ICFP Stat Physics 2016
From Werner KRAUTH
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[[ICFP Stat Physics 2016 infos| Look here for practical information]] | [[ICFP Stat Physics 2016 infos| Look here for practical information]] | ||
==Week 12 (23 November 2016): Phases and phase transitions: From van der Waals theory (and beyond) to liquid crystals== | ==Week 12 (23 November 2016): Phases and phase transitions: From van der Waals theory (and beyond) to liquid crystals== | ||
- | * [http://www.lps.ens.fr/~krauth/images/e/e3/TD11_ICFP_2016.pdf Tutorial 11: Real-space renormalization group for percolation] | + | * [http://www.lps.ens.fr/~krauth/images/d/dc/TD12_ICFP_2016.pdf Tutorial 12: First-order phase transitions: The case of liquid crystals] |
References for Week 12: | References for Week 12: | ||
==Week 11 (16 November 2016): The renormalization group - an introduction== | ==Week 11 (16 November 2016): The renormalization group - an introduction== |
Revision as of 14:31, 23 November 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 12 (23 November 2016): Phases and phase transitions: From van der Waals theory (and beyond) to liquid crystals
References for Week 12:
Week 11 (16 November 2016): The renormalization group - an introduction
References for Week 11:
- H. J. Maris & L. P. Kadanoff "Teaching the renormalization group" American Journal of Physics 46, 652 (1978)
- l. P. Kadanoff "Scaling Laws for Ising models near T_c" Physics 2, 263 (1966)
- K. G. Wilson "The renormalization group: Critical phenomena and the Kondo problem" Reviews of Modern Physics 47, 773 (1975)
- K. G. Wilson "The renormalization group and critical phenomena" Reviews of Modern Physics 55, 583 (1983) (Nobel lecture 1982)
- P. J. Reynolds, H. E. Stanley and W. Klein "A Real-space renormalization group for site and bond percolation" J. of Phys. C, 10, L167 (1977) (Tutorial)
- D. Stauffer, A. Aharony, "Introduction to Percolation Theory", 2nd rev. ed., Taylor & Francis, 2003 (Tutorial)
Week 10 (09 November 2016): Mean-Field theory 2/2 - Landau theory (definite formulation of MFT (1937), Ginzburg criterium (validity of MFT (1960))
References for Week 10:
- R. J. Baxter: "Exactly solved models in Statistical Mechanics" (1982) (Chapter 3: We expanded the free energy of the Ising model on a fully connected graph to motivate Landau theory)
- J. Als‐Nielsen and R. J. Birgeneau: "Mean field theory, the Ginzburg criterion, and marginal dimensionality of phase transitions" Am. Journal of Physics 45, 554 (1977) (Elementary discussion of the Ginzburg criterion, although we avoided the Fourier transform)
- L. D. Landau, E. M. Lifshitz, "Statistical Physics", Chap 147 (Ginzburg criterion).
Week 9 (02 November 2016): Mean-Field theory 1/2 - The three pillars: Self-consistency, absence of fluctuations, infinite-dimensional limit
- Tutorial 09: The Bethe lattice
- Homework 09: Mean-field theory as easy as 1-2-3. NB: Not graded, but please study. Check out Mean_field_self_consistency_single_site.py and Mean field gen d Ising lattice.py. These programs are useful for treating the homework. Also check out the program Ising mean field 1d.py which determines the self-consistent mean field solution of a one-dimensional Ising chain, as well as its linearized approximation.
References for Week 9:
- R. J. Baxter: "Exactly solved models in Statistical Mechanics" (1982) (Chapter 3, for the solution of the Ising model on a fully connected graph)
- M Plischke, B Bergersen, "Equilibrium Statistical Physics" (World Scientific) section 3.1, pp 63 - 65 (Self-consistency à la Weiss, development for small m)
Week 8 (26 October 2016): Kosterlitz-Thouless physics (physics in two dimensions) 2/2: Melting theory in two dimensions (KTHNY theory)
References for Week 8:
- J. M. Kosterlitz, D. M. Thouless "Ordering, Metastability and phase transitions in two-dimensional systems" J. Phys. C: Solid State Physics 6, 1181 (1973) (First two pages: Motivation of KT theory <=> 2D melting. Origin of KT theory <=> dislocation theory of melting).
- N. D. Mermin, "Crystalline Order in 2 Dimensions", Phys. Rev. 176, 250 (1968) (Discovery of the dissociation of positional and orientational ordering in the two-dimensional harmonic model: see eqs 32 & 33).
- D. R. Nelson, B. I. Halperin, "Dislocation-mediated melting in two dimensions" Phys. Rev. B 19, 2457 (1979) (THE theory paper on the 2D melting theory, very advanced).
- A. P. Young "Melting and the vector Coulomb gas in two dimensions" Phys. Rev. B 19, 1855 (1979) (Vector nature of the dislocation-dislocation interaction, very advanced material).
- D. R. Nelson, J. M. Kosterlitz, "Universal Jump in the Superfluid Density of Two-Dimensional Superfluids" Phys. Rev. Lett. 39, 1201 (1977) (We did not yet treat in class this most striking prediction of KT theory).
- S. T. Chui and J. D. Weeks, Phys. Rev. B 23, 2438 (1981) (Tutorial 08).
- J. M. Yeomans, Statistical Mechanics of Phase Transitions (Oxford, 1992), chapter 5 (Tutorial 08).
Week 7 (19 October 2016): Kosterlitz-Thouless physics (physics in two dimensions) 1/2: The XY (planar rotor) model
- Tutorial 07: The harmonic solid
- Homework 07: Topological excitation and their interactions in the XY model - See for yourself!. NB: Please check out the program Vortex_pair.py - complete since 23 Oct 2016 that will allow you to generate configurations without vortices, with one vortex, and with a vortex-antivortex pair.
NNB: There was a question about part 1B: "When you ask to explain that the minimum energy of a configuration with a vortex ar the center is stable under local fluctuations what do you really mean? What kind of fluctuations?". Answer: "Well, it's just the question to know why the vortex does not go away if you modify single spins one after the other..."
WK 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). The XY model and its variant can have KT transitions or else first-order transitions.)
- 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 is the final one of a long series of computational-physics papers that have established that the transition in the XY model is indeed of the Kosterlitz-Thouless type. It computes the critical temperature to 5 significant digits: β_KT = 1.1199).
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 are not yet available. A few essential references are given each week. ICFP students can access these references from within the Department (you may for example connect to Web of Science, and download them from there). You may also ask the library staff at 29 rue d'Ulm.
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)