# Proof that Statistical Mechanics is a model of Themodynamics

The laws of thermodynamics are essentially four axioms of a mathematical theory.

The expectation values of a statistical ensemble are supposed to satisfy the axioms of thermodynamics (under the assumption that the ergodic hypothesis holds).

How is it proved that statistical mechanics satisfies the axioms of thermodynamics?

The question is important to me because the abstract theory of thermodynamics is used to develop things like the Maxwell relations, which are then applied to statistical mechanics ensembles. For this to give valid results it statistical mechanics must be a model of thermodynamics. I would appreciate pointers to proofs of this and references if you have any. Thanks a lot.

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Let me first state that it's impossible to derive the result in full generality, i.e that it holds for any statistical model whatsoever, simply because this is false. One needs to place certain physical assumptions on the system in order for the result to even hold. But in order to actually prove anything (in the mathematical sense) one also needs to require additional technical assumption (the math is just hard here, no way around it).

The most common way is to work on a lattice $S$ (such as $S = \mathbb Z^d$), i.e. with discretized space (this makes your life much easier though still pretty hard) and also suppose we are given a space of fields $(E, \mathcal E, \lambda_0)^S$ where $(E, \mathcal E, \lambda_0)$ is some nice measure space. Finally, let the system be described by a collection of potentials $(\Phi_A)_{A \subset S}$ describing interactions in some subset $A$ of the lattice such that the energy $H_x(\cdot) = \sum_{A \ni x} \Phi_A(\cdot)$ at every site of the lattice is finite $\int H_x(\omega) \lambda(d\omega) < \infty$ where $\lambda$ is the product measure on $E^S$ (e.g. finite-range spin models such as the Ising model will do), we can prove that the any shift-invariant ergodic Gibbs measure (here ergodic refers to the shift translation on the lattice, not to the time evolution you are referring to) then satisfies a variational principle (of maximizng an entropy functional on some reasonable space of measures) and consequently the macroscopic quantities computed from it will have all the nice thermodynamic properties.

Two standard references for this would be

• Ruelle's Thermodynamic formalism. Note that the book is aimed at mathematicians, is very obtuse and hard to follow and requires working knowledge of topology, functional analysis and measure theory.
• Georgii's Gibbs measures and phase transitions where one whole part of the book is dedicated to the formal treatment of the variational problem for measures. He also studies the geometry of the Gibbs measure space as one varies the potential (since there is one extreme Gibbs measure for each phase for the given potential, this is nothing else than the phase diagram with coexistence lines, etc.; although here the phase diagram is actually infinite-dimensional). Here one can get by with just rudimentary knowledge of the above areas since Georgii develops some parts of topology and probability theory he needs.
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Part II: Statistical Mechanics of the book Classical and Quantum Mechanics via Lie algebras shows how the traditional form of thermodynamics follows from statistical mechanics, including Maxwell relations, and without assuming the ergodic hypothesis (which is not satisfied for most complex systems).

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