("Method of the most probable distribution")
For fermions in a square box the mu-plane is spanned
by integers nx,ny; each quantum state
is represented by a point.
A specific state of a system of N fermions is represented by a set of
N inhabited points on that plane.
The following game serves to find the average (and also most probable!)
distribution of particles on states:
To this date, this game has never actually been played; rather,
its outcome was calculated as seen in the textbooks.
Assign N particles randomly to the states on mu-plane
Make sure that the sum of the particle energies equals the given system
Discard all trials in which a state is inhabited by more than one particle
Determine the mean number of particles in each state;
sort the result according to the state energies