11.6 Low Temperature Behavior

The three-shelf simple model used to illustrate the basic ideas of quantum statistics qualitatively can also be used to illustrate the low temperature behavior that was discussed in chapter 6. To do so, however, the first shelf must be taken to contain just a single, nondegenerate ground state.

In that case, figure 11.7 of the previous section turns
into figure 11.8. Neither of the three systems sees much
reason to put any measurable amount of particles in the first shelf.
Why would they, it contains only one single-particle state out of 177?
In particular, the most probable shelf numbers are right at the
4

If the temperature is lowered however, as in figure 11.9
things change, especially for the system of bosons. Now the
mathematics of the most probable state wants to put a positive number
of bosons on shelf 1, and a large fraction of them to boot,
considering that it is only one state out of 177. The most probable
distribution drops way below the 4

When the temperature is lowered still much lower, as shown in figure
11.10, almost all bosons drop into the ground state and
the most probable state is right next to the origin

If you still need convincing that temperature is a measure of hotness, and not of thermal kinetic energy, there it is. The three systems of figure 11.10 are all at the same temperature, but there are vast differences in their kinetic energy. In thermal contact at very low temperatures, the system of fermions runs off with almost all the energy, leaving a small morsel of energy for the system of distinguishable particles, and the system of bosons gets practically nothing.

It is really weird. Any distribution of shelf numbers that is valid for distinguishable particles is exactly as valid for bosons and vice/versa; it is just the number of eigenfunctions with those shelf numbers that is different. But when the two systems are brought into thermal contact at very low temperatures, the distinguishable particles get all the energy. It is just as possible from an energy conservation and quantum mechanics point of view that all the energy goes to the bosons instead of to the distinguishable particles. But it becomes astronomically unlikely because there are so few eigenfunctions like that. (Do note that it is assumed here that the temperature is so low that almost all bosons have dropped in the ground state. As long as the temperatures do not become much smaller than the one of Bose-Einstein condensation, the energies of systems of bosons and distinguishable particles remain quite comparable, as in figure 11.9.)