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 referreeing 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
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"
I have some notes from the advanced stat mech course on percolation,
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 simulation or exact enumeration is needed.
Some additional material discussing these numerical methods will be given
Several projects have been made in this area by students in this course,
as well as by undergraduate project students.
Some that I especially recommend are:
- The best way to simulate percolation is with the use of the Hoshen-Kopelman algorithm.
Ido Braslavsky and Zaher Salman
PGPLOT versions of the
program given by Gould and Tobochnik. These can be downloaded
compile the source file pro.f with pgplotcl.
(The executables are for older systems, please ignore them; some may run on phelafel/aluf.)
- Percolation does not only apply to uniform balls on a lattice. It is
also relevant for
continuum disks of spheres of different sizes. David Saada prepared
a routine to model this, which can be downloaded
Look at perco.f, compiled as above.
- The objects that do or don't percolate may not be distributed uniformly,
There could be correlations in their positions.
An application of correlated percolation to Analysis and Presentation of
was made by Tamar Tepper using MATLAB. I would like the Materials Science and students to look at these files.
- A special form of correlated percolation is bootstrap percolation, (see the
project by Uri Lev), also recommended for the project report.
Here, sites remain occupied only if there also sufficient occupied neighbours.
A new boostrap site is the project of Omri Warshawski, here.
In the inverse problem, diffusion percolation, sites become occupied if there are
enough occupied neighbours. Diffusion percolation was developed by J. Adler
and A. Aharony to model crack development when water flows thru porous rocks.
A project that simulates models of this type was made by Amit Kanigel and the
routines can be downloaded
here. You need the files per, perc.c perc.h perhelp and percmath.c
and compile them with gcc -O -o perc.ex -L/usr/X11R6/lib perc.c -lm -lX11
- Nir Yefet has made three dimensional models of percolation clusters
using mesa. His webpage with lots of good links (including a link to another
percolation program in MATLAB) is
here. From these links you can find other links, at last count I
found about 10 distinct sites, many with
java applets. These will run
over netscape at the Technion if it is set correctly (enable java under the
edit preferences advanced option.) They may overtax the system resources
however (and crash your session), be warned. The algorithm description in this project has a bug in it, be warned.
- A really nice site by Dana Hoffman describes Invasion Percolation. Also recommended for the project report but currently not accessible - trying to get it restord.
- Exact enumeration (series expansions) is also a powerful technique
for studying percolation.
Nir Schreiber has made a study of this, and calculated some new terms for
one of these expansions. More on this when we discuss series
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