# ICFP Stat Physics 2019

(Difference between revisions)
 Revision as of 12:07, 3 January 2020Werner (Talk | contribs)← Previous diff Revision as of 12:18, 3 January 2020Werner (Talk | contribs) Next diff → Line 24: Line 24: ==Week 1 (09 September 2019): Probability theory== ==Week 1 (09 September 2019): Probability theory== + + * [http://www.lps.ens.fr/~krauth/images/5/5c/TD01_ICFP_2019_sol.pdf Tutorial 01: Characteristic functions / Stable distributions (with solutions)] + * [http://www.lps.ens.fr/~krauth/images/b/b9/HW01_ICFP_2019.pdf Homework 01: Chebychev inequality / Rényi formula / Lévy distribution (with solutions)] References for Week 1: References for Week 1: Line 29: Line 32: * W. Krauth, "[http://www.lps.ens.fr/~krauth/images/7/73/SMAC_1_3_4.pdf Statistical Mechanics: Algorithms and Computations]" (Oxford, 2006) Section 1.3.4 only - Error intervals from Chebychev inequality. * W. Krauth, "[http://www.lps.ens.fr/~krauth/images/7/73/SMAC_1_3_4.pdf Statistical Mechanics: Algorithms and Computations]" (Oxford, 2006) Section 1.3.4 only - Error intervals from Chebychev inequality. - * [http://www.lps.ens.fr/~krauth/images/5/5c/TD01_ICFP_2019_sol.pdf Tutorial 01: Characteristic functions / Stable distributions] + - * [http://www.lps.ens.fr/~krauth/images/b/b9/HW01_ICFP_2019.pdf Homework 01 (with solutions): Chebychev inequality / Rényi formula / Lévy distribution] + ** Useful program: [[renyi_problem_HW01_ICFP_2018.py| Renyi.py: Probability distribution of a sum of uniform random numbers]] ** Useful program: [[renyi_problem_HW01_ICFP_2018.py| Renyi.py: Probability distribution of a sum of uniform random numbers]] Line 37: Line 39: ==Week 2 (16 September 2019): Statistical inference== ==Week 2 (16 September 2019): Statistical inference== - * [http://www.lps.ens.fr/~krauth/images/a/a8/TD02_ICFP_2019_sol.pdf Tutorial 02: Inference - from maximum likelihood to Bootstrap] + * [http://www.lps.ens.fr/~krauth/images/a/a8/TD02_ICFP_2019_sol.pdf Tutorial 02: Inference - from maximum likelihood to Bootstrap (with solutions)] - * [http://www.lps.ens.fr/~krauth/images/a/a1/HW02_ICFP_2019.pdf Homework 02: Estimation, from maximum likelihood to Bayes] + * [http://www.lps.ens.fr/~krauth/images/a/a1/HW02_ICFP_2019.pdf Homework 02: Estimation, from maximum likelihood to Bayes (with solutions)] + * [http://www.lps.ens.fr/~krauth/images/1/1a/BootstrapConfidenceInterval.pdf Fact sheet: Bootstrap confidence interval] + * [http://www.lps.ens.fr/~krauth/images/e/ed/FactSheet_DKW.pdf Fact sheet: The Dvoretzky–Kieffer–Wolfowitz inequality or the incredible power of non-parametric statistics] + ** [[bayes_tank_problem_HW02_ICFP_2019.py| Bayes_tank.py: Bayesian approach to solving the German Tank problem]] ** [[bayes_tank_problem_HW02_ICFP_2019.py| Bayes_tank.py: Bayesian approach to solving the German Tank problem]] - * [http://www.lps.ens.fr/~krauth/images/1/1a/BootstrapConfidenceInterval.pdf Fact sheet on the Bootstrap confidence interval] + References for Week 2: References for Week 2: Line 49: Line 54: * B. Efron, "[https://projecteuclid.org/euclid.aos/1176344552 Bootstrap methods: another look at the jackknife]" The Annals of Statistics, 1-26, 1979. * B. Efron, "[https://projecteuclid.org/euclid.aos/1176344552 