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Lecture Note
30/2006
Mathematical Statistical Mechanics
Stefan Adams
Abstract
In this notes we give an introduction to mathematical statistical mechanics, based on the six lectures given at the Max Planck institute for Mathematics in the Sciences February/March 2006. The material covers more than what has been said in the lectures, in particular examples and some proofs are worked out as well the Curie-Weiss model is discussed in Section 9.3. The course partially grew out of lectures given for final year students at the University College Dublin in spring 2004. Parts of the notes are inspired from notes of Joe Pulé at University College Dublin.
The aim is to motivate the theory of Gibbs measures starting from basic principles in classical mechanics.
The first part covers Sections 1 to 5 and gives a route from physics to the mathematical concepts of Gibbs ensembles and the thermodynamic limit. The Sections 6 to 8 develop a mathematical theory for Gibbs measures. In Subsection 6.4 we give a proof of the existence of phase transitions for the two-dimensional Ising model via Peierls arguments. Translation invariant Gibbs measures are characterised by a variational principle, which we outline in Section 7. Section 8 gives a quick introduction to the theory of large deviations, and Section 9 covers some models of statistical mechanics. The part about Gibbs measures is an excerpt of parts of the book by Georgii 1988. In these notes we do not discuss Boltzmann's equation, nor fluctuations theory and nor quantum mechanics at all.
I hope these lectures will motivate further reading and even more further research in this interesting field of mathematical physics and stochastics.
Many thanks to Thomas Blesgen for reading the manuscript.
