11.4 Par­ti­cle-En­ergy Dis­tri­b­u­tion Func­tions

The ob­jec­tive in this sec­tion is to re­late the Maxwell-Boltz­mann, Bose-Ein­stein, and Fermi-Dirac par­ti­cle en­ergy dis­tri­b­u­tions of chap­ter 6 to the con­clu­sions ob­tained in the pre­vi­ous sec­tion. The three dis­tri­b­u­tions give the num­ber of par­ti­cles that have given sin­gle-par­ti­cle en­er­gies.

In terms of the pic­ture de­vel­oped in the pre­vi­ous sec­tions, they de­scribe how many par­ti­cles are on each en­ergy shelf rel­a­tive to the num­ber of sin­gle-par­ti­cle states on the shelf. The dis­tri­b­u­tions also as­sume that the num­ber of shelves is taken large enough that their en­ergy can be as­sumed to vary con­tin­u­ously.

Ac­cord­ing to the con­clu­sion of the pre­vi­ous sec­tion, for a sys­tem with given en­ergy it is suf­fi­cient to find the most prob­a­ble set of en­ergy shelf oc­cu­pa­tion num­bers, the set that has the high­est num­ber of sys­tem en­ergy eigen­func­tions. That gives the num­ber of par­ti­cles on each en­ergy shelf that is the most prob­a­ble. As the pre­vi­ous sec­tion demon­strated by ex­am­ple, the frac­tion of eigen­func­tions that have sig­nif­i­cantly dif­fer­ent shelf oc­cu­pa­tion num­bers than the most prob­a­ble ones is so small for a macro­scopic sys­tem that it can be ig­nored.

There­fore, the ba­sic ap­proach to find the three dis­tri­b­u­tion func­tions is to first iden­tify all sets of shelf oc­cu­pa­tion num­bers $\vec{I}$ that have the given en­ergy, and then among these pick out the set that has the most sys­tem eigen­func­tions $Q_{\vec{I}}$. There are some tech­ni­cal is­sues with that, {N.24}, but they can be worked out, as in de­riva­tion {D.57}.

The fi­nal re­sult is, of course, the par­ti­cle en­ergy dis­tri­b­u­tions from chap­ter 6:

\iota^{\rm {b}}=\frac{1}{e^{({\vphantom' E}^{\rm p}- \mu)/{...
...=\frac{1}{e^{({\vphantom' E}^{\rm p}- \mu)/{k_{\rm B}}T} + 1}.

Here $\iota$ in­di­cates the num­ber of par­ti­cles per sin­gle-par­ti­cle state, more pre­cisely, $\iota$ $\vphantom0\raisebox{1.5pt}{$=$}$ $I_s$$\raisebox{.5pt}{$/$}$$N_s$. This ra­tio is in­de­pen­dent of the pre­cise de­tails of how the shelves are se­lected, as long as their en­er­gies are closely spaced. How­ever, for iden­ti­cal bosons it does as­sume that the num­ber of sin­gle-par­ti­cle states on a shelf is large. If that as­sump­tion is prob­lem­atic, the more ac­cu­rate for­mu­lae in de­riva­tion {D.57} should be con­sulted. The main case for which there is a real prob­lem is for the ground state in Bose-Ein­stein con­den­sa­tion.

It may be noted that $T$ in the above dis­tri­b­u­tion laws is a tem­per­a­ture, but the de­riva­tion in the note did not es­tab­lish it is the same tem­per­a­ture scale that you would get with an ideal-gas ther­mome­ter. That will be shown in sec­tion 11.14.4. For now note that $T$ will nor­mally have to be pos­i­tive. Oth­er­wise the de­rived en­ergy dis­tri­b­u­tions would have the num­ber of par­ti­cles be­come in­fin­ity at in­fi­nite shelf en­er­gies. For some weird sys­tem for which there is an up­per limit to the pos­si­ble sin­gle-par­ti­cle en­er­gies, this ar­gu­ment does not ap­ply, and neg­a­tive tem­per­a­tures can­not be ex­cluded. But for par­ti­cles in a box, ar­bi­trar­ily large en­ergy lev­els do ex­ist, see chap­ter 6.2, and the tem­per­a­ture must be pos­i­tive.

The de­riva­tion also did not show that $\mu$ in the above dis­tri­b­u­tions is the chem­i­cal po­ten­tial as is de­fined in gen­eral ther­mo­dy­nam­ics. That will even­tu­ally be shown in de­riva­tion {D.61}. Note that for par­ti­cles like pho­tons that can be read­ily cre­ated or an­ni­hi­lated, there is no chem­i­cal po­ten­tial; $\mu$ en­tered into the de­riva­tion {D.57} through the con­straint that the num­ber of par­ti­cles of the sys­tem is a given. A look at the note shows that the for­mu­lae still ap­ply for such tran­sient par­ti­cles if you sim­ply put $\mu$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0.

For per­ma­nent par­ti­cles, in­creas­ingly large neg­a­tive val­ues of the chem­i­cal po­ten­tial $\mu$ de­crease the num­ber of par­ti­cles at all en­er­gies. There­fore large neg­a­tive $\mu$ cor­re­sponds to sys­tems of very low par­ti­cle den­si­ties. If $\mu$ is suf­fi­ciently neg­a­tive that $e^{({\vphantom' E}^{\rm p}-\mu)/{k_{\rm B}}T}$ is large even for the sin­gle-par­ti­cle ground state, the $\pm1$ that char­ac­ter­ize the Fermi-Dirac and Bose-Ein­stein dis­tri­b­u­tions can be ig­nored com­pared to the ex­po­nen­tial, and the three dis­tri­b­u­tions be­come equal:

The sym­metriza­tion re­quire­ments for bosons and fermi­ons can be ig­nored un­der con­di­tions of very low par­ti­cle den­si­ties.
These are ideal gas con­di­tions, sec­tion 11.14.4

De­creas­ing the tem­per­a­ture will pri­mar­ily thin out the par­ti­cle num­bers at high en­er­gies. In this sense, yes, tem­per­a­ture re­duc­tions are in­deed to some ex­tent as­so­ci­ated with (ki­netic) en­ergy re­duc­tions.