Collective Effects in Equilibrium and Nonequilibrium Physics

Beijing Normal University

Professor: Michael Cross

Science and Technology Building: Room 223

Announcements

Look here for class announcements etc.

The course of lectures is now completed!
Course Outline (revised 1 June)
Prof. Xintian Wu has kindly compiled a list of references in Chinese that provide introductory and supplemental material for my lectures.

Class Slides

These are copies of the slides used in the lectures. The "slides" are as used in class. The "handouts" are printed two to a page: you might want to print these out and make notes on them during the class.

Lecture 1: Introduction: When is physics simple? - slides, handout, supplementary reading (March 17)
Lecture 2: The simplest magnet: the Ising model - slides, handout, supplementary reading (March 24)
Lecture 3: Magnets with continuous symmetry - slides, handout, supplementary reading (March 31)
Lecture 4: Superfluidity and superconductivity - slides, handout, supplementary reading (April 7)
Lecture 5: Onsager theory and the fluctuation-dissipation theorem - slides, handout, supplementary reading (April 14)
Lecture 6: Hydrodynamics - slides, handout, supplementary reading (April 21)
Lecture 7: Instability far from equilibrium - slides, handout, supplementary reading (April 28)
Lecture 8: Nonlinear theory of patterns near onset - slides, handout, supplementary reading (June 2)
Lecture 9: Symmetry aspects of patterns - slides, handout, supplementary reading (June 9)
Lecture 10: Pattern formation in biology - slides, handout, supplementary reading (June19)

Supplementary Reading

(Some of these notes are extracted from other courses that I teach, and so may have references to material that is not given here.)

