Statistical Mechanics

Aim of the course is to present some advanced topics of statistical mechanics of classical and quantum systems in equilibrium, with particular focus upon the algebraic approach to quantum systems with infinitely many degrees of freedom.

The course is divided into two parts.

In the first part, the algebraic formalism is introduced which allows for a comprhensive description of notions and tools common to both classical and quantum statistical systems,like equilibrium states, dynamics, entropy and representations. The corresponding basic tools are then applied to concrete models of spin chains and of Bosonic and Fermionic systems.

Reference books

C. Kittel: Elementary Statistical Physics, Wiley and Sons

L.D. Landau, E.M. Lifschitz: Statistical Physics, Part 1. Vol. 5, Butterworth–Heinemann

 F. Strocchi: Elements of quantum mechanics of infinite systems. Singapore, World Scientific, 1985

In the second part, the fundamentals are provided of the algebraic approach to quantum systems with infinitely many degrees of freedom like the GNS representation of Gelfand, Naimark and Segal that allows for the extension to quantum systems of the notions of ergodicity and mixing and for the discussion of the existence of inequivalent representations.The relative techniques are then applied to spontaneous symmetry breaking phenomena and phase-transitions with particular focus upon Goldstone's theorem.

Reference books

F. Strocchi: Elements of quantum mechanics of infinite systems. Singapore, World Scientific, 1985

F. Strocchi: Symmetry Breaking, Lecture Notes in Physics 732, Berlin, Springer verlag, 2008

Program

1. Quantum and classical statistical Mechanics of equilibrium and non equilibrium

1.1 Entropy

1.2 Thermal states

1.3 Spin chains

1.4 Bosonic and Fermionic systems

2. Quantum systems with infinitely many degrees of freedom

2.1 Algebraic approach: states, observables, dynamics

2.2 Ergodicity and mixing

2.3 Spontaneuous symmetry breaking and phase-transitions

2.4 Goldstone theorem