# ICFP Stat Physics 2017

### From Werner KRAUTH

This is the homepage for the ICFP course: Statistical Physics: Concepts and Applications that is running from 4 September 2017 through 18 December 2017. Lectures start at 8:30, on Monday mornings, and tutorials at 10:45.

Lectures: Werner KRAUTH

Tutorials (TD): Olga PETROVA, Jacopo DE NARDIS

Look here for practical information

## Week 1 (4 September 2017): Probability theory

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

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 (11 September 2017): Statistical inference

- Tutorial 02: Maximum likelihood, Bootstrap and Bayes without a computer
- Homework 02: From Maximum Likelihood to Bayes statistics Useful program:

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 (18 September 2017): Statistical mechanics and Thermodynamics

- Tutorial 03: Two-level systems and the entropy of ice. Maxwell's distribution.
- NB: No Homework this week. Homeworks will continue 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 4 (25 September 2017): Phases and phase transitions: Van der Waals theory

- Tutorial 4: First-order phase transitions: The case of liquid crystals
- Homework 04: Van-der-Waals theory and Beyond-van-der-Waals Theory, an introduction
- Here is the python program for the van der Waals equation of state that you may use 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 (02 October 2017): Hard spheres and the Ising model in one dimension (Transfer matrix 1/2)

- Tutorial 05: One-dimensional short-range interacting systems with phase transitions (!).
- Homework 05: Transfer matrices for hard spheres, variations on a theme.

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 (09 October 2017): 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.
- Homework 06: Thouless (!) argument; transfer matrix for the two-dimensional Ising model on a stripe.
- PDF of Mathematica notebook file useful 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 (16 October 2017): Two-dimensional Ising model: From Kramers & Wannier to Kac & Ward (Low- and high-temperature expansions)

- Lecture Notes 07 Van der Waerden, Kac-Ward solution of the two-dimensional Ising model.
- Tutorial 07: High-temperature expansion of the Ising model - Free energy and correlation functions
- Homework 07: 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 HW07, we suppose N=even. Furthermore, note that an identity cycle is a cycle of length 1.

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 (23 October 2017): The three pillars of mean-field theory (Transitions and order parameters 1/2)

- Tutorial 08: The Bethe lattice
- 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 gen d Ising lattice.py. . Also check out the program
- Ising mean field 1d.py (self-consistent mean field solution of a one-dimensional Ising chain, as well as its linearized approximation)

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)

## Week 9 (06 November 2017): 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 (13 November 2017): Kosterlitz-Thouless physics in two dimensions: The XY model (Transitions without order parameters 1/2)

- Tutorial 10: The roughening transition
- Homework 10: Topological excitation and their interactions in the XY model - 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).
- 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 11 (20 November 2017): 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 (27 November 2017): 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 (04 December 2017): Quantum statistics 1/2: Ideal Bosons

- Tutorial 13: The Toric code
- Homework 13: Ideal Bosons in the canonical and grand-canonical formulation Please study carefully.
- Here is the python program for enumerating all N-body states of 5 bosons in 35 single-particle states the partition function of canonic bosons. This program is useful for homework 13.
- Here is the python program for rigorously computing the partition function of canonic bosons, using integration over a chemical potential, that you may use for homework 13.

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 (11 December 2017): Quantum statistics 2/2: 4He and the 3D Heisenberg model, Non-classical rotational inertia

Here is part of section 3.1.4 of SMAC about the disturbing non-classical rotation of a quantum particle on a ring. The discussion is due to A. J. Leggett (1973, see below).

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 (18 December 2017): The Fluctuation-Dissipation theorem (an introduction)

References for Week 15:

NB: In week 15 of the 2017/18 course, we finished the discussion of non-classical rotational inertia. We did not treat the fluctuation--dissipation theorem.

- 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)