (1/r)}. \] Here \(r<1\) denotes the (common) length scaling factor for the self-similar pieces comprising \(F\), \(M\) denotes the number of such pieces, \(\rho>1\) denotes the energy scaling factor appearing in the definition of the Dirichlet form on \(F\), and \(T\) denotes the mean crossing time of a random walk through the generating graph of \(F\) between initial vertices. (In the case of \(SG\), one has \(r=\tfrac12\), \(M=3\), \(\rho=\tfrac53\) and \(T=5\).) In view of the formulas for these three dimensions, it is easy to see that the Einstein relation in this context is equivale…
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 34B45
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 6015396
zbMATH Keywords: fractals
zbMATH Keywords: self-similarity
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Dirichlet form
zbMATH Keywords: Laplacian
zbMATH Keywords: spectral dimension
zbMATH Keywords: (strong) Markovian process
zbMATH Keywords: random walk
zbMATH Keywords: walk dimension
zbMATH Keywords: Einstein relation
reviewed by: Jeremy T. Tyson
MaRDI profile type: Publication
full work available at URL: dcdsb.2012.17.509
OpenAlex ID: W2321138147
DOI: 10.3934/DCDSB.2012.17.509
description: scientific article; zbMATH DE number 6015396