Statistical mechanics and fractals
type: Item
instance of: scholarly article
title: Statistical mechanics and fractals
zbMATH Open document ID: 0787.60119
author: R. L. Dobrushin
author: Shigeo Kusuoka
published in: Lecture Notes in Mathematics
publication date: 1994-01-17T00:00:00Z
review text: The book consists of two lecture notes given respectively by R. L. Dobrushin and S. Kusuoka during the special year (1988-1989) in Probability and Statistics at Nankai Institute of Mathematics. In Part I, entitled ``On the way to the mathematical foundations of statistical mechanics'', Dobrushin begins an introduction with his personal experience: The original reason why he got interested in the blend of mathematics and physics, the estrangement situation of mathematics from physics at the end of forties and the long way to arrive the modern mathematical physics. The aim of his lectures is to present a mathematical exposition on the foundations of classical statistical mechanics. He introduces a program to construct the mathematical theory, explains the underlying physical idea of the mathematical formulations, sketches the proofs for a few of the known results and lists a large number of open problems. The lectures cover a very wide range of the subject: classical fluid, dynamics of finite or infinite systems, random evolutions, Gibbsian states in finite or infinite volumes and the hydrodynamics. As the author addresses, the subject ``is (now still) something like a mix of some continents of well-developed mathematical theory with islands of separate mathematical results, amid a sea of open problems and conjectures.'' Many of the open problems seem quite difficult at the moment but they are certainly very attractive for the researchers. Part II of the book is S. Kusuoka's lectures on the ``Diffusion processes on nested fractals''. The time when Kusuoka talked was right after T. Lindstrøm's work on the same topic. Here, Kusuoka presents his own construction of the processes in terms of Dirichlet forms. Roughly speaking, the construction starts from a symmetric random walk on the ``lattice'' or ``pre''-nested fractals and then goes to the original fractal by studying the convergence of the corresponding sequence of the lattice Dirichlet forms. The approach is in a se…
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 82Bxx
Mathematics Subject Classification ID: 82B31
Mathematics Subject Classification ID: 82-02
zbMATH DE Number: 492393
zbMATH Keywords: diffusion processes on nested fractals
zbMATH Keywords: statistical mechanics
zbMATH Keywords: random evolutions
zbMATH Keywords: hydrodynamics
zbMATH Keywords: open problems and conjectures
zbMATH Keywords: Dirichlet forms
zbMATH Keywords: Sierpinski gasket
zbMATH Keywords: ergodicity
reviewed by: Mu-Fa Chen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0074238
OpenAlex ID: W610506688
DOI: 10.1007/BFB0074238
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description: scientific article; zbMATH DE number 492393