New Approaches to Modern Physics/ISING MODEL


118093 - New Approaches to Modern Physics
WINTER 2005

Advanced Statistical Mechanics


[Course Summary][Links to Lecture Notes and Homework][Announcements] [For Lecturers]
[JOAN ADLER's lectures]

4 hours on Percolation.

The best overall coverage of this material is in the monograph by Stauffer and Aharony - ``Introduction to Percolation Theory'', Taylor and Francis, 1992. Other references given specifically below.

1st hour - Background and basics.

  1. Short revision of phase transitions
    1. the Ising model
    2. first and second order phase transitions
    3. critical exponents
    4. upper and lower critical dimensions

    (based on material given in Stat Mech II and assumed known, see here for a reminder)

  2. Introduction of concepts of importance to percolation -
    1. common lattices
    2. universality
    3. Potts models
    4. duality

    (this material may or may not be known and will be discussed)

  3. Percolation basics - bond vs site

    Additional References

    1. Revision - Pliscke and Bergenson - Chapter 3 (3.1-3.7, 3.10)
    2. Concepts - Yeomans - Chapter 1 and 2.
    3. Percolation introduction - Plischke and Bergerson, p. 446-454,

2nd hour - Percolation and its variations and applications.

  1. definitions
  2. exact solution for the Bethe lattice

3rd and 4th hours - applications and algorithms

  1. series expansions
  2. simulations
  3. higher neighbours, etc
  4. clusters, ants and termites
  5. bootstrap percolation
  6. directed percolation
  7. invasion percolation
  8. percolation and fractals

Additional References

  1. bootstrap - Adler and Lev

  2. Gould and Tobochnik, draft of 3rd Edition

Conclusions and additional website links here

Homework assignment: select one of the percolation variations listed above or another one.

  1. Write a detailed description of the rules, discuss whether and why the critical point is lower/higher than that of regular nearest neighbour bond percolation on the same lattice.
  2. Use one of the references above or search the web to prepare a short summary of results and applications of this model in your own words.

No less than two and no more than five pages in total are recommended. You must hand in a piece of paper which may contain your discussion or a link to a website with your discussion. If a website then html or pdf formats only.