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.

Figure 11.8: Probabilities of shelf-number sets for the simple 64 particle model system if shelf 1 is a nondegenerate ground state. Left: identical bosons, middle: distinguishable particles, right: identical fermions. The temperature is the same as in the previous figures.
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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 45$\POW9,{\circ}$ limiting line through the points $I_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $I$, $I_3$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 and $I_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, $I_3$ $\vphantom0\raisebox{1.5pt}{$=$}$ $I$ on which $I_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. Actually, the mathematics of the system of bosons would like to put a negative number of bosons on the first shelf, and must be constrained to put zero on it.

Figure 11.9: Like the previous figure, but at a lower temperature.
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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 45$\POW9,{\circ}$ limiting line. The mathematics for distinguishable particles and fermions does not yet see any reason to panic, and still leaves shelf 1 largely empty.

Figure 11.10: Like the previous figures, but at a still lower temperature.
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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 $I_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $I_3$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. In contrast, while the system of distinguishable particles does recognize that high-energy shelf 3 becomes quite unreachable with the available amount of thermal energy, it still has a quite significant fraction of the particles on shelf 2. And the system of fermions will never drop to shelf 1, however low the temperature. Because of the Pauli exclusion principle, only one fermion out of the 64 can ever go on shelf one, and only 48, 75%. can go on shelf 2. The remaining 23% will stay on the high-energy shelf however low the temperature goes.

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.)