20410419 - MS410-Statistical Mechanics

To acquire the mathematical basic techniques of statistical mechanics for interacting particle or spin systems, including the study of Gibbs measures and phase transition phenomena, and apply them to some concrete models, such as the Ising model in dimension d = 1,2 and in the mean field approximation.
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Mutuazione: 20410419 MS410-MECCANICA STATISTICA in Scienze Computazionali LM-40 GIULIANI ALESSANDRO

Programme

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Existence of a phase transition in the long range 1D Ising model with interaction |x-y|^{-p}, 1

Core Documentation

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Disponibile online in preprint version su https://www.unige.ch/math/folks/velenik/smbook/index.html

Type of delivery of the course

Blackboard lectures in presence

Type of evaluation

Oral presentation of a selection of arguments to be agreed with the professor