Statistical Mechanics II

Interacting Systems, Emergent Properties and Critical Phenomena

Sitabhra Sinha

Class Schedule:

4/8/17: Introduction

Steven N. Durlauf, "How can statistical mechanics contr ibute to social science?", Proc. Natl. Acad. Sci. USA, 96: 10582, 1999 (4/8/17)
See also the video lecture by Steven N. Durlauf on Statistical Mechanics as a Tool for Economic Theory in YouTube.

Philip Ball, "The Physical Modelling of Human Social Systems", Complexus, 1:190, 2003 (4/8/17)
You may also want to see the book-length treatment by Philip Ball, Critical Mass (Heinemann, 2004).

Extract from Thomas C. Schelling, Micromotives and Macrobehavior (W. W. Norton, 1978), pp 137-166 (4/8/17)

L. Gauvin, J. Vannimenus, and J.-P. Nadal, "Phase diagram of a Schelling segregation model",Eur. Phys. J. B, 70: 293, 2009 (4/8/17)

7/8/17: Enumeration and Probabilities

See Linda E. Reichl, A Modern Course in Statistical Physics (Wiley-VCH, 2016), Chapter 2.

Sean O'Donoghue (CSIRO Mathematics, Informatics & Statistics, Australia): The Genetic Code Redisplayed, with Amino Acid Abundance(7/8/17)

Eugene V. Koonin and Artem S. Novozhilov, "Origin and evolution of the genetic code: the universal enigma", IUBMB Life 61 (2):99, 2009.

9/8/17: Probabilities and their Distribution: Bayes Rule, the Monty Hall problem, Moments and all that

Wikipedia entry on the "Monty Hall Problem" (7/8/17)
You may also want to see the book-length treatment by Jason Rosenhouse, The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brainteaser (Oxford University Press, 2009).

To see how the Normal distribution can be obtained from the Binomial distribution in the limit of large N (number of trials) and large N p (p being the probability of a particular outcome), refer to Reichl, Appendix A.1.3

For Shockley's derivation of the log-normal distribution from a multiplicative random process and applying it to explain how research productivity varies across individual scientists, see William Shockley, "On the statistics of individual variations of productivity in research laboratories", Proc. IRE 45: 279, 1957.

11/8/17: Probability Distributions (stable & unstable), Phase Space, Microcanonical Ensemble and Ergodic Hypothesis

For a nice treatment of the pendulum as a dynamical system in phase space, see Steven Strogatz, Nonlinear Dynamics and Chaos, pp. 176-181.
Unfortunately it is not always possible to associate an energy function with dynamical systems - only when we are dealing with conservative systems. See Steven Strogatz, Nonlinear Dynamics and Chaos, pp. 166-170.

The ergodic theorem was proved in a mathematical tour-de-force by G. D. Birkhoff in 1931 - for a relatively simpler (but still daunting to the mathematically uninitiated) exposition see G. D. Birkhoff, "What is the Ergodic Theorem ?", American Mathematical Monthly 49(4): 222 (1942).

14/8/17: Poisson Distribution (from Binomial distribution; derivation from first principles); Ideal Gas in a finite volume

There is a nice brief discussion about Bortkiewicz's fitting of the Poisson distribution to the number of soldiers dying in the Prussian army from horse kicks in Andy Cheshire's blogpost: "No Horsing Around with the Poisson Distribution, Troops"

16/8/17: Ideal Gas in an infinite volume and 1-dimensional chain of non-interacting spins (Magnetic equation of state)

18/8/17: Simple model of polymer, Ideal gas (equation of state) and the maximum entropy principle

What happens when there are interactions of some kind between the different parts of a polymer ? A nice example is the self-avoiding walk where a walker cannot visit a site already visited before. See the article "Self-avoiding walks" by Gordon Slade, Mathematical Intelligencer 16(1): 29 (1994).

For more on maximum entropy principle see "The principle of maximum entropy" by Silviu Guiasu and Abe Shenitzer, Mathematical Intelligencer 7(1): 42 (1985).

Also see E. T. Jaynes, "Information Theory and Statistical Mechanics", Physical Review 106(4): 620 (1957).

21/8/17: Shannon's measure of information entropy; Introduction to the Ising model

An accessible derivation of the form of the information entropy from first principles is given in Mario Rasetti, Modern Methods in Equilibrium Statistical Mechanics (World Scientific, 1986), pp 280-289.

A very readable introduction to the quirks of the Ising model is Brian Hayes, "The World in a Spin", American Scientist, Sept-Oct 2000 (14/8/17)

22/8/17: Condition probability vs likelihood (the German tank problem); Exact solution of the Ising chain
The German tank problem which demonstrates that likelihood is not a probability distribution as the total may diverge is discussed in detail in Wikipedia - see Wikipedia: The German Tank Problem
The related problem of determining the size of the alphabet from a small sample of writing is discussed in Alan Mackay, ``On the type-fount of the Phaistos disc", Statistical Methods in Linguistics 4: 15 (1964).
For the exact solution of the 1-dimension spin chain, see Plischke and Bergersen.

