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Ive read a few courses on statistical mechanics, and while their textual explanations and example choices differ, the flow of information from microscopy to macroscopy seems the same, and reading between the lines you can see some mathematical construct. Has statistical mechanics been formalized in the sense that say analysis has been formalized (down to quantifiers, sets, functions,...) more rigorously? Where can one find a formal axiomatic approach to statistical mechanics as opposed to an introductory descriptory approach?

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I am thinking along the lines of Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics, do you know similar textbooks? –  propaganda Jan 22 '12 at 22:55
    
I find this kind of questions annoying. In this case, statistical mechanics is one of the most rigorously treated subjects of mathematical physics - mostly because there are no conceptually confusing issues in it. That's why it's studied by many people employed by maths departments who can do these things very rigorously and formally. But the basic content is still the same as the content of the introductory books; they just get much further by analyzing more complex systems. In contrast to the OP's assumptions, stat. mech. isn't about some formal subtleties because there aren't any. –  Luboš Motl Jan 23 '12 at 6:42
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Here you have a monography on axiomatic classical statistical mechanics: amazon.com/… - it isn't a particularly well-known book and given the dull topic, there's no reason why it should be. –  Luboš Motl Jan 23 '12 at 7:06
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Some comments showing that physicists - in this case Feynman - don't consider the axiomatization of physics useful or important: sciencehouse.wordpress.com/2009/07/23/more-on-feynman - In particular, Feynman often said that there's no reason to try to "minimize" a set of axioms completely. A finite set of rules that is self-consistent is just equally OK. Which of the axioms are "more fundamental" than others is often just a convention, anyway. At any rate, physics has done an amazing job in reducing billions of diverse observations to simple rules. –  Luboš Motl Jan 23 '12 at 7:16
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To say that physics is just a collection of facts - like botany (and it's not quite true even for botany) - means to misunderstand physics completely. Physics doesn't have to become a nitpicking formalized branch of mathematics if it wants to unify the phenomena in the natural world. It does it without formalities; it really does. –  Luboš Motl Jan 23 '12 at 7:17
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2 Answers

Thermodynamics today is subsumed as the continuum limit of statistical mechanics. For statistical mechanics, the closest to an axiomatic deduction of the laws is Jaynes's approach, detailed in a series of papers starting in the 1950s. The basic law is that for every conserved quantity, you have a thermodynamic conjugate, and the statistical ensemble is the maximum entropy consistent with the thermodynamic conjugate values, if you don't fix the conserved quantity, or the maximum entropy distribution consistent with the value of the conserved quantity.

The philosophy behind this is that statistical mechanics is really a calculus regarding our knowledge of the microscopic state of a macroscopic body. It is in many ways a rigorous completion of the formalism of 19th century thermodynamics. It has been discussed here before--- you can find three classic reference (freely available--- thank you Physical Review) linked in Jaynes's Wikipedia article

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Also, you can read all of Jaynes' papers for free here: bayes.wustl.edu/etj/node1.html and most of his book on probability theory here: www-biba.inrialpes.fr/Jaynes/prob.html (the book is a useful reference for understanding his approach to stat. mech., although he died before he could write the second volume, which would have covered the subject in detail). –  Nathaniel May 31 '12 at 8:25
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All valid statements in the equilibrium thermodynamics of standard systems can be deduced from the following definition.

7.1.2 Definition. (Phenomenological thermodynamics)

(i) Temperature T, pressure P, and volume V are positive, mole numbers $N_j$ are nonnegative. The extensive variables H, S, V, $N_j$ are additive under the composition of disjoint subsystems. We combine the $N_j$ into a column vector with these components.

(ii) There is a convex system function ∆ of the intensive variables T, P, µ which is monotone increasing in T and monotone decreasing in P. The intensive variables are related by the equation of state

∆(T, P, µ) = 0.$~~~~~~~~~~~~~~~~~$ (7.1)

The set of (T, P, µ) satisfying T > 0, P > 0 and the equation of state is called the state space.

(iii) The Hamilton energy H satisfies the Euler inequality

H ≥ TS − PV + µ · N $~~~~~~$ (7.2)

for all (T, P, µ) in the state space.

(iv) Equilibrium states have well-defined intensive and extensive variables satisfying equality in (7.2). A system is in equilibrium if it is completely characterized by an equilibrium state.

This is the complete list of assumptions defining phenomenological equilibrium thermodynamics for standard systems; the system function ∆ can be determined either by fitting to experimental data, or by calculation from a more fundamental description, cf. Theorem 9.2.1. All other properties follow from the system function. Thus, all equilibrium properties of a material are characterized by the system function ∆.

This is from the beginning of Part II of Classical and Quantum Mechanics via Lie algebras.

Later comes statistical mechanics proper, in a similar, but more technical style.

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