### D.58 The par­ti­cle en­ergy dis­tri­b­u­tions

This note de­rives the Maxwell-Boltz­mann, Fermi-Dirac, and Bose-Ein­stein en­ergy dis­tri­b­u­tions of weakly in­ter­act­ing par­ti­cles for a sys­tem for which the net en­ergy is pre­cisely known.

The ob­jec­tive is to find the shelf num­bers for which the num­ber of eigen­func­tions is max­i­mal. Ac­tu­ally, it is math­e­mat­i­cally eas­ier to find the max­i­mum of , and that is the same thing: if is as big as it can be, then so is . The ad­van­tage of work­ing with is that it sim­pli­fies all the prod­ucts in the ex­pres­sions for the de­rived in de­riva­tion {D.57} into sums: math­e­mat­ics says that equals plus for any (pos­i­tive) and .

It will be as­sumed, fol­low­ing de­riva­tion {N.24}, that if the max­i­mum value is found among all shelf oc­cu­pa­tion num­bers, whole num­bers or not, it suf­fices. More dar­ingly, er­rors less than a par­ti­cle are not go­ing to be taken se­ri­ously.

In find­ing the max­i­mum of , the shelf num­bers can­not be com­pletely ar­bi­trary; they are con­strained by the con­di­tions that the sum of the shelf num­bers must equal the to­tal num­ber of par­ti­cles , and that the par­ti­cle en­er­gies must sum to­gether to the given to­tal en­ergy :

Math­e­mati­cians call this a con­strained max­i­miza­tion prob­lem.

Ac­cord­ing to cal­cu­lus, with­out the con­straints, you can just put the de­riv­a­tives of with re­spect to all the shelf num­bers to zero to find the max­i­mum. With the con­straints, you have to add penalty terms that cor­rect for any go­ing out of bounds, {D.48}, and the cor­rect func­tion whose de­riv­a­tives must be zero is

where the con­stants and are un­known penalty fac­tors called the La­grangian mul­ti­pli­ers.

At the shelf num­bers for which the num­ber of eigen­func­tions is largest, the de­riv­a­tives must be zero. How­ever, that con­di­tion is dif­fi­cult to ap­ply ex­actly, be­cause the ex­pres­sions for as given in the text in­volve the fac­to­r­ial func­tion, or rather, the gamma func­tion. The gamma func­tion does not have a sim­ple de­riv­a­tive. Here typ­i­cal text­books will flip out the Stir­ling ap­prox­i­ma­tion of the fac­to­r­ial, but this ap­prox­i­ma­tion is sim­ply in­cor­rect in parts of the range of in­ter­est, and where it ap­plies, the er­ror is un­known.

It is a much bet­ter idea to ap­prox­i­mate the dif­fer­en­tial quo­tient by a dif­fer­ence quo­tient, as in

This ap­prox­i­ma­tion is very mi­nor, since ac­cord­ing to the so-called mean value the­o­rem of math­e­mat­ics, the lo­ca­tion where is zero is at most one par­ti­cle away from the de­sired lo­ca­tion where is zero. Bet­ter still, will be no more that half a par­ti­cle off, and the analy­sis al­ready had to com­mit it­self to ig­nor­ing frac­tional parts of par­ti­cles any­way. The dif­fer­ence quo­tient leads to sim­ple for­mu­lae be­cause the gamma func­tion sat­is­fies the con­di­tion for any value of , com­pare the no­ta­tions sec­tion un­der !.

Now con­sider first dis­tin­guish­able par­ti­cles. The func­tion to dif­fer­en­ti­ate is de­fined above, and plug­ging in the ex­pres­sion for as found in de­riva­tion {D.57} pro­duces

For any value of the shelf num­ber , in the limit , tends to neg­a­tive in­fin­ity be­cause tends to pos­i­tive in­fin­ity in that limit and its log­a­rithm ap­pears with a mi­nus sign. In the limit , tends once more to neg­a­tive in­fin­ity, since for large val­ues of is ac­cord­ing to the so-called Stir­ling for­mula ap­prox­i­mately equal to , so the term in goes to mi­nus in­fin­ity more strongly than the terms pro­por­tional to might go to plus in­fin­ity. If tends to mi­nus in­fin­ity at both ends of the range 1 , there must be a max­i­mum value of some­where within that range where the de­riv­a­tive with re­spect to is zero. More specif­i­cally, work­ing out the dif­fer­ence quo­tient:

and is in­fin­ity at 1 and mi­nus in­fin­ity at . Some­where in be­tween, will cross zero. In par­tic­u­lar, com­bin­ing the log­a­rithms and then tak­ing an ex­po­nen­tial, the best es­ti­mate for the shelf oc­cu­pa­tion num­ber is

The cor­rect­ness of the fi­nal half par­ti­cle is clearly doubt­ful within the made ap­prox­i­ma­tions. In fact, it is best ig­nored since it only makes a dif­fer­ence at high en­er­gies where the num­ber of par­ti­cles per shelf be­comes small, and surely, the cor­rect prob­a­bil­ity of find­ing a par­ti­cle must go to zero at in­fi­nite en­er­gies, not to mi­nus half a par­ti­cle! There­fore, the best es­ti­mate for the num­ber of par­ti­cles per sin­gle-par­ti­cle en­ergy state be­comes the Maxwell-Boltz­mann dis­tri­b­u­tion. Note that the de­riva­tion might be off by a par­ti­cle for the lower en­ergy shelves. But there are a lot of par­ti­cles in a macro­scopic sys­tem, so it is no big deal.