Bootstrap methods: another look at the jackknife]" The Annals of Statistics, 1-26, 1979. * P. Diaconis and B. Efron, "[http://web.cecs.pdx.edu/~cgshirl/Documents/Demonstrations/1983%20Diaconis%20Efron%20Computer-Intensive%20Methods%20in%20Statistics%20Sci%20Am%20May%201983.pdf Computer intensive methods in statistics]" Scientific American 248, no. 5, pp. 116-130, 1983. * P. Diaconis and B. Efron, "[http://web.cecs.pdx.edu/~cgshirl/Documents/Demonstrations/1983%20Diaconis%20Efron%20Computer-Intensive%20Methods%20in%20Statistics%20Sci%20Am%20May%201983.pdf Computer intensive methods in statistics]" Scientific American 248, no. 5, pp. 116-130, 1983. - - Factsheet for Week 2: - * [http://www.lps.ens.fr/~krauth/images/e/ed/FactSheet_DKW.pdf The Dvoretzky–Kieffer–Wolfowitz inequality or the incredible power of non-parametric statistics] ==Week 3 (23 September 2019): Statistical mechanics and Thermodynamics== ==Week 3 (23 September 2019): Statistical mechanics and Thermodynamics== + + * [http://www.lps.ens.fr/~krauth/images/1/15/TD03_ICFP_2019.pdf Tutorial 03: Basic statistical mechanics - spin ice, pressure, etc (with solutions)] I finished the discussion of error intervals in statistics, following the discussion in I finished the discussion of error intervals in statistics, following the discussion in Line 64: Line 68: - In the discussion of the foundations of statistical mechanics, I used the following references * D. A. Levin, Y. Peres "Markov Chains and Mixing Times, second edition" (for the discussion of the ergodic theorem) * D. A. Levin, Y. Peres "Markov Chains and Mixing Times, second edition" (for the discussion of the ergodic theorem) * Landau Lifshitz V, chapters 3,4,5 * Landau Lifshitz V, chapters 3,4,5 - - - * [http://www.lps.ens.fr/~krauth/images/1/15/TD03_ICFP_2019.pdf Tutorial 03: Basic statistical mechanics - spin ice, pressure, etc] - - - - References for the TD of week 3: - - * K. Huang, "Statistical Mechanics 2nd edition" (1987) (Tutorial Problem 1). * K. 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). * L. Pauling, J. Am. Chem. Soc. 12 (2680-2684), 1935.(Tutorial Problem 2 on residual entropy of ice). Line 82: Line 76: ==Week 4 (30 September 2019): Phases and phase transitions: Van der Waals theory== ==Week 4 (30 September 2019): Phases and phase transitions: Van der Waals theory== - * [http://www.lps.ens.fr/~krauth/images/3/3f/HW04_ICFP_2019.pdf Homework 04 (with solution): Van-der-Waals theory and beyond] + * [http://www.lps.ens.fr/~krauth/images/3/3f/HW04_ICFP_2019.pdf Homework 04: Van-der-Waals theory and beyond (with solutions)] - + * [http://www.lps.ens.fr/~krauth/images/6/66/TD04_ICFP_2019.pdf Tutorial 04: Clapeyron's equation, and first-order transitions in liquid crystals (with solutions)] - ** Useful program: [[Van_der_Waals.py| Van_der_Waals.py - Liquid-gas equation of state]] + * Useful program for Homework 04: [[Van_der_Waals.py| Van_der_Waals.py - Liquid-gas equation of state]] - + - * [http://www.lps.ens.fr/~krauth/images/6/66/TD04_ICFP_2019.pdf Tutorial 04: Clapeyron's equation, and first-order transitions in liquid crystals] + - + References for Week 4: References for Week 4: Line 103: Line 94: ==Week 5 (07 October 2019): Hard spheres and