Gas Velocities Plotted

	2D Velocity Distribution Evolution
The intent of this animation is to show the evolution of the occupancy of the energy states of a 2 dimensional Bose-Einstein gas as it is cooled down to become a condensate.  Bose particles (Bosons)have the quality that any number of them can be in the same momentum state consistent with the temperature of the gas.  The functional expression for the occupancy number (called the occupancy here) for a given momentum state is computed by the equation
O(E)=Occupancy(E)=1/{Exp[(E-mu)/kT]-1}
where E is the kinetic energy a particle that state, mu is the chemical potential, k is Boltzmann's constant, and T is the absolute temperature (E must always exceed mu).  What is noticeable here is that, when E-mu is much less than kT, the occupancy can be much greater than 1 and an approximate expression for O(E) is kT/(E-mu).  For our 2D case, E can be written
E=(px*px+py*py)/(2*m)
where px and py are the momentum components in the x and y directions and m is the mass of the Bose particle.  
As we cool down the whole group of Bosons by reducing their kinetic energy, some of the momentum cells will have experimental occupancies N greater than expected from the O(E) equation but this is expected because this is a dynamic process.  During cooling we will need a measure of the temperature and for that we can the average kinetic energy <T> so may re-write O(E) by substituting <T> for kT:
O(E)=1/{Exp[(E-mu)/<E>]-1}

I chose to compute and apply color values to circular sector cells.  The reason is that these cells represent the individual momentum (px,py) states of the 2D Boson gas. 
To initiate the animation I choose all of the starting energies the same so you will see that the initial momenta form a circle in the (px,py) plane.  

The particles will usually collide with each other so that their energies get dispersed, some going higher and some going lower.  In order to condense into the expected Boson-Einstein distribution, the lower energy particles will participate less in collisions since they are moving more slowly. 
I show a 2D color plot of state occupancy Vs px and py.  Below that, in black,  I show a line plot that represents the occupancy along the px axis.  

Just for reference, I also show color plots of what would be the distribution of momentum values for both a Fermi-Dirac and a Maxwell-Boltzmann distribution.  
The learner has access to sliders that allow adjustment of mu and kT and the starting Fermion particle energy, E0.   The rate of cooling (by reducing average energy) is also adjustable.
The learner must understand that this animation is a case where many processes must take place in order for the resulting state to reach equilibrium so a maximum run time for the program for a PC is on the order of several minutes.  When the average kinetic energy becomes less than 0.0001, the animation will stop and the learner can change parameters and restart it.
The learner has access to sliders with which to adjust the program parameters:
1. The Average Initial Kinetic Energy, E0, per particle
2. A nominal value of the temperature factor kT
3. The value of the chemical potential, mu
4. The factor f by which the total kinetic energy is reduced each animation period
Velocity Distribution Derivation
Sliders
Disc Radius Blue Radius Total Discs Starting energy