2D Velocity Distribution Evolution

For our 2D momentum case, the kinetic energy `E` can be written

`E=(p_x^2+p_y^2)/(2m)`

where `p_x` and `p_y` are the momentum components in the x and y directions and `m` is the mass of the a disc. From the Maxwell-Boltzmann distribution we can write for the energy distribution probability

`(dN(E))/(dE)=Cexp(-E/E_(avg))`

where `C` is a constant and `E_(avg)` is the average kinetic energy.

Here we plot the `p_x` and `p_y` momentum distributions.

`(dN)/(dp_x)=p_x/|p_x|exp(-p_x^2/(p_x^2+p_y^2))`

where we recall that `p_x^2+p_y^2` is a constant due to energy conservation.

To initiate the animation I choose all of the starting kinetic energies the same so you would see that the initial momenta form a circle in the `(p_x,p_y)` 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 Maxwell-Boltzmann distribution, the lower energy particles will participate less in collisions since they are moving more slowly.

The learner has access to sliders with which to adjust the program parameters:
1. The Initial Kinetic Energy, `E_0`, per particle
2. Red Particle Radius
3. Blue Particle Radius
4. Total Number of Particles

Note that increasing any one of these parameters increases the rate of equilibration of the momentum distribution.

Graphics Description

I have provided

1.Animated illustrations of the disc movements

2.Histogram type linear plots of the `p_x` and `p_y` momentum bin occupancy as they are updated

1.Histogram color coded circular plot in the `(p_x,p_y)`plane of momentum bin occupancy as they are updated

The interested reader may like to see my discussion of Maxwell's derivation of the velocity distribution