# Statistical Physics : Concepts and applications

### Accès rapides

Prochain Séminaire de la FIP :
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Retrouvez toutes les informations pour vos stages :
Stages L3
Stages M1 ICFP

Actualités : Séminaire de Recherche ICFP
du 14 au 18 novembre 2022 :

Retrouvez le programme complet

Contact - Secrétariat de l’enseignement :
Tél : 01 44 32 35 60
enseignement@phys.ens.fr

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Faculty : Werner Krauth
Tutor : Maurizio Fagotti et Olga Petrova
Tutor : Olga Petrova
ECTS credits : 6
Language of instruction : English
Web site :

### Description

Scope of the course
This lecture course on statistical mechanics will take students from the foundations of probability theory and statistical inference to the important models and the central concepts and techniques of statistical mechanics. The main focus will be on equilibrium and on classical systems, but we will also treat transport and dissipation, and discuss quantum statistical mechanics for Boson systems and quantum spin models.
There will be 15 lectures and tutorial sessions, 8 graded homeworks (50% of the grade), and a written final exam (50%, also).
Planning
Lectures and tutorial sessions each Wednesday morning, from 7 September 2016 through 14 December 2016 (Lectures : 8:30 - 9:25 am, 9:35 - 10:30 am ; tutorials : 10:45 - 11:40 am, 11:50 am - 12:45 pm).

Final exam : 22 January 2017 8:30 - 12:30
Prerequisites
This course will be self-contained. Some prior exposure to elementary statistical or thermal physics on the undergraduate level may be useful.

Computing requirements

Probability, statistics, and statistical physics are today closely linked to computing. Students should be able to download, run and modify elementary Python programs. Many such programs will be provided, and some will have to be written for the homework sessions.

Syllabus

1. Probability theory
• Probabilities, probability distributions, sampling
• Random variables
• Expectations
• Inequalities (Markov, Chebychev, Hoeffding)
• Convergence of random variables (Laws of large numbers, CLT)
• Lévy distributions
2. Statistics (statistical inference, estimation, learning)
• Point estimation, confidence intervals
• Bootstrap
• Method of moments
• Maximum likelihood, Fisher information
• Parametric Bootstrap
• Bayes statistics
3. Statistical mechanics and Thermodynamics
• Rapid overview on the connection between statistical mechanics and thermodynamics
• lightning review of ensembles and
• lightning review of the main physical quantities (partition function, energy, free energy, entropy, chemical potential, correlation functions, etc).
4. Physics in one dimension
• One-dimensional hard spheres, virial expansion, partition function
• One-dimensional Ising model
• Transfer matrix
• Kittel model
•  Chui-Weeks model : Infinite-dimensional transfer matrix
• One-dimensional Ising model with 1/r^2 interactions
5. Two-dimensional Ising model : From Ising to Onsager
• Peierls argument, Kramers-Wannier relation
• Two-dimensional transfer matrix (following Schultz et al)
• Jordan-Wigner transformation
• Free energy calculation
• Spontaneous magnetization, zero-field susceptibility
• Kaufman, Ferdinand-Fisher, Beale
6. Two-dimensional Ising model : From Kac and Ward to Saul and Kardar
• Van der Waerden, low-temperature and high-temperature expansions
• Duality
7. Physics in two dimensions ( Kosterlitz-Thouless physics) : XY (planar rotor) model
• Peierls argument
•  Mermin-Wagner theorem
• Non-universality
8. Physics in two dimensions ( Kosterlitz-Thouless physics) : Particle systems, superfluids
9. Physics in infinite dimensions : Mean-field theory, Scaling
10. Physics in infinite dimensions : Landau theory
11. Renormalization group
12. The Solid state : Order parameters, correlation functions
13. Quantum systems - bosons
14. Quantum systems - spin systems
15. Equilibrium and transport, Fluctuation-dissipation theorem.

References
Lecture notes will be available before each course.
Books

1. L. Wasserman, "All of Statistics, A Concise Course in Statistical Inference" (Springer, 2005)
2. W. Krauth, "Statistical Mechanics : Algorithms and Computations" (Oxford, 2006)
3. M Plischke, B Bergersen, "Equilibrium Statistical Physics" (World Scientific)
4. L. D. Landau, E. M. Lifshitz, "Statistical Physics" (Pergamon)

### Accès rapides

Prochain Séminaire de la FIP :
Accéder au programme

Retrouvez toutes les informations pour vos stages :
Stages L3
Stages M1 ICFP

Actualités : Séminaire de Recherche ICFP
du 14 au 18 novembre 2022 :

Retrouvez le programme complet

Contact - Secrétariat de l’enseignement :
Tél : 01 44 32 35 60
enseignement@phys.ens.fr

r>