The case of iden­ti­cal fermi­ons is next. The func­tion to dif­fer­en­ti­ate is now

This time is mi­nus in­fin­ity when a shelf num­ber reaches 1 or . So there must be a max­i­mum to when varies be­tween those lim­its. The dif­fer­ence quo­tient ap­prox­i­ma­tion pro­duces

which can be solved to give

The fi­nal term, less than half a par­ti­cle, is again best left away, to en­sure that 0 as it should. That gives the Fermi-Dirac dis­tri­b­u­tion.

Fi­nally, the case of iden­ti­cal bosons, is, once more, the tricky one. The func­tion to dif­fer­en­ti­ate is now

For now, as­sume that 1 for all shelves. Then is again mi­nus in­fin­ity for 1. For , how­ever, will be­have like . This tends to mi­nus in­fin­ity if is pos­i­tive, so for now as­sume it is. Then the dif­fer­ence quo­tient ap­prox­i­ma­tion pro­duces

which can be solved to give

The fi­nal half par­ti­cle is again best ig­nored to get the num­ber of par­ti­cles to be­come zero at large en­er­gies. Then, if it is as­sumed that the num­ber of sin­gle-par­ti­cle states on the shelves is large, the Bose-Ein­stein dis­tri­b­u­tion is ob­tained. If is not large, the num­ber of par­ti­cles could be less than the pre­dicted one by up to a fac­tor 2, and if is one, the en­tire story comes part. And so it does if is not pos­i­tive.

Be­fore ad­dress­ing these nasty prob­lems, first the phys­i­cal mean­ing of the La­grangian mul­ti­plier needs to be es­tab­lished. It can be in­ferred from ex­am­in­ing the case that two dif­fer­ent sys­tems, call them and , are in ther­mal con­tact. Since the in­ter­ac­tions are as­sumed weak, the eigen­func­tions of the com­bined sys­tem are the prod­ucts of those of the sep­a­rate sys­tems. That means that the num­ber of eigen­func­tions of the com­bined sys­tem is the prod­uct of those of the in­di­vid­ual sys­tems. There­fore the func­tion to dif­fer­en­ti­ate be­comes

Note the con­straints: the num­ber of par­ti­cles in sys­tem must be the cor­rect num­ber of par­ti­cles in that sys­tem, and sim­i­lar for sys­tem . How­ever, since the sys­tems are in ther­mal con­tact, they can ex­change en­ergy through the weak in­ter­ac­tions and there is no longer a con­straint on the en­ergy of the in­di­vid­ual sys­tems. Only the com­bined en­ergy must equal the given to­tal. That means the two sys­tems share the same La­grangian vari­able . For the rest, the equa­tions for the two sys­tems are just like if they were not in ther­mal con­tact, be­cause the log­a­rithm in sep­a­rates, and then the dif­fer­en­ti­a­tions with re­spect to the shelf num­bers and give the same re­sults as be­fore.

It fol­lows that two sys­tems that have the same value of can be brought into ther­mal con­tact and noth­ing hap­pens, macro­scop­i­cally. How­ever, if two sys­tems with dif­fer­ent val­ues of are brought into con­tact, the sys­tems will ad­just, and en­ergy will trans­fer be­tween them, un­til the two val­ues have be­come equal. That means that is a tem­per­a­ture vari­able. From here on, the tem­per­a­ture will be de­fined as = 1/, so that 1/, with the Boltz­mann con­stant. The same way, for now the chem­i­cal po­ten­tial will sim­ply be de­fined to be the con­stant . Chap­ter 11.14.4 will even­tu­ally es­tab­lish that the tem­per­a­ture de­fined here is the ideal gas tem­per­a­ture, while de­riva­tion {D.62} will es­tab­lish that is the Gibbs free en­ergy per atom that is nor­mally de­fined as the chem­i­cal po­ten­tial.

Re­turn­ing now to the nasty prob­lems of the dis­tri­b­u­tion for bosons, first as­sume that every shelf has at least two states, and that is pos­i­tive even for the ground state. In that case there is no prob­lem with the de­rived so­lu­tion. How­ever, Bose-Ein­stein con­den­sa­tion will oc­cur when ei­ther the num­ber den­sity is in­creased by putting more par­ti­cles in the sys­tem, or the tem­per­a­ture is de­creased. In­creas­ing par­ti­cle den­sity is as­so­ci­ated with in­creas­ing chem­i­cal po­ten­tial be­cause

im­plies that every shelf par­ti­cle num­ber in­creases when in­creases. De­creas­ing tem­per­a­ture by it­self de­creases the num­ber of par­ti­cles, and to com­pen­sate and keep the num­ber of par­ti­cles the same, must then once again in­crease. When gets very close to the ground state en­ergy, the ex­po­nen­tial in the ex­pres­sion for the num­ber of par­ti­cles on the ground state shelf 1 be­comes very close to one, mak­ing the to­tal de­nom­i­na­tor very close to zero, so the num­ber of par­ti­cles in the ground state blows up. When it be­comes a fi­nite frac­tion of the to­tal num­ber of par­ti­cles even when is macro­scop­i­cally large, Bose-Ein­stein con­den­sa­tion is said to have oc­curred.

