Thursday, June 6, 2013

Gaussian:Pascal's Triange <->Maxwell-Boltzmann:???

I think I've got something very cool here. I was working on the problem of total spin states of multiple electron systems, and a guy on stackexchange posted a function showing the distribution of the states. I'm not exactly smart enough to read his math, but I can pick out his numbers, and I figured out how to generate them out of a kind of modified Pascal's Triangle.

I'm not going to explain Pascal's Triangle here, but there's something about it everyone should know. If you take any row, the higher the better, it starts to look a lot like a Gaussian distribution. There are all kinds of reasons why this makes sense. Now, this new function I'm working with, this modified Pascal's triangle, supposedly describes the number of ways to generate a particular spin state from a random collection of electrons. So I'm it like Pascal's triangle? Is the distribution somehow Gaussian?

Well, in three dimensions you shouldn't really expect a Gaussian distribution. For example, in an ideal gas, the x-, y-, and z-velocities of the molecules are all individually described as Gaussians. But the total distribution described radially is then a Gaussian multiplied by r-squared. that how electron spins are distributed? I think it is.

It's not all that easy to run the numbers, but I've put it in a spreadsheet and I think it works out. The triangle goes like this:

     1  1
   1 2
  1 3 2
 1 4 5
1 5 9 5
etc. (You should be able to see how the numbers get generated.)

The way it works, for example, is if you have three electrons you look in row 3, and there is one way of generating the spin-3/2 states (four altogether) and two ways of generating each of the two spin-1/2 states. So there are eight states, counting duplicates.

The question then becomes: if you assume in a random collection of electrons that each state is equally probable, and you plot the resultant spins as  vectors in three dimensions, are the tips of those vectors distributed in space as a Maxwell-Boltzmann function? I think it turns out they are.


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