the Ising model in one dimension (Transfer matrix 1/2)== ==Week 5 (07 October 2019): Hard spheres and the Ising model in one dimension (Transfer matrix 1/2)== - * [http://www.lps.ens.fr/%7Ekrauth/images/0/0b/HW05_ICFP_2019.pdf Homework 05 (with solutions): Transfer matrices] + * [http://www.lps.ens.fr/%7Ekrauth/images/3/32/TD05_ICFP_2019.pdf Tutorial 05: One-dimensional classical models with phase transitions (with solutions)] - * [http://www.lps.ens.fr/%7Ekrauth/images/3/32/TD05_ICFP_2019.pdf Tutorial 05: One-dimensional classical models with phase transitions] + * [http://www.lps.ens.fr/%7Ekrauth/images/0/0b/HW05_ICFP_2019.pdf Homework 05: Transfer matrices (with solutions)] References for Week 5: References for Week 5: Line 116: Line 107: ==Week 6 (14 October 2019): Two-dimensional Ising model: From Ising to Onsager (Transfer matrix 2/2)== ==Week 6 (14 October 2019): Two-dimensional Ising model: From Ising to Onsager (Transfer matrix 2/2)== - * [http://www.lps.ens.fr/%7Ekrauth/images/8/8d/TD06_ICFP_2019.pdf Tutorial 06: Peierls argument for spontaneous symmetry breaking in two and higher dimensions (with solutions).] * [http://www.lps.ens.fr/%7Ekrauth/images/8/8d/TD06_ICFP_2019.pdf Tutorial 06: Peierls argument for spontaneous symmetry breaking in two and higher dimensions (with solutions).] - * [http://www.lps.ens.fr/%7Ekrauth/images/1/19/HW06_ICFP_2019.pdf Homework 06: Thouless (!) argument; transfer matrix for the two-dimensional Ising model on a stripe.] + * [http://www.lps.ens.fr/%7Ekrauth/images/1/19/HW06_ICFP_2019.pdf Homework 06: Thouless (!) argument; transfer matrix for the two-dimensional Ising model on a stripe (with solutions).] - * [http://www.lps.ens.fr/~krauth/images/2/2f/Transfer_2d_Ising.pdf PDF of Mathematica notebook file useful for Lecture 06 and Homework 06.] + * [http://www.lps.ens.fr/~krauth/images/2/2f/Transfer_2d_Ising.pdf Useful Mathematica notebook file for Lecture 06 and Homework 06.] Line 132: Line 122: * [http://www.lps.ens.fr/%7Ekrauth/images/f/f8/TD07_ICFP_2019.pdf Tutorial 07: Thermodynamic quantities and high-temperature expansions (with solutions)] * [http://www.lps.ens.fr/%7Ekrauth/images/f/f8/TD07_ICFP_2019.pdf Tutorial 07: Thermodynamic quantities and high-temperature expansions (with solutions)] + * [http://www.lps.ens.fr/%7Ekrauth/images/4/4b/HW07_ICFP_2019.pdf Homework 07: Graphical solution for the two-dimensional Ising model (with solutions)] * [[Ising dual 4x4.py| Python2 program centered around the question of duality in the two-dimensional Ising model]] * [[Ising dual 4x4.py| Python2 program centered around the question of duality in the two-dimensional Ising model]] Line 137: Line 128: * [http://www.lps.ens.fr/%7Ekrauth/images/2/2a/U_4x4.pdf PDF of Mathematica notebook file (setup of Kac-Ward matrix for a 4x4 Ising model) from Homework 07 (beautiful solution found by Botao LI).] * [http://www.lps.ens.fr/%7Ekrauth/images/2/2a/U_4x4.pdf PDF of Mathematica notebook file (setup of Kac-Ward matrix for a 4x4 Ising model) from Homework 07 (beautiful solution found by Botao LI).] * [[Combinatorial ising.py| Here is a Python2 program for the 2x2 and the 4x4 lattice]] showing that the determinant of the Kac-Ward matrix gives the square of the