Physique statistique : Des concepts aux applications

Enseignant : Werner Krauth
Chargé des TD : Victor Dagard
ECTS : 6
Langue d’enseignement : Anglais
Site web :


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.
Organization, grading
There will be 15 lectures and 14 tutorial sessions, 9 homeworks (with solutions, not graded), and one written intermediate (30%) and one written final exams (70%).

Lectures (CM) and tutorials (TD) : Each Monday morning, from 10 September 2018 through 07 January 2019 () (Lectures : 8:30 - 9:25 am, 9:35 - 10:30 am ; tutorials : 10:45 - 11:40 am, 11:50 am - 12:45 pm). Attention : No CM/TD on 05 November 2018
Intermediate exam : Monday, 12 November 2018 10:30 - 12:30 in room L357/359, third floor of 24 rue Lhomond.
Final exam : Monday, 14 January 2019 9:00 - 12:00 in room L357/359, third floor of 24 rue Lhomond.

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.


  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.

Lecture notes will be available before each course.

  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)