Lecture 1
- Notes on the dynamical systems motivation for the fundamental postulate of statistical mechanics
- Notes on equilibrium and nonequilibrium systems (read up to Sec. 2.2)
- If you are interested in more exotic quantum or topological ordering phenomenon you can look at the talk by Balents that I mentioned in the lecture, or the paper by Balents et al.
- Particles in box demonstration: 2 particles, many particles. (Challenge: starting from the second demonstration read initial conditions in_2 or in_4 and watch carefully what happens. Discuss!)
Lecture 2
- Notes on the Boltzmann factor.
- Notes on the mean field theory of the Ising model.
- Notes on Landau theory of second order phase transitions.
- If you would like to see a more formal derivation of mean field theory you can look at these notes on the Hubbard-Stratonovich transformation. This however is an advanced topic. (Note that I use Q for the partition function and A for the free energy in these notes.)
- You can investigate a Monte-Carlo simulation of the 2d Ising model using this Java demonstration. A Monte-Carlo simulation produces a series of snapshots which are representative samples of the appropriate ensemble (usually the canonical) so that an average over a very large number of snapshots gives the correct thermodynamic averages.
Lecture 3
- A java demonstration showing the quenching from random initial conditions in a 2d XY model
- The canonical reference for the Mermin-Wagner-Hohenberg theorem is Mermin and Wagner Phys. Rev. Lett. 17, 1133 (1966). They refer to work of Hohenberg that stimulated theirs, but was actually published later, Phys. Rev. 158, 383 (1967)
- There is a very nice general (but mathematical) discussion of topological defects by David Mermin in Rev. Mod. Phys. 51, 591 (1979)
- You can find a discussion of the Kosterlitz-Thouless transition in many text books, for example Principles of Condensed Matter Physics by Chaikin and Lubensky (Cambridge University Press, Cambridge, 1995)
- For a more formal derivation of the "hydrodynamic equations" for the easy-plane magnet see Haleperin and Saslow Phys. Rev. B16, 2154 (1977). We'll be returning to these ideas later in the course as well.
- For an application of spin wave theory to the study of nuclear antiferromagnetism in solid He³ at mK temperatures see the paper I wrote with Osheroff and Fisher, Nuclear Antiferromagnetic Resonance in Solid 3He, Phys. Rev. Lett. 44, 792 (1980)
- A web page on topological defects in cosmology.
Lecture 4
- Superfluidity: There are many resources on superfluidity that you can find on the internet. Particular ones that I have found useful are:
  1. Review by P. W. Anderson Rev. Mod. Phys. 38, 298 (1966) - maybe the only paper on broken symmetry with a picture of a steam engine (figure 4)!
  2. The Physics Nobel Prize website is a good place to find pedagogical lectures on modern subjects (the talks of the prize winners at the symposium honoring them). The website for the 2001 prize given for "the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates" has some nice articles relevant to this week's topic.
- Josephson effect in superconductors
  1. Brian Josephson's 1973 Nobel Prize lecture
  2. The Nobel Laureate Versus the Graduate Student: a discussion of the early controversy of the Josephson effect
  3. Paper by Rowell, Phys. Rev. Lett. 11, 200 (1963), on the experimental discovery of the Josephson effect.
  4. Popular article by John Clarke who played a key role in developing the SQUID, on applications "from diagnosis of brain tumors to tests of relativity": Scientific American, August 1994
- Josephson effect in superfluids: The search for the corresponding Josephson effect in superfluids eluded experimentalists for decades. In the last few years the goal has been accomplished. The website of Richard Packard at Berkeley is a good resource. Many of the group's recent publications are on this topic. In particular look at:
  1. Quantum interference of superfluid He-3: Nature 412, 6842 (2001)
  2. Superfluid He-3 Josephson weak links: Rev. Rod. Phys. 74, 741 (2002)
  3. Quantum Whistling in Superfluid He-4: Nature 433, 376 (2005)
Lecture 5
- General References
  - More detailed notes on the derivation of Onsager regression and the fluctuation dissipation theorem.
  - You can find a discussion of Onsager regression and fluctuation-dissipation in many statistical mechanics textbooks. My discussion is modelled on the chapter in Introduction to Modern Statistical Mechanics by David Chandler (Oxford University Press, 1987)
  - Original references on fluctuation dissipation etc.:
    1. H.B. Callen and R.F. Greene, Phys. Rev. 86, 702 (1952)
    2. R. Kubo, J. Phys. Soc. Japan 12, 570 (1957)
- Application to a Single Biomolecule Detector
  1. J. L. Arlett, M. R. Paul, J. E. Solomon, M. C. Cross, S. E. Fraser, and M. L. Roukes, "BioNEMS: Nanomechanical Systems for Single-Molecule Biophysics" Nobel Symposium 131, August 2005. (Contact me if you would like a preprint of the paper.)
  2. M. R. Paul and M. C. Cross, "The stochastic dynamics of nanoscale mechanical oscillators immersed in a viscous fluid", Phys. Rev. Lett. 92, 235501 (2004)
- Application to LIGO: LIGO is the Caltech-MIT gravity wave detector project, now up and running. The fluctuation-dissipation theorem is an important tool in understanding the noise that limits the sensitivity of the experiment - at the operation frequency thermal noise is the dominant source. Therefore "understand the dissipation and you understand the noise". Some references are:
