3 - Large deviations and statistical mechanics

Summary

Having in mind the discussion on metastability, we need a brief introduction to statistical mechanics. This will be done in Section 3.3. Considerations on the connection between large deviation theory and the statistical description of (equilibrium) thermodynamical systems appear naturally and are the content of Section 3.4, focusing on basic aspects rather than generality, with references to more advanced literature.

Apart from such motivations, the extension of the results of Chapter 1 beyond independent variables is natural in many contexts. We discuss the Gärtner–Ellis method briefly and apply it to finite Markov chains, as the simplest situation to start with. While discussing large deviations for (equilibrium) statistical mechanics models, it is important to stress the role of subadditivity and convexity, already illustrated in Section 1.5, where the Ruelle–Lanford method was considered in the context of i.i.d. variables. Some basic results are taken from Section 3 of Pfister's lecture notes [243], simplified for our situation.

Large deviations for dependent variables. Gärtner–Ellis theorem

Examining the proof of the Cramér theorem in Section 1.4, we are naturally led to allow a moderate dependence among the X_i variables. This extension is due to Gärtner [132] and Ellis [108].