Show all 26 classical distributions

How many ways can you distribute 9 units of energy among 6 identical, indistinguishable fermions? Fermi-Dirac statistics differ dramatically from the classical Maxwell-Boltzmann statistics in that fermions must obey the Pauli exclusion principle. Considering the particles in this example to be electrons, a maximum of two particles can occupy each spatial state since there are two spin states each. Whereas there were 26 possible configurations for distinguishable particles, these are reduced to the 5 states which have no more than two particles in each state.

Evaluating the average occupancy of each energy state is much simpler than in the Maxwell-Boltzmann example since each macrostate has a weight of 1. The average occupancy is just the sum of the numbers of particles in a given energy state over all the 5 distributions divided by 5.

The average distribution of 9 units of energy among 6 identical particles

E Average number Maxwell- Boltzmann Average number Bose- Einstein Average number Fermi- Dirac 0 2.143 2.269 1.8 1 1.484 1.538 1.6 2 0.989 0.885 1.2 3 0.629 0.538 0.8 4 0.378 0.269 0.4 5 0.210 0.192 0.2 6 0.105 0.115 0 7 0.045 0.077 0 8 0.015 0.038 0 9 0.003 0.038 0

For fermions, there are only 5 possible distributions of 9 units of energy among 6 particles compared to 26 possible distributions for classical particles. To get a distribution function of the number of particles as a function of energy, the average population of each energy state must be taken. The average for each of the 9 states is shown above compared to the results obtained by Maxwell-Boltzmann statistics and Bose-Einstein statistics .

Low energy states are less probable with Fermi-Dirac statistics than with the Maxwell-Boltzmann statistics while mid-range energies are more probable. While that difference is not dramatic in this example for a small number of particles, it becomes very dramatic with large numbers and low temperatures. At absolute zero all of the possible energy states up to a level called the Fermi energy are occupied, and all the levels above the Fermi energy are vacant.