# Where does a math person go to learn statistical mechanics?

The more math I read, the more I see concepts from statistical mechanics popping up -- all over the place in combinatorics and dynamical systems, but also in geometric situations. So naturally I've been trying to get a grasp on statistical mechanics for a while, but I haven't been very successful. I've skimmed through a couple of textbooks, but they tended to be heavy on the physical consequences and light on the mathematical underpinnings (and even to an extent light on the physical/mathematical intuition, which is inexcusable!)

I suspect that part of the problem is that, unlike the analogous situation with quantum mechanics, I'm not sure what mathematics I can fall back on if I don't "get" some statistical model. So, is there a good resource for statistical mechanics for the mathematically-minded?

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I don't know. But I would like an answer to this question as well. – Michael Lugo Nov 5 '09 at 0:49
I've had that problem with physics books as well. Idk why they're so... unmathematical... and even not great at helping people get an intuition for the subject!! – Michael Hoffman Nov 5 '09 at 1:24
There are good ones - even great ones - out there. You just have to look hard for them (see my answer below for examples). One of the problems is that it is very difficult to strike the right balance between mathematical rigor and conceptual (i.e. physical) understanding, particularly when writing. – Ian Durham Feb 19 '10 at 21:05
I would recommend ams.org/bookstore-getitem/item=ULECT-19 for a start. – William Mar 28 '13 at 3:44

## 29 Answers

A classic book on solvable two-dimensional models is Baxter's "Exactly Solved Models in Statistical Mechanics" now available in a new edition from Dover. The Yang-Baxter equation, of course, has many connections with important branches of mathematics. This book explains its origins and use in solving certain physically motivated models.

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Baxter's book is available in PDF form from his website: tpsrv.anu.edu.au/Members/baxter/book – j.c. Jan 16 '10 at 4:39
That's great, I didn't realize it was online and free. Wish I could upvote it twice for that reason! – Emily Peters Mar 6 '10 at 16:02
The above link seems to have migrated here: physics.anu.edu.au/theophys/baxter_book.php . – HJRW Apr 24 '12 at 10:25

So, is there a good resource for statistical mechanics for the mathematically-minded?

If you are looking for a book, the real answer is "not really". As a mathematician masquerading as a physicist (more often than not of a statistical-physical flavor) I have looked long, hard, and often for such a thing. The books cited above are some of the best for what you want (I own or have read at least parts of many of them), but I would not say that any are really good for your purposes.

Many bemoan the lack of The Great Statistical Physics text (and many cite Landau and Lifshitz, or Feynman, or a few other standard references while wishing there was something better), and when it comes to mathematical versions people naturally look to Ruelle. But I would agree that the Minlos book (which I own) is better for an introduction than Ruelle (which I have looked at, but never wanted to buy).

Other useful books not mentioned above are Thompson's Mathematical Statistical Mechanics, Yeomans' Statistical Mechanics of Phase Transitions and Goldenfeld's Lectures On Phase Transitions And The Renormalization Group. None of them are really special, though if I had to recommend one book to you it would be one of these or maybe Minlos.

You might do better in relative terms with quantum statistical mechanics, where some operator algebraists have made some respectable stabs at mathematical treatments that still convey physics. But really that stuff is at a pretty high level (and deriving the KMS condition from the Gibbs postulate in the Heisenberg picture can be done in a few lines) so the benefit is probably marginal at best.

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One book that I've recently acquired that looks quite good (I haven't read much except for the last chapter) is Gallavotti's short treatise: books.google.com/books?id=2jDuMESyj6MC – Steve Huntsman May 21 '10 at 13:27

There are also Feynman's lecture notes: "Statistical mechanics: a set of lectures". I haven't used them myself.

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+1, they are quite readable. – Ilya Nikokoshev Nov 5 '09 at 8:29

"Statistical Mechanics: Entropy, Order Parameters, and Complexity" by James Sethna (my favorite) and "Statistical Mechanics" by Kerson Huang are both really good books. Sethna's book is very readable and engaging; Huang's is more of a syatematic textbook.

Ruelle's book is mathematically rigorous but is aimed at people who already know something about the field. Baxter's book is incredibly valuable for what it covers, but it is highly specialized and provides no motivation for people who aren't already comfortable with the basic formalism.

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Free to download from the author's page here: pages.physics.cornell.edu/~sethna/StatMech/… – durum Feb 23 at 15:09

If the goal is to really learn statistical physics, I suspect Steve Huntsman's answer is probably correct.

It sounds like your goal is primarily to get a quick overview of how mathematicians use statistical physics, with lots of intuition, and lots of applications to nearby areas, but not necessarily much physics or any 'best possible' results. If that is the case, the Montanari/Mezard book `Information, Physics and Computation' is excellent. The emphasis is on teaching many different techniques, rather than on the statistical physics itself, and it is really written as a textbook rather than a reference book (i.e. the theorems are the easiest ones to understand, not the most powerful ones to use). In other words, it has exactly the mathematical underpinnings, but doesn't cover what many of the other books treat as central material.

