Summary

The first, simulated annealing, is an often-used method for finding global maxima and minima of complicated multi-dimensional functions. It can be used any time you have a function that gets big when things are bad and small when things are good (or vice-versa). The problem being solved in SimSurface is a minimization of potential energy. The question we are trying to answer is: given n stationary electrons/protons confined to a 2-dimensional surface by four charged walls, what arrangement of electrons has the minimum total energy and is therefore the configuration preferred by nature? Simulated annealing works by randomly varying each parameter, and storing the configuration with the lowest energy. We use that as a starting point for the next random process, but each time we change the parameters a little less. The collection of parameters that we are changing is sometimes refered to as an amoeba, as each change we make in the parameters is like a pod of an amoeba, spreading out, and then moving in the direction of least resistance.
 

Typical initial configuration

 Typical final configuration 

The second commonly used numerical technique is the relaxation method. Once the system has reached a minimum energy (to within a specified tolerance), we would like to know what the electrical potential looks like around the electrons. This could of course just be calculated directly at each point, but a much faster and widely used technique called relaxation is actually used to solve Laplace's equation. Laplace's equation says that the net curvature of the potential surface is zero everywhere (del^2 phi=0) on the surface (except at the point charges themselves), but the relaxation method is very easy to understand, even for those that do not fully understand the equation it is solving. It is an iterative process, which at each iteration sets the value of a grid point equal to the average of its 4 neigbors. The charges and walls are held constant throughout this process. SimSurface displays the end result of this process: the potential on the plate.

 Note that the colored bands in the image are equipotential lines of the field.

Running SimSurface on Windows & Macintosh

Simulation Menu

Settings Dialog Box

Toolbox

Start SimSurface, and randomize the charges. Select Anneal form the Simulation menu, and watch the electrons move by random motion, at a lower and lower temperature with each move. Notice the configuration that the electrons finally reach. Depending on which version you have, the toolbox may or may not show the energy of any given configuration, but you can always open the notebook before annealing, and see the change in energy with each iteration.

Assorted Notes and Frequently Asked Questions

Occasionally the final energy obtained is different, or the pattern is slightly varied. Why?

The simulated annealing process used by SimSurface is inherently random in nature. Thus when run with different random seeds, the program may produce different output patterns. However the differences in energies should be very small.

SimSurface is great, but I'm looking for an example of simulated annealing that does not require as much understanding of physics. Any suggestions?

Why yes. We also have a program titled SimElevator which uses the same annealing process to model the way in which people arrange themselves in an elevator. If there's sufficient interest in this (mail us), we would consider creating a Mosaic front end to this as well.

Is source code for SimSurface publicly available?

In a word, no. In two words, not yet . Multiple versions of SimSurface have been written to run on UNIX environments, over the web, and on both PC and Macintosh. If you are interested in collaborating with us to create a version of the code to run on a different platform, or to improve the features of the code, contact David Joiner at djoiner@shodor.org.

What's the coolest thing about SimSurface?

Our opinion: when we find the minimum energy configuration of 64 electrons, we are in essence solving a global minimum problem in 128 dimensions. If instead we just tested each coordinate at each of 10 values (a very coarse sampling), we would have 1.0e128 energy calculations. Needless to say this would take longer than the age of the universe. Instead, our program runs in about 6 seconds. Thus as you can see the algorithm used is far more important than whether the program is run on a Cray, an SGI or a Macintosh.