# 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.

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.