Note that un­der rea­son­able as­sump­tions, it will only be the ground state shelf that ever ac­quires a fi­nite frac­tion of the par­ti­cles. For, as­sume the con­trary, that shelf 2 also holds a fi­nite frac­tion of the par­ti­cles. Us­ing Tay­lor se­ries ex­pan­sion of the ex­po­nen­tial for small val­ues of its ar­gu­ment, the shelf oc­cu­pa­tion num­bers are

For to also be a fi­nite frac­tion of the to­tal num­ber of par­ti­cles, must be sim­i­larly small as . But then, rea­son­ably as­sum­ing that the en­ergy lev­els are at least roughly equally spaced, and that the num­ber of states will not de­crease with en­ergy, so must be a fi­nite frac­tion of the to­tal, and so on. You can­not have a large num­ber of shelves each hav­ing a fi­nite frac­tion of the par­ti­cles, be­cause there are not so many par­ti­cles. More pre­cisely, a sum roughly like , (or worse), sums to an amount that is much larger than the term for 2 alone. So if would be a fi­nite frac­tion of , then the sum would be much larger than .

What hap­pens dur­ing con­den­sa­tion is that be­comes much closer to than is to the next en­ergy level , and only the ground state shelf ends up with a fi­nite frac­tion of the par­ti­cles. The re­main­der is spread out so much that the shelf num­bers im­me­di­ately above the ground state only con­tain a neg­li­gi­ble frac­tion of the par­ti­cles. It also fol­lows that for all shelves ex­cept the ground state one, may be ap­prox­i­mated as be­ing . (Spe­cific data for par­ti­cles in a box is given in chap­ter 11.14.1. The en­tire story may of course need to be mod­i­fied in the pres­ence of con­fine­ment, com­pare chap­ter 6.12.)

The other prob­lem with the analy­sis of the oc­cu­pa­tion num­bers for bosons is that the num­ber of sin­gle-par­ti­cle states on the shelves had to be at least two. There is no rea­son why a sys­tem of weakly-in­ter­act­ing spin­less bosons could not have a unique sin­gle-par­ti­cle ground state. And com­bin­ing the ground state with the next one on a sin­gle shelf is surely not an ac­cept­able ap­prox­i­ma­tion in the pres­ence of po­ten­tial Bose-Ein­stein con­den­sa­tion. For­tu­nately, the math­e­mat­ics still partly works:

im­plies that 0. In other words, is equal to the ground state en­ergy ex­actly, rather than just ex­tremely closely as above.

That then is the con­densed state. With­out a chem­i­cal po­ten­tial that can be ad­justed, for any given tem­per­a­ture the states above the ground state con­tain a num­ber of par­ti­cles that is com­pletely un­re­lated to the ac­tual num­ber of par­ti­cles that is present. What­ever is left can be dumped into the ground state, since there is no con­straint on .

Con­den­sa­tion stops when the num­ber of par­ti­cles in the states above the ground state wants to be­come larger than the ac­tual num­ber of par­ti­cles present. Now the math­e­mat­ics changes, be­cause na­ture says “Wait a minute, there is no such thing as a neg­a­tive num­ber of par­ti­cles in the ground state!” Na­ture now adds the con­straint that 0 rather than neg­a­tive. That adds an­other penalty term, to and takes care of sat­is­fy­ing the equa­tion for the ground state shelf num­ber. It is a sad story, re­ally: be­low the con­den­sa­tion tem­per­a­ture, the ground state was awash in par­ti­cles, above it, it has zero. None.

A sys­tem of weakly in­ter­act­ing he­lium atoms, spin­less bosons, would have a unique sin­gle-par­ti­cle ground state like this. Since be­low the con­den­sa­tion tem­per­a­ture, the el­e­vated en­ergy states have no clue about an im­pend­ing lack of par­ti­cles ac­tu­ally present, phys­i­cal prop­er­ties such as the spe­cific heat stay an­a­lyt­i­cal un­til con­den­sa­tion ends.

It may be noted that above the con­den­sa­tion tem­per­a­ture it is only the most prob­a­ble set of the oc­cu­pa­tion num­bers that have ex­actly zero par­ti­cles in the unique ground state. The ex­pec­ta­tion value of the num­ber in the ground state will in­clude neigh­bor­ing sets of oc­cu­pa­tion num­bers to the most prob­a­ble one, and the num­ber has nowhere to go but up, com­pare {D.62}.