Statistical mechanics and fractals
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 sense close to the author's previous construction for Sierpinski gasket. The sound seems natural but the technique is still quite hard and the uniqueness problem still remains open as far as I known. The technique enables him not only to figure out a nice expression of the Dirichlet form but also to study further the martingale dimension, the stationary distribution for the diffusions and even ergodicity in some special cases. Kusuoka's construction, though the publication was delayed for years, is considered as a fundamental work and has been used many times by the researchers in the field. Finally, even though I do not know the reason but it should be pointed out that the pages 43 and 44 of the book was unfortunately labeled in the opposite order.
scientific article; zbMATH DE number 492393
scientific article