Band Width and Wave localization as a function of Disorder

Links for program, Mathematica Script, etc.:
simluation.c
Mathematica Package
etc

First I'd like to showcase my results using the simulation and band widths all using disorder in increments of 0.02 from 0 to 1 scaling with t, the tunneling energy.

I was a bit unsure at the time if it was physical that the energy of the well could be greater than the value of t. The Bounded data represents the wells that were capped at this value, or in other words generated from a cutoff normal distribution at the ends. I eventually dismissed the idea, and allowed the wells to have energy relative to each one. This is the unbounded data, and seems to fit a quadratic equation given as \(4x^2+x+4\). \(x\) and the band width is on the scale of \(t\). Here's a link to what the distribution tends to look like. 12 Samples were used. These take the longest to solve.

Next I ran a simulation with a sample rate of 50 comparing the width of the wave through the lattice on the same scale of disorder. As you can see, it seems to fit very well with an inverse relationship. Something about the nature of Normal distributions tell me this should have been expected, but I can't remember at the moment.

These next images show both the nature of the simulation, and how the width is changed through out the simulation.

Here a small sample was taken over a short span of time. You can clearly see how at higher disorder, the width converges more rapidly then lower disorder. With no disorder, the width never converges.

And here is nearly the same graph with a much higher sample rate and over a much longer time. You can see by the error bars that even with 500 samples, the wave can behave quite surprisingly as it moves about finding the lowest energy state.

I'll try to run more simulations to get a more accurate Width vs. Disorder plot, but at 50,000 iterations and 500 samples the program ran more than 8 hours so I'll decrease the accuracy by a bit in the code.

If anyone wants to run the code themselves, I've included the current version as source. The inputs can only be changed by editing the file itself, and the output should be redirected to a file. It is formatted so that it can be instantly inserted into excel.

I'd like to thank the people over at Florida State University, from whom I used the Normal Random Number Generators package to do these simulations in C.

I'd like to discuss the actual physical constrains on these types of simulations, and understand the limitations.