Mathematical Statistical Physics

SUMMARY OF LECTURES

Lecture 1. Introduction to Newtonian mechanics. Conservative systems, energy conservation. Hamiltonian mechanics. Liouville theorem.

Lecture 2. Poincaré recurrence theorem. Maxwell's paradox. Introduction to equilibrium thermodynamics. Postulate of thermodynamics, existence of entropy, characterization of equilibrium. Intensive variables. Euler relation.

Lecture 3. Ergodic hypothesis. Microcanonical ensemble. Discretization of phase space. Canonical ensemble. Shannon entropy, variational formulation of the canonical distribution.

Lecture 4. Grand canonical ensemble. Examples: the continuum gas. Computation of the partition function. The lattice gas approximation.

Lecture 5. Boltzmann entropy. Characterization of equilibrium in terms of the Boltzmann entropy density for the hard core lattice gas. Stirling's formula. Fluctuations around equilibrium. Plausibility argument for concavity of the Boltzmann entropy for more general lattice gases.

Lecture 6. Characterization of equilibrium in the canonical and grand-canonical ensembles. The free energy and the pressure. Equation of state for the hard-core lattice gas. Introduction to magnetic systems.

Lecture 7. Paramagnetic/ferromagnetic transition in spin systems. Spontaneous magnetization. Ising model: zero temperature and infinite temperature. Mean-field approximation: the Curie-Weiss model.

Lecture 8. Hamiltonian and Gibbs distribution of the Curie-Weiss model. Zero magnetic field. Rate function, free energy, estimates for the probability of the magnetization.

Lecture 9. The CW in nonzero magnetic field. Pressure, rate function and spontaneous magnetization. Approach to criticality: critical exponents of the CW model. Introduction to the Ising model: Hamiltonian with free, periodic and \eta-boundary conditions.

Lecture 10. Existence of the thermodynamic limit for the pressure of the Ising model, and for the magnetization. One-dimensional Ising model.

Lecture 11. Infinite volume Gibbs states. Griffiths-Kelly-Sherman (GKS) inequalities, existence of the Gibbs states with + and - boundary conditions. Translation invariance, clustering of the correlations.

Lecture 12. Proof of GKS inequalities. First order phase transitions as lack of uniqueness of the Gibbs state. Phase diagram of the Ising model. Equivalence of lack of uniqueness and emergence of spontaneous magnetization.

Lecture 13. The phase transition of the Ising model in d \geq 2. Definition of critical temperature. Geometric representation of the Ising model in d=2. The Peierls argument: proof of spontaneous symmetry breaking for the Ising model with + boundary conditions and no magnetic field.

Lecture 14. Spontaneous symmetry breaking for the Ising model in d>2 via Griffiths inequalities. Uniqueness of the Gibbs state at high temperature: high temperature expansion for the Ising model. Absence of spontaneous magnetization.

Lecture 15. Absence of phase transitions for the Ising model in a nonzero magnetic field. Vitali and Hurwitz theorems for holomorphic functions. Lee-Yang theorem (proof based on duplication trick).

Lecture 16. Lattice gases, motivations and definitions. Canonical Gibbs distribution and canonical free energy. Existence of the thermodynamic limit (statement). Subadditive sequences.

Lecture 17. Proof of subadditivity of the canonical free energy, existence of the thermodynamic limit. Continuity and convexity of the free energy.

Lecture 18. Grand canonical description of the lattice gas. Strict convexity of the pressure. Properties of the pressure and of the average density as a function of the chemical potential.

Lecture 19. Equivalence of canonical and grand canonical ensemble for the lattice gas. Concentration of the grand canonical Gibbs distribution on the minima of the canonical free energy. Introduction to phase transitions for lattice gases.

Lecture 20. Relation between lattice gases and spin systems: phase transitions for the nearest neighbor lattice gas. Graph of the pressure as function of the average density. Long range potentials: the van der Waals model. Relation with the Curie-Weiss model.

Lecture 21. Lack of convexity for the free energy of the van der Waals model, lack of equivalence of ensembles. Kac interactions. How to recover convexity: Lebowitz-Penrose theorem.

Lecture 22. Proof of the Lebowitz-Penrose theorem.

SUGGESTED BOOKS

G. Gallavotti, Statistical Mechanics: A Short Treatise. Available online at: http://ricerca.mat.uniroma3.it/ipparco/pagine/libri.html

S. Friedli and Y. Velenik. Statistical Mechanics of Lattice Systems. Available online at: http://www.unige.ch/math/folks/velenik/smbook/index.html

LECTURES

Tue - Thu 14 - 16, room N16

Exercise class: Wed 8 - 10, room S9. Teaching assistant: Dr. Giovanni Antinucci