# Does it help to learn statistical mechanics in order to learn thermodynamic formalism?

Does it help to learn statistical mechanics or thermodynamics (as in physics or mathematical physics) in order to learn thermodynamic formalism: the study of equilibrium states, Gibbs measure, maximal measures mostly on shift spaces or ℤn?

I've not taken a course on thermodynamics, but so far my learning of the concept of Gibbs states, pressures, equilibrium states on shift spaces seems to be going fine. But then maybe I'm missing physical intuitions that may be necessary later.

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Added [soft-question] tag. Actually this whole question is in danger of being closed. – Harald Hanche-Olsen Nov 19 '09 at 14:04

If you're learning thermodynamic formalism in order to apply it to physics then it's fairly clear that you ought to learn the physical context, so I infer from this that you are more interested in using it either to understand dynamical systems, or for the sake of its applications to other areas of mathematics entirely (such as those explored by Pollicott, Baladi, Lalley, Urbanski, Sharp, and so on). Of the twenty or so people I know who are active in thermodynamic formalism, only two or three have any background whatsoever in mathematical physics - so I think it's safe to say that understanding the physical background isn't necessary. For example, I've learned enough thermodynamic formalism to write a paper on it, and I know no statistical mechanics whatsoever.

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Here is what you really can benefit from in a nutshell: the microcanonical and canonical ensembles, and the fact that a phase transition is a singularity in the partition function. A bit of Hamiltonian mechanics (e.g., Liouville's theorem and phase space partitioning a la Poincare) is also relevant as it informs stuff like SRB measures and Markov partitions. The rest will come up organically if and when you need it.

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I work occasionally on combinatorial problems where these formalisms are useful. I double-majored in chemistry and mathematics as an undergraduate, so I took a couple physical chemistry classes and learned some thermodynamics and statistical mechanics. I find that occasionally something I learned in that scenario gives me intuition on apparently nonphysical problems. But the time I invested there is probably not worth it in terms of its usefulness for my mathematics.

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Try Feynman's Lectures on Statistical Mechanics, you'll have a good chance to learn both.

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