1/9/17: Peierls argument for spontaneous magnetization in the 2-dimensional Ising model The argument by Peierls can be found in Rudolf Peierls, ``On Ising's model of ferromagnetism'', Mathematical Proceedings of the Cambridge Philosophical Society 32: 477 (1936).
In class however I used the cleaner statement of the argument given later by Robert Griffiths, ``Peierls proof of spontaneous magnetization in a two-dimensional Ising ferromagnet'', Physical Review 136: A437 (1964).

4/9/17: Correspondence of Ising model with other systems: Binary alloys and lattice gas; Mean-field model of fluids: Van der Waals model

6/9/17: Dynamical systems, Codimension-2 bifurcation and the P-V-T & H-M-T surfaces of fluids & magnetic systems; Spin-correlation function in the Ising chain; Behavior arpund the critical point: Critical exponents alpha, beta, gamma, delta, eta and nu
See the section in Strogatz on Pitchfork bifurcation and Imperfect bifurcations and catastrophes.

8/9/17: Dynamical systems and the approach to equilibrium; Example of constructing a mean-field theory: SIS model of infection spreading; Maxwell construction for the coexistence region in van der Waals model

11/9/17: Critical phenomena: Critical exponents, Correlation function and Correlation length; Critical behavior in mean-field theory

13/9/17: Landau phenomenological theory of phase transitions: first order and second order

21/9/17: Rushbrooke and other inequalities relating critical exponents; Homogeneous function of single and multiple variables; Generalized homogeneous function; Widom scaling

12/10/17: Scaling behavior around a critical point: Static scaling hypothesis; Equation of state near critical point; Kadanoff Block Spin construction

13/10/17: Renormalization group for 1-dimensional Ising model

16/10/17: Wilson RG; Relating eigen properties of RG with critical exponents

17/10/17: Differential form of RG transformation

18/10/17: Renormalization group for 2-dimensional triangular lattice Ising model

27/10/17: Fractals: Introduction
For a nice introduction to fractals see Steven Strogatz, Nonlinear Dynamics and Chaos, pp. 398-413.
The analysis of Jackson Pollock paintings using fractals is discussed in Alison Abbott, ``Fractals and art: In the hands of a master'', Nature 439: 648 (2006).
The original article of Richard Taylor et al is available at ``Fractal analysis of Pollock's drip paintings", Nature 399: 422 (1999).
Note that this analysis has been challenged by Katherine Jones-Smith and Harsh Mathur, ``Fractal Analysis: Revisiting Pollock's drip paintings", Nature 444: E9 (2006)

30/10/17: Fractals: Similarity and Box-counting dimensions; Frenkel-Kontorova model

31/10/17: Ising spins on the fractal Koch curve

2/11/17: Percolation: Introduction; Forest fire model

10/11/17: Percolation in 1-dimensional lattice

15/11/17: Percolation in Bethe lattice

16/11/17: Heisenberg and XY spin model; Vortices and anti-vortices in the XY model

17/11/17 (morning): XY spin model: Topological phase transition
A brief discussion on how to simulate the XY model is given in Planar model

17/11/17 (afternoon): Disorderd systems: Spin glass; Hopfield model


Assignment 1 (due on 16/08/2017)

Assignment 2 (due on 31/08/2017)


Mid-term take-home examination (due on 12/10/2017)

End-term examination question paper (29/11/2017)

Additional Readings

There is, unfortunately, no one book that everyone agrees on as the best text on statistical mechanics of interacting systems and critical phenomena.
We will consult
Michael Plischke and Birger Bergersen, Equilibrium Statistical Physics,
which covers most of the topics in equilibrium phenomena that we will deal with in this course - but probably is not that good from a pedagogical point of view.
A book which is close to perfect in terms of approaching the subject matter (IMHO) but is possibly too terse for students is
James P. Sethna, Entropy, Order Parameters and Complexity.
A compromise is the book by Linda E. Reichl, A Modern Course in Statistical Physics (4th revised edition, 2016).

For renormalization group theory, I will mostly use
Finn Ravndal, Scaling and Renormalization Groups: Introductory Lectures.

In later part of the course we will consult for the topic of networks
Mark Newman, Networks: An Introduction,
possibly the best textbook on the subject.
There is a strong connection between the concepts of advanced statistical mechanics and the concepts in the theory of dynamical bifurcations, and for the latter you can consult
Steven H Strogatz, Nonlinear Dynamics and Chaos,
a pedagogical marvel (I wish there was a Strogatz equivalent for advanced statistical mechanics).
For non-equilibrium systems (specfically for pattern formation in such systems), a very good non-technical book is
Philip Ball, The Self-Made Tapestry.
For a more technical introduction see
Malte Henkel, Haye Hinrichsen and Sven Lubeck, Non-Equilibrium Phase Transitions (Springer, 2008)
A very concise version of the above is Haye Hinrichsen, "Non-equilibrium phase transitions", Lecture notes for International Summer School on Problems in Statistical Physics XI, 2005.

Web resources for Advanced Statistical Mechanics:
Michael Cross's three semester course on Statistical Physics, CalTech, 2006
Michael Cross's course on Collective Effects in Equilibrium and Nonequilibrium Physics, Beijing Normal University, 2008
Gareth Alexander's course on Statistical Mechanics of Complex Systems, University of Warwick, 2011
David Tong's course on Statistical Mechanics and Thermodynamics, University of Cambridge, 2012