corresponding partition functions. * [[Combinatorial ising.py| Here is a Python2 program for the 2x2 and the 4x4 lattice]] showing that the determinant of the Kac-Ward matrix gives the square of the corresponding partition functions. - * [http://www.lps.ens.fr/%7Ekrauth/images/4/4b/HW07_ICFP_2019.pdf Homework 07 (with solution): Graphical solution for the two-dimensional Ising model] + References for Week 7: References for Week 7: Line 148: Line 139: * [http://www.lps.ens.fr/%7Ekrauth/images/5/58/TD08_ICFP_2019.pdf Tutorial 08: Physics in infinite dimensions---Ising model on a Bethe lattice (with solutions)] * [http://www.lps.ens.fr/%7Ekrauth/images/5/58/TD08_ICFP_2019.pdf Tutorial 08: Physics in infinite dimensions---Ising model on a Bethe lattice (with solutions)] + * [http://www.lps.ens.fr/%7Ekrauth/images/5/50/HW08_ICFP_2019.pdf Homework 08: Mean-field theory as easy as 1-2-3 (with solutions)]. - * [http://www.lps.ens.fr/%7Ekrauth/images/5/50/HW08_ICFP_2019.pdf Homework 08: Mean-field theory as easy as 1-2-3]. In relation with homework 08, please check out the following programs: ** [[Mean_field_self_consistency_single_site.py]] and ** [[Mean_field_self_consistency_single_site.py]] and ** [[Mean field gen d Ising lattice.py]]. . Also check out the program ** [[Mean field gen d Ising lattice.py]]. . Also check out the program Line 169: Line 160: * [http://www.lps.ens.fr/%7Ekrauth/images/1/17/TD10_ICFP_2019.pdf Tutorial10: The roughening transition (with solutions)] * [http://www.lps.ens.fr/%7Ekrauth/images/1/17/TD10_ICFP_2019.pdf Tutorial10: The roughening transition (with solutions)] + * [http://www.lps.ens.fr/~krauth/images/5/58/HW10_ICFP_2018.pdf Homework 10: Topological excitation and their interactions in the XY model (with solutions)]] + * [http://www.lps.ens.fr/%7Ekrauth/images/c/cb/Wegner_XY_finite.pdf Wegner_XY_finite.pdf] Factsheet: Wegner's solution of the d-dimensional harmonic model.] + * Please check out [[Wegner 1d Exact.py|the program Wegner 1d Exact.py]], [[Wegner 1d Direct.py|the program Wegner 1d Direct.py]], as well as [[Wegner 2d Exact.py|the program Wegner 2d Exact.py]]. + + - See for yourself!]. NB: Please check out [[Vortex pair.py| the program Vortex_pair.py]] that will allow you to generate configurations without vortices, with one vortex, and with a vortex-antivortex pair. References for Week 10: References for Week 10: - - - * [http://www.lps.ens.fr/~krauth/images/5/58/HW10_ICFP_2018.pdf Homework 10: Topological excitation and their interactions in the XY model - See for yourself!]. NB: Please check out [[Vortex pair.py| the program Vortex_pair.py]] that will allow you to generate configurations without vortices, with one vortex, and with a vortex-antivortex pair. - - * [http://www.lps.ens.fr/%7Ekrauth/images/c/cb/Wegner_XY_finite.pdf Wegner_XY_finite.pdf] Factsheet: Wegner's solution of the d-dimensional harmonic model. Please check out [[Wegner 1d Exact.py|the program Wegner 1d Exact.py]], [[Wegner 1d Direct.py|the program Wegner 1d Direct.py]], as well as [[Wegner 2d Exact.py|the program Wegner 2d Exact.py]]. - * 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). See factsheet. * 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). See factsheet.

