ICFP Stat Physics 2016
From Werner KRAUTH
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* F. Wegner, "Spin-Ordering in a Planar Classical Heisenberg Model" Z. Phys 206, 465 (1967) (Exact solution of the harmonic approximation to the XY model, algebraic long-range correlations) | * F. Wegner, "Spin-Ordering in a Planar Classical Heisenberg Model" Z. Phys 206, 465 (1967) (Exact solution of the harmonic approximation to the XY model, algebraic long-range correlations) | ||
* J. M. Kosterlitz, D. M. Thouless "Ordering, Metastability and phase transitions in two-dimensional systems" J. Phys. C: Solid State Physics 6, 1181 (1973) (Nobel-prize winning paper, proposing topological excitations. For the free-energy argument for the XY model see p. 1190 ff). | * J. M. Kosterlitz, D. M. Thouless "Ordering, Metastability and phase transitions in two-dimensional systems" J. Phys. C: Solid State Physics 6, 1181 (1973) (Nobel-prize winning paper, proposing topological excitations. For the free-energy argument for the XY model see p. 1190 ff). | ||
- | * J. Fröhlich, T. Spencer "The Kosterlitz-Thouless Transition in Two-Dimensional Abelian Spin Systems and the Coulomb Gas" Comm. Math. Phys. 81, 527 (1981) (Famous paper proving the existence of a low-temperature phase with algebraic correlations. Nuance: This paper proves the existence of a low-temperature phase but not of a KT transition). | + | * J. Fröhlich, T. Spencer "The Kosterlitz-Thouless Transition in Two-Dimensional Abelian Spin Systems and the Coulomb Gas" Comm. Math. Phys. 81, 527 (1981) (Important paper proving the existence of a low-temperature phase with algebraic correlations. Nuance: This paper proves the existence of a low-temperature phase but not the presence of a KT transition). |
+ | * E. Domany, M. Schick, and R. H. Swendsen "First-Order Transition in an xy Model with Nearest-Neighbor Interactions Phys. Rev. Lett. 52, 1535 (1984) (Paper explaining the two-energy scales J (for a first-order transition) and J_R (for the KT transition)) | ||
+ | * M. Hasenbusch, "The two-dimensional XY model at the transition temperature: a high-precision Monte Carlo study" J. Phys. A: Math. Gen. 38, 5869 (2005) (This paper establishes beyond reasonable doubt that the transition in the XY model is indeed of the Kosterlitz-Thouless type. It also computes the critical temperature to 5 significant digits). | ||
==Week 6 (12 October 2016): Two-dimensional Ising model: From Kramers & Wannier to Kac & Ward== | ==Week 6 (12 October 2016): Two-dimensional Ising model: From Kramers & Wannier to Kac & Ward== |
Revision as of 12:47, 19 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
Week 7 (19 October 2016): Kosterlitz-Thouless physics (physics in two dimensions) 1/2: The XY (planar rotor) model
References for Week 7:
- F. Wegner, "Spin-Ordering in a Planar Classical Heisenberg Model" Z. Phys 206, 465 (1967) (Exact solution of the harmonic approximation to the XY model, algebraic long-range correlations)
- J. M. Kosterlitz, D. M. Thouless "Ordering, Metastability and phase transitions in two-dimensional systems" J. Phys. C: Solid State Physics 6, 1181 (1973) (Nobel-prize winning paper, proposing topological excitations. For the free-energy argument for the XY model see p. 1190 ff).
- J. Fröhlich, T. Spencer "The Kosterlitz-Thouless Transition in Two-Dimensional Abelian Spin Systems and the Coulomb Gas" Comm. Math. Phys. 81, 527 (1981) (Important paper proving the existence of a low-temperature phase with algebraic correlations. Nuance: This paper proves the existence of a low-temperature phase but not the presence of a KT transition).
- E. Domany, M. Schick, and R. H. Swendsen "First-Order Transition in an xy Model with Nearest-Neighbor Interactions Phys. Rev. Lett. 52, 1535 (1984) (Paper explaining the two-energy scales J (for a first-order transition) and J_R (for the KT transition))
- M. Hasenbusch, "The two-dimensional XY model at the transition temperature: a high-precision Monte Carlo study" J. Phys. A: Math. Gen. 38, 5869 (2005) (This paper establishes beyond reasonable doubt that the transition in the XY model is indeed of the Kosterlitz-Thouless type. It also computes the critical temperature to 5 significant digits).
Week 6 (12 October 2016): Two-dimensional Ising model: From Kramers & Wannier to Kac & Ward
- Tutorial 06: High-temperature expansion of the Ising model - Magnetic field, susceptibilities, and correlation functions
- 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. Furthermore, note that an identity cycle is a cycle of length 1. Text minimally modified on 16/10/2016.
- 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.
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
- Tutorial 05: Peierls argument for spontaneous symmetry breaking in two and higher dimensions.
- Homework 05: Thouless (!) argument; transfer matrix for the two-dimensional Ising model on a stripe.
- PDF of Mathematica notebook file useful for Lecture 05 and Homework 05.
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
- Tutorial 04: One-dimensional short-range interacting systems with phase transitions (!).
- Homework 04: Transfer matrices for hard spheres, variations on a theme. NB: Figures updated for clarity on 30 Sep 2016
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
- Tutorial 03: Two-level systems and the entropy of ice. Maxwell's distribution.
- NB: Homework break for this week, graded homeworks will start in week 4.
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
- Tutorial 02: Maximum likelihood, Bootstrap and Bayes without a computer
- Homework 02: From Maximum Likelihood to Bayes statistics Useful program:
- NB: Homework 02 will be corrected, but does not count for the final grade ("dry run"). Due on 21 September 2016, return of corrected copies: 28 September 2016.
- Bayes_tank.py: Bayesian approach to solving the German Tank problem
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
- Tutorial 01: Characteristic functions / Stable distributions
- Homework 01: Chebychev inequality / Rényi formula / Lévy distribution
- NB: Homework 01 will be corrected, but does not count for the final grade ("dry run"). Due on 14 September 2016, return of corrected copies: 21 September 2016.
- Useful program: Renyi.py: Probability distribution of a sum of uniform random numbers
- Useful program: Levy.py: Probability distribution of a sum of random variables that may (or may not) have an infinite variance
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)