Customer Reviews

Statistical Physics I: Equilibrium Statistical Mechanics (Springer Series in Solid-State Sciences)

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Statistical Mechanics (1&2) by Toda and Kubo is the best textbook on statistical mechanics I've read. This book has some material such as Linear response theory(in vol 2), ergotic problems(in vol 1) which is difficult to find in other ones. I like the way the author write in linear response part. It is beautiful.

Statistical Mechanics (1&2) by Toda and Kubo is the best textbook on statistical mechanics I've read. This book has some material such as Linear response theory(in vol 2), ergotic problems(in vol 1) which is difficult to find in other ones. I like the way the author write in linear response part. It is beautiful.

This text provides a good, readable introduction to Markov processes, including Fokker-Planck equations, from the standpoint of typical physical examples. A weakness is that (by only mentioning and not developing Ito calculus) the book does not make it clear to the reader that most stochastic processes are nonstationary. This is important: today, we are interested in far from equilibrium dynamics, much less so in dynamics near equilibrium where the fluctuation-dissipation theorem holds. On the other hand, the standard financial math texts (Baxter and Rennie, Steele, ...) do us no service in this direction either. The book goes beyond the older reference by Wax, which is still a very good introduction to Markov processes. In any case, no existing reference treats the general case of a space-time-dependent diffusion coefficient adequately, the case of most interest for the dynamics of financial markets. Now for details of the weak spots.

There are two mistakes on pages 65-68. The discussion is based on the sde dx=-R(x)dt+D(x,t)^1/2dB(t) where B(t) is a Wiener process. First, it is claimed that the random force D(x,t)^1/2dB is Gaussian with a white spectrum. In general, the random force is not even stationary unless D is independent of x. The unstated assumption is that the random force is always stationary, so that with R(x)<0 there is an approach to equilibrium. When the diffusion coefficient depends on x (or more generally on (x,t)) then there is no approach to equilibrium for the case of unbounded x even with R<0, as the lognormal model of standard finance theory so vividly shows. Second, even if an equilibrium solution of the corresponding Fokker-Planck equation 'exists', it cannot be reached dynamically when the force is nonstationary. Again, the lognormal model illustrates this point. Arguments (typical in economics) that an equilibrium solution 'exists' are meaningless are useless if the dynamics can't approach that solution.