Università Roma Tre

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– The goals of statistical mechanics
– Review of thermodynamics. Convex functions and Legendre transform
– Models of statistical mechanics: canonical ensemble, grand canonical and Gibbs states.
– The Ising model and the lattice gas models. Existence of the thermodynamic limit for the free energy of the Ising or lattice gas model.
– The general structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs state.

THE ISING MODEL
– Review of known results on the Ising model in one or more dimensions.
– The solution of the one-dimensional Ising model via the transfer matrix method.
– The mean field Ising model: exact solution. Phase transition and loss of equivalence between statistical ensembles
– Ising with long-range interactions (Kac potentials) in the mean-field limit. The Maxwell construction.
– FKG and Griffiths inequalities. Existence of the infinite volume correlation functions of states with + and − conditions in the ferromagnetic Ising model.
- The geometric representation of the Ising model: high and low temperature contours.
- Existence of a phase transition in the low temperature Ising model: the Peierls's argument.
– Absence of a phase transition at high temperature and exponential decay of boundary effects boundary conditions.
– Lee-Yang theorem and analyticity of the pressure at non-zero magnetic field.
– Existence of a phase transition in the one dimensional Ising model with power law interaction |x − y|^{−p}, 1

S. Friedli, Y. Velenik: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press, 2017.

G. Gallavotti: Statistical Mechanics. A short treatise, ed. Springer-Verlag, 1999.