# ICFP Stat Physics 2016

### From Werner KRAUTH

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 15 (14 December 2016): 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 treated the first 10 pages, or so, in the lecture.
- H. Risken "The Fokker-Planck equation (Springer Verlag, 1996 we did not actually get that far).

## Week 14 (07 December 2016): 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:

- J. A. Lipa, J. A. Nissen, D. A. Stricker, D. R. Swanson, T. C. P. Chui "Specific heat of liquid helium in zero gravity very near the lambda transition", Phys. Rev. B 68, 174518 (2003) (Final account of a 1992 space shuttle (!) experiment to measure the critical indices of the normal-superfluid transition in liquid 4He).
- M. Campostrini, M. Hasenbusch, A. Pelissetto, E. Vicari "Theoretical estimates of the critical exponents of the superfluid transition in 4He by lattice methods" (3d XY model calculations to compare with 4He experiments (!)).
- 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 13 (30 November 2016): Quantum statistics 1/2: Ideal Bosons

- Tutorial 13: The Toric code
- Homework 13: Ideal Bosons in the canonical and grand-canonical formulation Not graded, but please study
- 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:

- 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 12 (23 November 2016): Phases and phase transitions: From van der Waals theory (and beyond) to liquid crystals

- Tutorial 12: First-order phase transitions: The case of liquid crystals
- Homework 12: Van-der-Waals theory and Beyond-van-der-Waals Theory, an introduction due on December 15, 2016

NOTE ADDED: I corrected a bug in formulas (4)- (6) on 7 Dec 2016

- Here is the python program for the van der Waals equation of state that you may use for homework 12.
- The following question came up: "Which pressures we have to compare and which volumes?" My answer was as follows: "Hello, try to divide 373K by 647K, and you will understand ;)" In other words, go get a hold of the equation of state of water, for example from wikipedia, and interpret it in terms of vdW theory.

References for Week 12:

- 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 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 that will allow you to generate configurations without vortices, with one vortex, and with a vortex-antivortex pair.

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

## 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)