# ICFP Stat Physics 2019

### From Werner KRAUTH

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

Look here for practical information

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

- Tutorial 01: Characteristic functions / Stable distributions (with solutions)
- Homework 01: Chebychev inequality / Rényi formula / Lévy distribution (with solutions)

References for Week 1:

- L. Wasserman, "All of Statistics, A Concise Course in Statistical Inference" (Springer, 2005) part 1
- W. Krauth, "Statistical Mechanics: Algorithms and Computations" (Oxford, 2006) Section 1.3.4 only - Error intervals from Chebychev inequality.

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

- Tutorial 02: Inference - from maximum likelihood to Bootstrap (with solutions)
- Homework 02: Estimation, from maximum likelihood to Bayes (with solutions)
- Fact sheet: Bootstrap confidence interval
- Fact sheet: The Dvoretzky–Kieffer–Wolfowitz inequality or the incredible power of non-parametric statistics

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) Section 1.3.4 only

Further References for Week 2:

- 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 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

- Tutorial 04: Clapeyron's equation, and first-order transitions in liquid crystals (with solutions).
- Homework 04: Van-der-Waals theory and beyond (with solutions).
- Useful program: Van_der_Waals.py (for Homework 04).

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)

- Tutorial 05: One-dimensional classical models with phase transitions (with solutions)
- Homework 05: Transfer matrices (with solutions)

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)

- Tutorial 06: Peierls argument for spontaneous symmetry breaking in two and higher dimensions (with solutions).
- Homework 06: Thouless (!) argument; transfer matrix for the two-dimensional Ising model on a stripe (with solutions).
- Useful Mathematica notebook file for Lecture 06 and Homework 06.

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)

- Tutorial 07: Thermodynamic quantities and high-temperature expansions (with solutions)
- Homework 07: Graphical solution for the two-dimensional Ising model (with solutions)

- Python2 program centered around the question of duality in the two-dimensional Ising model
- PDF of Mathematica notebook file (setup of Kac-Ward matrix for a 2x2 Ising model) essential for Homework 07.
- PDF of Mathematica notebook file (setup of Kac-Ward matrix for a 4x4 Ising model) from Homework 07 (beautiful solution found by Botao LI).
- 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.

References for Week 7:

- 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 07).

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

- Tutorial 08: Physics in infinite dimensions---Ising model on a Bethe lattice (with solutions)
- Homework 08: Mean-field theory as easy as 1-2-3 (with solutions).
- Useful prgogram: Mean_field_self_consistency_single_site.py (for Homework 08).
- Useful program: Mean field gen d Ising lattice.py (for Homework 08).
- Useful program: Ising mean field 1d.py (for Homework 08).

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)

- Tutorial 10: The roughening transition (with solutions).
- Homework 10: Topological excitation and their interactions in the XY model (with solutions).
- Factsheet: Wegner's solution of the d-dimensional harmonic model.
- Useful program: Wegner 1d Exact.py (for Homework 10).
- Useful program: Wegner 1d Direct.py (for Homework 10).
- Useful program: Wegner 2d Exact.py (for Homework 10).
- Useful program: Vortex_pair.py (for Lecture 10).

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)