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For a much lower level answer, I found the relevant chapters in volume 1 of the Feynman lectures to be a helpful introduction.

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Volume 2 and Volume 3 both contain applications - vol 3 to Quantum Mechanics. Feynman is good on trying to motivate physical intuition too, and if you connect with his way of thinking then you get a lot out of it. What you don't really get is all the alternative "wrong ways" of thinking about physical situations which don't work - that comes from trying to figure it out yourself. – Mark Bennet Feb 22 '11 at 18:51

McCoy and Wu's "The Two-Dimensional Ising Model" is, of course, devoted mostly to the Ising model, but one can learn a lot about statistical mechanics in general by reading it.

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Well, how everyone said, there is not yet a didactic book for the entire subject but, there exists some texts and books about topics of the theory such that you can read rapidly than Ruelle.

Some examples:(all are rigorous)

The classical: Gibbs Measures and Phase Transitions. De Gruyter Studies in Mathematics Vol. 9. Berlin: de Gruyter 1988. by H.-O. Georgii

Entropy, Large Deviations and Statistical Mechanics. Springer-Verlag. by Richard S. Ellis.

The statistical mechanics of lattice gases, volume 1 by Barry Simon

And, there are many surveys and currently appears new reviews about some topics, some examples(I included pages of mathematicians that work on the subject, in their pages you can get some introductory texts):

http://arxiv.org/abs/0712.1171

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There are apparently only a few books on rigorous results in statistical mechanics. David Ruelle's books are apparently standard, though I found them difficult to digest when I picked them up. One which I found more accessible is "Introduction to Mathematical Statistical Physics" by Minlos.

A personal note is that I found statistical mechanics very unintuitive and difficult to learn at first. I felt that the formalism didn't come together for me until I was familiar with a multitude of physical systems.

A more advanced book which I really love for giving physical intuition is Cardy's book on the renormalization group (though this thin book is very light on rigor!)

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If you know some statistics, you will probably find J. Rau Statistical mechanics in a nutshell 22 pages long essay very clear and motivating.

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It's certainly not a basic book, but Itzykson and Drouffe's "Statistical Field Theory" gives a good overview of the use of quantum field theory techniques in statistical mechanics. Some of the chapters are quite readable.

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MIT's OpenCourseWare has excellent resources for statistical physics. See, for instance, http://ocw.mit.edu/OcwWeb/Physics/8-044Spring-2008/Syllabus/index.htm.

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Start here: An Introduction to Thermal Physics by Dan Schroeder is an excellent introduction that provides a single, consistent mathematical underpinning providing the insight necessary to truly understand things like entropy and the differences between the classical and quantum cases of statistical mechanics. I can't emphasize enough how important the core ideas of that book are to understanding the foundations of statistical mechanics. (As an aside, Schroeder also happens to be the coauthor, with Michael Peskin, of An Introduction to Quantum Field Theory which has purportedly replaced Bjorken and Drell as the standard in that field, though this is only hearsay.)

Supplement with: Foundations of Statistical Mechanics by Oliver Penrose which is another consistent approach, and the still relevant Statistical Physics by Landau and Lifshitz.

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This topic is old, but I'll still add my 2¢. I usually don't really like statistical mechanics books aimed at physicists, as they are often much more focused on computational techniques than on concepts. There are however very good lecture notes by Yoshi Oono, available on his page:

http://web.me.com/oono/Y_OONO_official_site/tutorial.html

The relevant file is SecondSMR.pdf . Notice that this is a password-protected pdf file, unfortunately. The password is given on the page, though.

This book is aimed at physicists, but contains a very unusual amount of mathematical content, esp. in footnotes, with many references to the math. phys. literature. This book assumes that the reader already has some knowledge of this field. There are other lecture notes, aimed at undergraduates, on his page as well (I haven't looked closely at those, so I cannot comment on their quality):

http://web.me.com/oono/Y_OONO_official_site/tutorial_(undergrad).html

Yvan

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Two other books which are worthwhile I find R.B. Israel: Convexity in the Theory of Lattice gases. It has wonderful introduction by Wightman , which is like book in itself. it is limited in scope but is excellent in what it treats. T.C. Dorlas Statistical mechanics, is written by a mathematical physicist and covers many topics in a more rigorous way than most physics textbooks do.

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A really good first textbook for statistical mechanics is David Chandler's Introduction to Modern Statistical Mechanics. It's written by a physical chemist for senior undergraduates and does an excellent job distilling down the very fundamental material into a one-semester course at Berkeley. It's not at all math heavy.

