ICFP Stat Physics 2016

From Werner KRAUTH

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* [http://www.lps.ens.fr/~krauth/images/8/89/TD06_ICFP_2016.pdf Tutorial 06: High-temperature expansion of the Ising model - Magnetic field, susceptibilities, and correlation functions] * [http://www.lps.ens.fr/~krauth/images/8/89/TD06_ICFP_2016.pdf Tutorial 06: High-temperature expansion of the Ising model - Magnetic field, susceptibilities, and correlation functions]
* [http://www.lps.ens.fr/~krauth/images/4/45/HW06_ICFP_2016.pdf Homework 06: Inside the Kac-Ward solution of the two-dimensional Ising model.] Note that we make the connection between the Kac-Ward matrix and the architecture of highway crossings! * [http://www.lps.ens.fr/~krauth/images/4/45/HW06_ICFP_2016.pdf Homework 06: Inside the Kac-Ward solution of the two-dimensional Ising model.] Note that we make the connection between the Kac-Ward matrix and the architecture of highway crossings!
-NB: In the text of the HW06, we suppose N=even. Text minimally modified on 16/10/2016.+NB: In the text of the HW06, we suppose N=even. Furthermore, note that an identity cycle is a cycle of length 1. Text minimally modified on 16/10/2016.
* [http://www.lps.ens.fr/~krauth/images/8/85/U_2x2_nb.pdf PDF of Mathematica notebook file (setup of Kac-Ward matrix for a 2x2 Ising model) essential for Homework 06.] * [http://www.lps.ens.fr/~krauth/images/8/85/U_2x2_nb.pdf PDF of Mathematica notebook file (setup of Kac-Ward matrix for a 2x2 Ising model) essential for Homework 06.]
* Please check out the program [[Ising dual 4x4.py]], which takes a very close look at the Kramers-Wannier duality for a finite lattice. The material here is related to the first module of lecture 06, and is also discussed in the Feynman lectures on Statistical Mechanics. * Please check out the program [[Ising dual 4x4.py]], which takes a very close look at the Kramers-Wannier duality for a finite lattice. The material here is related to the first module of lecture 06, and is also discussed in the Feynman lectures on Statistical Mechanics.

Revision as of 20:31, 16 October 2016

This is the homepage for the ICFP course: Statistical Physics: Concepts and Applications that is running from 7 September 2016 through 14 December 2016.

Lectures: Werner KRAUTH

Practicals & Homeworks: Maurizio FAGOTTI, Olga PETROVA

Look here for practical information

Contents

Week 6 (12 October 2016): Two-dimensional Ising model: From Kramers & Wannier to Kac & Ward

NB: In the text of the HW06, we suppose N=even. Furthermore, note that an identity cycle is a cycle of length 1. Text minimally modified on 16/10/2016.

References for Week 6:

  • W. Krauth, "Statistical Mechanics: Algorithms and Computations" (Oxford, 2006) section 5.1.3 (high-temperature expansion, following van der Waerden (1941)), and section 5.1.4 (Kac-Ward solution)).
  • R. P. Feynman "Statistical Mechanics: A set of Lectures" (Benjamin/Cummings, 1972) (thorough discussion of Kramers-Wannier duality which yields the value of T_c, some discussion of the Kac-Ward solution).
  • M. Kac, J. C. Ward, "A combinatorial solution of the two-dimensional Ising model" Physical Review 185, 832 (1952) (NB: The paper contains the explicit diagonalization of the matrix U).
  • J. M. Yeomans, "Statistical Mechanics of Phase Transitions (Oxford, 1992), chapter 6 (for exercise 1 of tutorial 06).

Week 5 (5 October 2016): Two-dimensional Ising model: From Ising to Onsager

References for Week 5:

  • R. Peierls, Proceedings of the Cambridge Philosophical Society, 32, 477 (1936) (famous loop-counting argument establishing spontaneous symmetry breaking in the two-dimensional Ising model below a finite temperature)
  • C. Bonati, Eur. J. Phys. 35, 035002 (2014) (generalization of the Peierls argument to higher dimensions)
  • M Plischke, B Bergersen, "Equilibrium Statistical Physics" (World Scientific) section 6.1 (Transfer matrix for the two-dimensional Ising model, Onsager's solution)
  • T D Schultz, D C Mattis, E Lieb, "Two-dimensional Ising model as a soluble problem of many fermions" Reviews of Modern Physics (1964) (Authoritative account of Onsager's solution).

