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.

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.

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.

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.)