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Sorry, the last answer, it is mine. statmeca.fmns.rug.nl – Bissacot Feb 19 '10 at 7:48
I fixed the links for you. – David Speyer Feb 19 '10 at 15:31

One of the best books on statistical mechanics and its mathematical aspects is "Statistical Mechanics, A Short Treatise" by G. Gallavotti, Springer, 1999: Google Books link.

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Oops! I just saw the ref in a comment by Steve Huntsman, sorry for the repeat. – Abdelmalek Abdesselam May 1 '11 at 18:12

This is a quite old thread, but as the question remains relevant, please forgive the following plug. We are in the process of writing an introductory book on (some aspects of) equilibrium statistical mechanics, with mathematicians (and mathematically-inclined physicists) in mind. Early drafts of some chapters can already be downloaded from this page. More should be available soon (as soon as we consider them presentable).

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Maybe I didn't read the replies well enough, but apparently no one mentions A. Khinchin's book Mathematical Foundations of Statistical Mechanics. I just started reading this book. It's definitely mathematical and specifically says that it's written for a mathematician. The notation is a bit old, but the book is very readable, and far, far better than any book of physics I've seen lately.

It's only an introduction, so it doesn't contain the work done in recent decades, but a good one I would say. If there is anything really wrong with it, I would attribute it to the translator.

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It has been mentioned twice. – S. Carnahan Jul 5 '13 at 5:04

Many physics people like Pathria's Statistical Mechanics. It's good for physical intuition.

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Khinchin, Mathematical Foundations of Statistical Mechanics. \$2.94 used on Amazon.

http://www.amazon.com/Mathematical-Foundations-Statistical-Mechanics-Khinchin/dp/0486601471

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For quantum statistical mechanics the standard textbooks are:

O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics. Vol. 1: C* and W*-algebras. Symmetry Groups. Decomposition of States.

O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics. Vol. 2: Equilibrium States. Models in Quantum Statistical Mechanics.

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Statistical Mechanics of Disordered Systems, a Mathematical Perspective, by Anton Bovier. Part I provides a brief introduction to statistical mechanics, while the rest two parts focus on disordered systems. To my understanding, the author treats the problems from a mathematical point of view.

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Two short easy places to start. Of course after that, there are lots. Schroedinger's lecture notes on Statistical Thermodynamics are a clearly thought out gem of pedagogy, it was really his forte. And you don't need to finish the book either.

Khinchin's monograph on Mathematical Foundations of Statistical Mechanics is also incredibly insightful, although you can skip the proofs, and again, the first half of the book suffices.

This will give you a clear, logical, and physical foundation to then go on and read anything else.

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I tried a few texts mentioned here, but Hugo Touchette's arxiv article, 89 pages, is my favorite. Kept simple, well motivated, and with links to all the more rigorous/difficult stuff.

In his (and other's) opinion the mathematics of statistical mechanics is large deviation theory (or at least it is as close as it gets). He is quite convincing.

http://arxiv.org/abs/0804.0327

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Although large deviations theory is certainly deeply related to important aspects of statistical mechanics (the probabilistic interpretation of thermodynamic potentials, the equivalence of ensembles, the variational principle, etc.), the latter certainly does not reduce to the former. – Yvan Velenik Mar 23 '14 at 11:36
@YvanVelenik: you may well be right, I am an amateur on this topic. Could you give an example or a sketch of the difference? – tortortor Mar 24 '14 at 2:25
Well, large deviations theory is about logarithmic asymptotics of various probabilities, which only corresponds to a tiny piece of the statistical physics domain. Even keeping with the most probabilistic aspects of statistical physics, you can say that the latter is interested in studying properties of Gibbs measures. Many of these properties cannot be cast in a large deviations framework (just as large deviations theory only highlights some aspects of probability theory). – Yvan Velenik Mar 24 '14 at 8:23
As for explicit examples, I doubt that LD theory helps much if you want to determine, say, critical exponents, asymptotic behavior of correlation functions, etc. – Yvan Velenik Mar 24 '14 at 8:25
Notice also that LD theory provides mostly structural information: it allows you, for example, to relate rate functions (thermodynamic potentials) corresponding to different ensembles. However, by itself, it generally tells you basically nothing about these rate functions, or their minimizers (not even about uniqueness or not of the latter, i.e., the issue of phase transitions; to obtain such information, you have usually to go far beyond LD theory). – Yvan Velenik Mar 28 '14 at 7:58

I've only briefly looked at it but the book Statistical Physics of Particles by Mehran Kardar seems pretty good. It starts with an introduction to the relevant probability theory and then moves on to the basics of statistical mechanics. There is also a subsequent book Statistical Physics of Fields.

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You may check:

Phase transitions and critical phenomena (Vol. 11), 1987

This is an encyclopedic volume, different chapters being written by different selected authors.

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