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Subject: MECCANICA STATISTICA (A.A. 2020/2021)

master degree course in MATHEMATICS

Course year 1
CFU 6
Teaching units Unit Meccanica statistica
Models and Application (lesson)
  • TAF: Compulsory subjects, characteristic of the class SSD: MAT/07 CFU: 6
Teachers: Cristian GIARDINA'
Exam type oral
Evaluation final vote
Teaching language Italiano
Contents download pdf download

Teachers

Cristian GIARDINA'

Overview

This course aims at introducing the logic of mathematical modeling in statistical mechanics. The final
goal is to describe and explain reality through the
use of quantitative methods. The course presents an introduction to probabilistic techniques in the context of statistical mechanics. The focus will be on the
use of statistical methods and the interpretation of results.

Admission requirements

None formally required. However, the student is assumed to have a basic knowledge of mathematical-physics, analysis and geometry.

Course contents

1) Law of large numbers, central limit theorem
2) Large deviations theory: Cramer theorem for the empirical mean, Sanov theorem for the empirical measure, Gartner-Ellis theorem for dependent variables, Varadhan's lemma.
3) Principles of stat mech. Boltzmann Gibb measure. Ensemble equivalence. Perfect gas.
4) Phase transition: Curie-Weiss model, Ising model in 1 dimension, spin glasses.

Exam: End-of-course oral assessment.

Students Consultation: by appointment, Dept. of Mathematics.


Teaching methods

Video lectures.

Assessment methods

Exam: End-of-course oral assessment.

Learning outcomes

Knowledge and understanding:
At the end of the course the student should have acquired the elements of statistical mechanics and the understanding of the basic mechanisms that rule pahse transitions. The student will reach a full knowledge of the probabilistic and mathematical tools which are used in classical statistical mechanics. In particular, large deviations theory and stochastic processes are discussed. This allows the student to give a full rigorous description of phase transitions.

- Applying knowledge and understanding: the student at the end of the course should be able to apply the knowledge acquired to the analysis of simple statistcial mechanics systems.

- Making judgements: the student at the end of the course should be able to recognize by himself the appropriate approaches and calculation methods in Statistical Mechanics.

- Communicating skills: the student at the end of the course should be able to describe with appropriate technical language and mathematical formalism the subjects presented in the course.

- Learning skills: The lectures should stimulate the learning skills and to go deep in subjects connected with the ones presented in the lessons.

Readings

- Appunti del corso.
- Large Deviations, Frank den Hollander, Field Institute Monograph, American Mathematical Society, 2000
- Entropy, Large Deviations and Statistical Mechanics, Richard S. Ellis, Springer-Verlag 1985