For the case of Maxwell-Boltzmann (MB) statistics, the two particles are
considered to be distinguishable. Let us denote them and .
Furthermore, any number of particles can occupy the same quantum state.
The possible different states of the gas are shown in Tab. 6.

There are clearly 9 distinct states.

For the case of Bose-Einstein (BE) statistics, the two particles are
considered to be indistinguishable. Let us denote them both as .
Furthermore, any number of particles can occupy the same quantum state.
The possible different states of the gas are shown in Tab. 7.

There are clearly 6 distinct states.

Finally, for the case of Fermi-Dirac (FD) statistics, the two particles are
considered to be indistinguishable. Let us again denote them both as .
Furthermore, no more than one particle can occupy a given quantum state. The possible different states of the gas are shown in Tab. 8.

There are clearly only 3 distinct states.

It follows, from the above example, that Fermi-Dirac statistics are
more restrictive (*i.e.*, there are less possible states of the
system) than Bose-Einstein statistics, which are, in turn, more restrictive
than Maxwell-Boltzmann statistics. Let

(575) |

(576) | |||

(577) | |||

(578) |

We conclude that in Bose-Einstein statistics there is a greater relative tendency for particles to cluster in the same state than in classical statistics. On the other hand, in Fermi-Dirac statistics there is less tendency for particles to cluster in the same state than in classical statistics.