Einstein relation on fractal objects
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 equivalent to the identity \[ T = \rho \, M, \] that is, ``time\(=\)resistance\(\times\)mass''. In section 5, the author sketches an elementary proof of the identity \(T=\rho\,M\) via the theory of Markov chains on finite graphs. The paper is effectively organized around a single main message and does an effective job of communicating that message. However, this reviewer found the discussion to be occasionally unclear. In particular, the precise context is not always easy to ascertain: the discussion moves back and forth without notice from the particular case of the Sierpiński gasket to the more general case of PCF fractals. Nevertheless, the paper succeeds overall to present in a nontechnical and expository fashion some interesting features of modern fractal analysis.
scientific article; zbMATH DE number 6015396