Gas Energies Plotted

	Energy Evolution of a 2D Gas with Lennard-Jones Forces between Discs
Here we use repelling forces that are mediated by a Lennard-Jones (LJ) potential rather than the hard sphere scattering that was used before in the Gas Physics chapter.  The form of the LJ potential is
VLJ=4*epsilon*((sigma/r)^12-(sigma/r)^6)
where epsilon and sigma are explained below and r is the separation between atom centers.  
The LJ potential is strongly repelling at close range and weakly attracting at longer ranges.   The attracting forces can explain Joule-Thomson cooling by expansion of a gas through a nozzle since, by conservation of energy, increase in the usually negative potential results in a decrease of kinetic energy.
From the LJ potential a force, FLJ, can by computed by taking the derivative with respect to r:
FLJ=-dVLJ/dr
and this force is used iteratively in Newton's laws of motion to update the momentum and speed of the mobile atoms.
For this animation we set sigma equal to twice the hard sphere radius of the atoms.  For r>sigma we use the forces from the above equation to compute the speed changes. 
After a brief startup time, the least squares fit of an exponential to the histogram shows the energy frequency distribution is in excellent agreement.  A histogram of the frequency distribution of the potential energy of atoms is also plotted.  It is noted that this distribution is almost flat except near the maximum potential energy where the numbers per bin drop off precipitously.  This drop-off is narrow because the atoms spend very little time near the potential minimum.

The exponential is of the form
f(T)=exp(-T/<T>)
where T is the kinetic energy and <T> is the average kinetic energy.
The learner has access to sliders that  allow adjustments, within limits, of the ratio of the initial kinetic energy to the minimum potential energy, the minimum potential energy (epsilon), the number of atoms, and the value of epsilon for the forces that keep the atoms inside the rectangle (boundary forces).
The program is set up so that if any of the above values are changed, the animation is stopped and the energy bins are emptied and a new initial distribution is randomly chosen. When the Start button is pressed the energies start again to evolve toward their new final distributions.
Energy Distribution Details
Disc Radius Total Discs epsilon*100 Starting energy percent
Single Step