  1. P.R. Slauson, Phys. Rev. D42, 2437 (1990) - discussion of mechanical dissipation and noise (available
  2. A. Gillespie and F. Raab, Phys. Lett. A190, 213 (1994) - theory and measurements of suspension losses and noise in a LIGO test bed
  3. Y. Levin, Phys. Rev. D57, 659 (1998) - application of the fluctuation dissipation theorem leading to a novel calculation of mass-mode noise
Lecture 6
- The theory of hydrodynamics has been applied to many systems, including exotic superfluids such as He³-A, liquid crystals, electromagnetism in dielectric materials, and critical phenomenon. Some references are:
  1. Heat and mass flow in a fluid: Landau and Lifshitz, Fluid Mechanics, chapter 5
  2. Hydrodynamic theory of spin waves: Halperin and Saslow, Phys. Rev. B16, 2154 (1977)
  3. Two fluid hydrodynamics of a superfluid: Khalatnikov, An Introduction to the Theory of Superfluidity; Putterman, Superfluid Hydrodynamics
  4. Dynamic critical phenomenon: Hohenberg and Halperin, Rev. Mod. Phys. 49, 435 (1977)
  5. Mario Liu, a colleague of mine now at the University of Hanover, has applied the hydrodynamic approach to many diverse systems. You can find examples by searching on "Mario Liu hydrodynamics". A rather exotic example he and I worked on together is the Gauge Wheel of Superfluid 3He, Phys. Rev. Lett. 43, 296 (1979).
Lecture 7
- Notes on the calculation of the Rayleigh-Benard instability with the correct, no slip boundary conditions
- Notes on the Turing instability and the application to chemical reaction diffusion systems.
- Three classic (and readable) papers on instability in systems far from equilibrium are:
  1. Lord Rayleigh, On Convection Currents in a Horizontal Layer of Fluid, when the Higher Temperature is on the Under Side Phil. Mag. 32, 529 (1916)
  2. G. I. Taylor, Stability of a Viscous Liquid Contained between Two Rotating Cylinders, Phil. Tran. Roc. Soc. Lon. A, 223 289, (1923)
  3. A. M. Turing, The Chemical Basis of Morphogenesis, Phil. Trans. Roy. Soc. Lon. 237, (1952)
Lecture 8
- Detailed notes on the amplitude equation approach
- Notes on the multiple-scales perturbation method
- References to related papers
- Java demonstrations
- Lecture 9
  - Notes on the phase equation
  - Paper by Wesfried and Croquette on experimental test of the phase diffusion equation in convection [Phys. Rev. Lett. 45, 634 (1980)]
  - A review article The topological theory of defects in ordered media Rev. Mod. Phys. 51, 591 (1979) by Mermin discusses the mathematical theory of topological defects in ordered system. This discussion applies to both equilibrium and nonequilibrium systems.
  - A review article The world of the complex Ginzburg-Landau equation Rev. Mod. Phys. 74, 99 (2002) by Aranson and Kramer provides a comprehensive discussion of the Complex Ginzburg-Landau equation
  - Java demonstrations
    1. Complex Ginzburg-Landau equation
    2. Complex Ginzburg-Landau equation (larger system)
    3. Nonlinear waves in a reaction diffusion system
- Lecture 10
  - Java demonstrations
    - Variety of states at strong nonlinearity
    - Role of discreteness
  - References
    - General
      - Mathematical Biology by J. D. Murray, Springer-Verlag (1989)
      - Models of Biological Pattern Formation by H. Meinhardt, Academic (1992)
      - The Algorithmic Beauty of Sea Shells, H. Meinhardt, Springer (1995)
      - Pattern Formation Outside of Equilibrium, M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993), Section XI
    - Turing Patterns
      - The Chemical Basis of Morphogenesis, A. M. Turing, Phil. Trans. Roy. Soc. Lon. 237, (1952)
      - Transition from a Uniform State to Hexagonal and Striped Turing Patterns Patterns, Q. Ouyang and H. L. Swinney, Nature 352, 610 (1991)
      - Transition to Chemical Turbulence, Q. Ouyang and H. L. Swinney, Chaos 1, 411 (1991)
    - Segmentation in Drosophila
      - Mechanism of Eve Stripe Formation, J. Reinitz and D. H. Sharp, Mechanisms of Development 49, 133 (1995)
      - Establishment of Developmental Precision and Proportions in the Early Drosophila Embryo, B. Houchmandzadeh, E. Wieschaus and S. Leibler, Nature 415, 798 (2002)
      - Dynamical Modelling of Pattern Formation During Embryonic Development, D. Thieffry and L. Sanchez, Current Opinion in Genetics & Development 13, 326 (2003)
      - The FlyEx database
    - Fish Patterns
      - A Reaction-Diffusion Wave on the Skin of the Marine Angelfish Pomacanthus, S. Kondo and R. Asai, Nature 376, 765 (1995)
      - Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation, E. J. Crampin, E. A. Gaffney, and P. K. Maini, Bull. Math. Biology 61, 1093 (1999)
    - Patterns on the Visual Cortex
      - Geometry of Orientation and Ocular Dominance Columns in Monkey Striate Cortex, K. Obermayer and G. Blasde, J. Neuroscience 13, 4114 (1993)
      - Orientation Preference Patterns in Mammalian Visual Cortex: a Wire Length Minimization Approach, A. A. Koulakov and D. B. Chklovskii, Neuron 29, 519 (2001)
      - What Geometric Visual Hallucinations Tell Us about the Visual Cortex, P. C. Bressloff et al., Neural Computation 14, 473 (2002)
  All files are in Acrobat (pdf) format
  
  Last modified: March 13, 2007, Michael Cross