## Revision as of 12:18, 3 January 2020

This is the homepage for the ICFP course: Statistical Physics: Concepts and Applications that is running from 09 September 2019 through 13 January 2020. Lectures start at 8:30 on Monday mornings, and tutorials at 10:45.

Lectures: Werner KRAUTH

Tutorials (TD): Victor DAGARD, Valentina ROS

Homeworks, factsheets: Botao LI

Latest News:

• 02 January 2020: Solutions of TD08, TD10, TD11, and of HW08 uploaded.
• 30 December 2019: Here are lecture notes, that I am still continuing to work on all the time pages 1-75, pages 76-125, pages 126-end.
• THERE IS NO LECTURE ON 28 October 2019!! Next Lecture on 4 November 2019; Mid-term exam on 18 November 2019.
• 24 October 2019: Previous Midterm exams uploaded: 2017 2018
• 24 October 2019: Solution of Homework 06 available, Python program for Lecture 07 (Duality)
• 22 October 2019: Solution of Homework 05 available
• 07 October 2019: Lecture 5 (preliminary, without HW and TD)
• 30 September 2019: Lectures 1-3 (preliminary version)
• 29 September 2019: Factsheet Bootstrap method, Factsheet DKW inequality available, solution HW01 available.
• 09 September 2019: TD01 (with solution) and HW01 uploaded
• 31 August 2019: Website set up

## Week 1 (09 September 2019): Probability theory

References for Week 1:

## Week 2 (16 September 2019): Statistical inference

References for Week 2:

Further References for Week 2:

## Week 3 (23 September 2019): Statistical mechanics and Thermodynamics

I finished the discussion of error intervals in statistics, following the discussion in

• W. Krauth SMAC, pp 55-59
• L. D. Brown, T. Tony Cai, A. DasGupta "Interval Estimation for a Binomial Proportion" Statistical Science 16, 101–133 (2001). This highly cited paper started a big discussion on the use and mis-use of error bars.
• L. Wasserman "All of statistics", Section 6.3.2, p. 92 (on confidence sets).
• Useful program: Wald_Interval_Lecture3_ICFP_2019.py This program checks the probability with which the error bar (confidence interval) computed from the approximation of the binomial distribution actually contains the parameter of the Bernoulli distribution.
• Useful program: Chebychev_Hoeffding_Interval_Lecture3_ICFP_2019.py This program checks the probability with which the error bar (confidence interval) computed from the Chebychev and Hoeffding inequalities actually contains the parameter of the Bernoulli distribution.

• D. A. Levin, Y. Peres "Markov Chains and Mixing Times, second edition" (for the discussion of the ergodic theorem)
• Landau Lifshitz V, chapters 3,4,5
• K. 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 4 (30 September 2019): Phases and phase transitions: Van der Waals theory

References for Week 4:

• L. D. Landau, E. M. Lifshitz V, "Statistical Physics" (Pergamon, 1959, and later editions). NB: Chapter numbers and titles vary with edition. The following chapters all refer to the Lecture:
• Chap 73 "Conditions of phase equilibrium"
• Chap 79 "The critical point" (note that LL do not use the term "spinodal" for the points where dP/dV vanishes)
• Chap 71 "Deviations of gases from the ideal state"
• Chap 73 "Van der Waals' equation"
• Chap 82 "The law of corresponding states"
• Chap 152 (in some editions only) "Van der Waals theory of the critical point"
• Chap 21 "Thermodynamic inequalities" (dP/dV < 0 is not strictly valid (!!) in finite systems - see homework)
• R. A. Sauerwein, M. J. De Oliveira "Lattice model for biaxial and uniaxial nematic liquid crystals" J. of Chem. Phys. 144, 194904 (2016, Tutorial)
• J. E. Mayer, W. W. Wood, "Interfacial Tension Effects in Finite, Periodic, Two-Dimensional Systems", Journal of Chemical Physics, 42, 4268 (1965, for the homework)

## Week 5 (07 October 2019): Hard spheres and the Ising model in one dimension (Transfer matrix 1/2)

References for Week 5:

• 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 5)
• J. A. Cuesta and A. Sanchez, J. Stat. Phys. 115, 869 (2004) (Third exercise of Tutorial 5, generalized Kittel model)

## Week 6 (14 October 2019): Two-dimensional Ising model: From Ising to Onsager (Transfer matrix 2/2)

References for Week 6:

• 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 7 (21 October 2019): Two-dimensional Ising model: From Kramers & Wannier to Kac & Ward (Low- and high-temperature expansions)

References for Week 7:

