New Approaches to Modern Physics/DEFINITIONS


118093 - New Approaches to Modern Physics
WINTER 2005

Advanced Statistical Mechanics


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[JOAN ADLER's lectures]

DEFINITIONS

  1. Percolation, to quote a visitor who gave a colloquium many years ago at the Technion, ``is so simple that you could even explain it to your wife.'' (The audience reponse indicated some surprise and it was pointed out to a very embarassed visitor that the only member of the audience active in percolation theory was in fact someone's wife. Despite this, when referreing said visitors recent submission to PRL I recommended publication.)

    Here is an explanation suitable for husbands and other non-physicists:

    If a container is filled with metal balls an electric current can pass thru it. If the container were filled with glass beads, no current would pass thru. What percentage of metal beads is needed so that electricity can pass thru??? Instead of beads of metal and glass we could study alloys made of metallic and insulating atoms, and ask what percentage of the atoms must be metallic in order that the alloy will conduct electricity. The change from insulating to conducting state that occurs as the percentage of metal balls is increased is called the percolation phase transition.

    This is a phase transition very much like the phase transition between magnetic and nonmagnetic states in a ferromagnet or Ising model. In a magnet the phase transition occurs at Tc: below this temperature the system is ferromagnetic, above paramagnetic. The percolation transition occurs at the percolation threshold pc, below this there is no connection, above it the system is connected.

    Lets think about a one dimensional string of beads. How many must be metallic for a current to pass thru them? i.e. what is pc for this system.

    I have a special relationship to percolation theory because I selected it as a topic for my own undergraduate project more years ago than I care to remember, and later edited a book on "Percolation Structures and Processes". See an extract here .

    Percolation can be studied by analytic (probablistic methods) but a case as simple as the site problem on the square lattice cannot be solved with these methods. Thus in general simulation or exact enumeration (series expansions) is needed.

    However, there are certain cases about which some things can be said exactly, so lets start here.

  2. First lets define some basics. Percolation comes in many flavours but the simplest distinction is between bond percolation and site percolation. Lets revisit the square lattice - on the left are the sites of the lattice and on the right are the bonds.

    what what

  3. Bond percolation means that each bond is occupied with probability p , site percolation - each site is occupied with probablity p . Is pc going to be the same for both? if not which will be higher??????? (By the way was our glass/metal ball example site or bond percolation?)

  4. It is bond percolation that is the same as the limit of the Potts model and hence the one for whioch a lot of exact stuff is known. Site percolation is the one which is usually applied to real systems.

  5. Therefore, universality is something we need to explore here.