00695 - Statistical Mechanics

Learning outcomes

At the end of the course the student is acquainted with the basic notions of statistical mechanics like, for instance, the concept of equilibrium probability measure. He is able to deal with the study of new scientific results, theoretical and applied, where statistical mechanics is involved.

Course contents

- Introduction.

- Probability spaces, Entropy and its property.

- Probability spaces as simplexes.

- Interacting particle systems and Ising models. Ising model in d=1 with free and periodic boundary conditions.

- Dichotomic functions and their Fourier expansion.

- Simple notions for Ising models at d>1

- The thermodynamic limit and correlation inequalities.

- Mean field models. Curie-Weiss model.

- Thermodynamic limit for mean field models.

- The solution of the Curie Weiss model with bounds from above and below.

- Large number theorem and central limit theorem for the magnetization. Free model and model with interaction.

- Large deviation theory and solution of the Curie Weiss model.

- Theoretical inverse problem: from thermodynamic quantities to parameters.

- Phenomenological inverse problem: from real data to thermodynamic quantities.

- Maximum likelihood.

- The Sherrington and Kirkpatrick model. Parisi solution. Guerra Talagrand proof for the free energy. Panchenko proof for ultrametricity.

- Associative memory and the Hopfield model.

- The multi-specie generalization of the SK model.

- Deep-learning architecture with cilindrical and conical boundary conditions. Bounds for the free energy.

- Learning and retrieval equivalence.

- Seminars on recent advances.

Readings/Bibliography

Notes of the teacher.

Suggested books:

- "Perspectives on Spin Glasses", P.Contucci e C.Giardinà,CUP, 2012

- "Spin Glasses and Complexity (Primers in Complex Systems)" D.Stein and C.Newman, OUP, 2013

- "Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction", S.Friedli and Y.Velenik,CUP, 2017

Teaching methods

Class lectures that include theory and exercises. Computer simulations and illustrations.

Assessment methods

The exam is an oral examination or a research seminar, to be agreed with the teacher, on topics related to the course themes.

Teaching tools

Some lectures will be supported by illustrations of computer simulations of statistical mechanics systems.

Links to further information

http://www.dm.unibo.it/~contucci/lectiones.html

Office hours

See the website of Pierluigi Contucci