## Week 8 (04 November 2019): The three pillars of mean-field theory (Transitions and order parameters 1/2)

References for Week 8:

• 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)
• M. Plischke, B. Bergersen, "Equilibrium Statistical Physics" (World Scientific) section 3.1, pp 67 - 68 (Bragg-Williams theory)
• C. M. Bender, S. A. Orszag, "Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory" (Springer, 1999) (difference equations)

## Week 9 (18 November 2019): Landau theory / Ginzburg criterium (Transitions and order parameters 2/2) / Midterm exam

References for Week 9:

• 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 in the lecture we avoided the Fourier transform)
• L. D. Landau, E. M. Lifshitz, "Statistical Physics", Chap 147 (Ginzburg criterion).

## Week 10 (25 November 2019): Kosterlitz-Thouless physics in two dimensions: The XY model (Transitions without order parameters 1/2)

- See for yourself!]. NB: Please check out the program Vortex_pair.py that will allow you to generate configurations without vortices, with one vortex, and with a vortex-antivortex pair.

References for Week 10:

• 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). See factsheet.
• 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). See homework.
• J. Fröhlich, T. Spencer "The Kosterlitz-Thouless Transition in Two-Dimensional Abelian Spin Systems and the Coulomb Gas" Comm. Math. Phys. 81, 527 (1981) (Paper proving 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 transitiont. The title thus overstates the content of the paper).
• 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 11 (02 December 2019): Kosterlitz-Thouless physics in two dimensions: KTHNY Melting theory (Transitions without order parameters 2/2)

References for Week 11:

• 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, quite 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, quite advanced).
• 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).

## Week 12 (09 December 2019): The renormalization group - an introduction

References for Week 12:

• 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 13 (16 December 2019): Quantum statistics 1/2: Ideal Bosons

References for Week 13:

• J. A. Lipa et al, "Specific heat of liquid helium in zero gravity very near the lambda point", Phys. Rev. B 68, 174518 (2003) (Space-shuttle experiments)
• M. Hasenbusch, "The three-dimensional XY universality class: a high precision Monte Carlo estimate of the universal amplitude ratio A +/ A −" J. Stat. Mech. (2006) P08019 (Interpretation of space-shuttle experiments in 3d XY model).
• D. M. Ceperley, E. L. Pollock, "Path-integral computation of the low-temperature properties of liquid 4He" Phys. Rev. Lett. 56, 351 (1986) (First-principles numerical computation of the Lambda transition)
• W. Krauth, "Statistical Mechanics: Algorithms and Computations" (2006) Chap 5.1: The two formulations of the model of ideal bosons
• A. Kitaev, "Fault-tolerant quantum computation by anyons", Annals Phys. 303, 2-30 (2003, Tutorial)

## Week 14 (06 January 2020): Quantum statistics 2/2: 4He and the 3D Heisenberg model, Non-classical rotational inertia

References for Week 14:

• G. B. Hess and W. M. Fairbank "Measurement of angular momentum in superfluid helium" Phys. Rev. Lett. 19, 216 (1967) (Non-classical response of a quantum fluid to rotation - A slowly rotating 4He vessel accelerates when cooled (!)).
• W. Krauth "Statistical Mechanics: Algorithms and Computations" Sect 3.1.4. (Allows to understand non-classical rotational inertia by only considering an ideal quantum particles).
• A. J. Leggett "Topics in the theory of helium" Physica Fennica 8, 125 (1973) (Fundamental paper which explains Non-classical rotational inertia very similarly to how we proceeded in the lecture.)

## Week 15 (13 January 2020): The Fluctuation-Dissipation theorem (an introduction)

References for Week 15:

• R. Kubo "The fluctuation-dissipation theorem" Reports on Progress in Physics, 29, 255 (1966). This is a fundamental text, of which we treat the first 10 pages, or so, in the lecture.
• H. Risken "The Fokker-Planck equation (Springer Verlag, 1996).

## 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, 2006)
• L. D. Landau, E. M. Lifshitz, "Statistical Physics" (Pergamon, 1969)