Week 4 (28 September 2016): Physics in one dimension - Hard spheres and the Ising model

References for Week 4:

  • W. Krauth, "Statistical Mechanics: Algorithms and Computations" (Oxford, 2006) p. 269ff (hard-sphere partition function using the double substitution).
  • M Plischke, B Bergersen, "Equilibrium Statistical Physics" (World Scientific) p. 145f (some background material on the virial expansion), p. 77 ff (Ising chain, although our treatment was considerably different).
  • R. H. Swendsen, "Statistical mechanics of colloids and {Boltzmann's} definition of the entropy" American Journal of Physics 74, 187 (2006) (a good discussion of the Gibbs phenomenon)
  • D. J. Thouless, "Long-range order in one-dimensional Ising systems" Physical Review 187, 732 (1969) (Ingenious discussion of the 1/r^2 Ising model)
  • J. M. Kosterlitz, "Kosterlitz-Thouless physics: a review of key issues" Rep. Prog. Phys. 79 026001 (2016) (first two pages only, discussion and historical context for the Thouless paper. This is elementary to follow.).
  • C. Kittel, American Journal of Physics 37, 917 (1969) (First exercise of Tutorial 4)
  • J. A. Cuesta and A. Sanchez, J. Stat. Phys. 115, 869 (2004) (Third exercise of Tutorial 4, generalized Kittel model)

Week 3 (21 September 2016): Statistical mechanics and Thermodynamics

References for Week 3:

  • Kerson Huang, "Statistical Mechanics 2nd edition" (1987) (Tutorial Problem 1).
  • L. Pauling, J. Am. Chem. Soc. 12 (2680-2684), 1935.(Tutorial Problem 2 on residual entropy of ice).
  • Bramwell, Gingras, Science 294, 1495 ( 2001) (Spin ice in pyrochlore).

Week 2 (14 September 2016): Statistical inference

References for Week 2:

  • L. Wasserman, "All of Statistics, A Concise Course in Statistical Inference" (Springer, 2005) part 2
  • W. Krauth, "Statistical Mechanics: Algorithms and Computations" (Oxford, 2006) p. 58 only ;)
  • B. Efron, "Maximum likelihood and decision theory" Ann. Statist. 10, 340, 1982.
  • B. Efron, "Bootstrap methods: another look at the jackknife" The Annals of Statistics, 1-26, 1979.
  • P. Diaconis and B. Efron, "Computer intensive methods in statistics" Scientific American 248, no. 5, pp. 116-130, 1983.

Week 1 (7 September 2016): Probability theory


Syllabus

  • Week 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
  • Week 2: Statistics (statistical inference, estimation, learning)
    • Point estimation, confidence intervals
    • Bootstrap
    • Method of moments
    • Maximum likelihood, Fisher information
    • Parametric Bootstrap
    • Bayes statistics
  • Week 3: Statistical mechanics and Thermodynamics
    • Rapid overview on the connection between statistical mechanics and thermodynamics
    • Ensembles and physical observables (partition function, energy, free energy, entropy, chemical potential, correlation functions, etc).
  • Week 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
  • Week 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
  • Week 6: Two-dimensional Ising model: From Kac and Ward to Saul and Kardar
    • Van der Waerden, low-temperature and high-temperature expansions
    • Duality
  • Week 7: Physics in two dimensions (Kosterlitz-Thouless physics): XY (planar rotor) model
    • Peierls argument
    • Mermin-Wagner theorem
    • Non-universality
  • Week 8: Physics in two dimensions (Kosterlitz-Thouless physics): Particle systems, superfluids
  • Week 09: Physics in infinite dimensions: Mean-field theory, Scaling
  • Week 10: Physics in infinite dimensions: Landau theory
  • Week 11: Renormalization group
  • Week 12: The Solid state: Order parameters, correlation functions
  • Week 13: Quantum systems - bosons.
  • Week 14: Quantum systems - spin systems
  • Week 15: Equilibrium and transport, Fluctuation-dissipation theorem.

References

Lecture notes will be available before each course.

Books

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