Textbooks about Statistical Mechanics don't seem to mention Markov Chains, but they intuitively use the concept. Is there a introductory text unifying both views?
The Markov chains provide efficient computational approach for statistical mechanics. For introductory insights I can recommend the coursera course: https://www.coursera.org/course/smac
Statistical Mechanics: Algorithms and Computations
Markov chains can indeed be used to mimic systems approaching equilibrium, but also for non equilibrium situations. On the opposite the concept of Gibbs distributions is closely related to Markov chains (where the present state depends upon a finite past) or chains with complete connections (where the present state depends upon an infinite past). In mathematical physics Gibbs distributions are defined as conditional probabilities where the state inside e.g. a d-dimensional lattice depends upon boundary conditions. If you replace one dimension of the lattice by time you obtain a process which is conditioned by the past ('left' boundary) but can also be conditioned by the future (right boundary). If the potential involved has finite range, the corresponding process is Markov; if the potential has infinite range the process is a chain with complete connection. Note that a Gibbs distribution is not necessarily invariant by translation along the lattice: the corresponding notion as a process is a non stationary process. In this sense there is a deep connection between Markov process (or their extension to infinite memory) and statistical mechanics. See e.g. the beautiful lecture by Gregory Maillard and Roberto Fernandez. http://www.latp.univ-mrs.fr/~maillard/Greg/Publications_files/main.pdf or by Arnaud Le Ny http://www.math.u-psud.fr/~leny/articles/em_15_leny.pdf
No, for Markov chains you need the additional postulate of the Stosszahlansatz of Boltzmann
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