Elements of Statistical Mechanics and Large Deviation Theory
Abstract:
These notes contain an introduction to rigorous classical statistical mechanics and large deviation theory, and
presents some
tight existing relations between these two theories.
To start, the Shannon entropy of a random variable is defined, and distributions of maximal entropy (such as the Gibbs distribution)
are studied.
Then, the Theorem of Sanov is introduced, and used in various ways, in particular to show the Gibbs contitionning principle and to introduce
the microcanonical distribution for the ideal gas.
Abstract Large Deviation Theory is then presented, including in particular Varadhan's Lemma and the contraction principle.
Then, the Curie Weiss model is introduced, and provides an example of the use of
Varadhan's Lemma (to compute and study the free energy).
The following chapter gives a general description of infinite systems in equilibrium, in the so-called DLR formalism.
In particular, conditions are given in order to guarantee existence of a Gibbs measure, compatible with a quasilocal specification.
The Uniqueness criterium of Dobrushin is presented.
The Ising model is introduced as a particular case of Gibbsian specification, and its phase diagram constructed in relative details.
The notes end with the variational characterization of Gibbs measures, and with the Large Deviation interpretation of the set of
invariant Gibbs measures compatible with a quasilocal specification (the so-called level-3 LDP).
Key words: Shannon entropy, maximal entropy distribution, Gibbs distribution, equilibrium, Sanov Theorem, ideal gas,
Gibbs conditionning principle, large deviation principle, Varadhan lemma, contraction principle, Curie Weiss model, free energy,
quasilocal specification, Gibbs measure, DLR formalism, Dobrushin uniqueness, Ising model, phase diagram, variational principle, level 3 LDP.
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