6.14 Maxwell-Boltz­mann Dis­tri­b­u­tion

The pre­vi­ous sec­tions showed that the ther­mal sta­tis­tics of a sys­tem of iden­ti­cal bosons is nor­mally dra­mat­i­cally dif­fer­ent from that of a sys­tem of iden­ti­cal fermi­ons. How­ever, if the tem­per­a­ture is high enough, and the box hold­ing the par­ti­cles big enough, the dif­fer­ences dis­ap­pear. These are ideal gas con­di­tions.

Un­der these con­di­tions the av­er­age num­ber of par­ti­cles per sin­gle-par­ti­cle state be­comes much smaller than one. That av­er­age can then be ap­prox­i­mated by the so-called

$$\fbox{$\displaystyle \mbox{Maxwell-Boltzmann distribution:... ...{\rm p}- \mu)/{k_{\rm B}}T}} \qquad \iota^{\rm{d}} \ll 1 $} %$$ (6.21)

Here ${\vphantom' E}^{\rm p}$ is again the sin­gle-par­ti­cle en­ergy, $\mu$ the chem­i­cal po­ten­tial, $T$ the ab­solute tem­per­a­ture, and $k_{\rm B}$ the Boltz­mann con­stant. Un­der the given con­di­tions of a low par­ti­cle num­ber per state, the ex­po­nen­tial is big enough that the $\pm1$ found in the Bose-Ein­stein and Fermi-Dirac dis­tri­b­u­tions (6.9) and (6.19) can be ig­nored.

Fig­ure 6.16 gives a pic­ture of the dis­tri­b­u­tion for non­in­ter­act­ing par­ti­cles in a box. The en­ergy spec­trum to the right shows the av­er­age num­ber of par­ti­cles per state as the rel­a­tive width of the red re­gion. The wave num­ber space to the left shows a typ­i­cal sys­tem en­ergy eigen­func­tion; states with a par­ti­cle in them are in red.

Fig­ure 6.16: Par­ti­cles at high-enough tem­per­a­ture and low-enough par­ti­cle den­sity.

Since the (anti) sym­metriza­tion re­quire­ments no longer make a dif­fer­ence, the Maxwell-Boltz­mann dis­tri­b­u­tion is of­ten rep­re­sented as ap­plic­a­ble to dis­tin­guish­able par­ti­cles. But of course, where are you go­ing to get a macro­scopic num­ber of, say, 10$\POW9,{20}$ par­ti­cles, each of a dif­fer­ent type? The imag­i­na­tion bog­gles. Still, the d in $\iota^{\rm {d}}$ refers to dis­tin­guish­able.

The Maxwell-Boltz­mann dis­tri­b­u­tion was al­ready known be­fore quan­tum me­chan­ics. The fac­tor $e^{-{\vphantom' E}^{\rm p}/{k_{\rm B}}T}$ in it im­plies that the num­ber of par­ti­cles at a given en­ergy de­creases ex­po­nen­tially with the en­ergy. A clas­si­cal ex­am­ple is the de­crease of den­sity with height in the at­mos­phere. In an equi­lib­rium (i.e. isother­mal) at­mos­phere, the num­ber of mol­e­cules at a given height $h$ is pro­por­tional to $e^{-mgh/{k_{\rm B}}T}$ where $mgh$ is the grav­i­ta­tional po­ten­tial en­ergy of the mol­e­cules. (It should be noted that nor­mally the at­mos­phere is not isother­mal be­cause of the heat­ing of the earth sur­face by the sun and other ef­fects.)

The ex­am­ple of the isother­mal at­mos­phere can be used to il­lus­trate the idea of in­trin­sic chem­i­cal po­ten­tial. Think of the en­tire at­mos­phere as build up out of small boxes filled with par­ti­cles. The walls of the boxes con­duct some heat and they are very slightly porous, to al­low an equi­lib­rium to de­velop if you are very pa­tient. Now write the en­ergy of the par­ti­cles as the sum of their grav­i­ta­tional po­ten­tial en­ergy plus an in­trin­sic en­ergy (which is just their ki­netic en­ergy for the model of non­in­ter­act­ing par­ti­cles). Sim­i­larly write the chem­i­cal po­ten­tial as the sum of the grav­i­ta­tional po­ten­tial en­ergy plus an in­trin­sic chem­i­cal po­ten­tial:

$${\vphantom' E}^{\rm p}= mgh + {\vphantom' E}^{\rm p}_{\rm i} \qquad \mu = mgh + \mu_{\rm i}$$

Since ${\vphantom' E}^{\rm p}-\mu$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\vphantom' E}^{\rm p}_{\rm {i}}-\mu_{\rm {i}}$, the Maxwell-Boltz­mann dis­tri­b­u­tion is not af­fected by the switch to in­trin­sic quan­ti­ties. But that im­plies that the re­la­tion­ship be­tween ki­netic en­ergy, in­trin­sic chem­i­cal po­ten­tial, and num­ber of par­ti­cles in each in­di­vid­ual box is the same as if grav­ity was not there. In each box, the nor­mal ideal gas law ap­plies in terms of in­trin­sic quan­ti­ties.

How­ever, dif­fer­ent boxes have dif­fer­ent in­trin­sic chem­i­cal po­ten­tials. The en­tire sys­tem of boxes has one global tem­per­a­ture and one global chem­i­cal po­ten­tial, since the porous walls make it a sin­gle sys­tem. But the global chem­i­cal po­ten­tial that is the same in all boxes in­cludes grav­ity. That makes the in­trin­sic chem­i­cal po­ten­tial in boxes at dif­fer­ent heights dif­fer­ent, and with it the num­ber of par­ti­cles in the boxes.

In par­tic­u­lar, boxes at higher al­ti­tudes have less mol­e­cules. Com­pare states with the same in­trin­sic, ki­netic, en­ergy for boxes at dif­fer­ent heights. Ac­cord­ing to the Maxwell-Boltz­mann dis­tri­b­u­tion, the num­ber of par­ti­cles in a state with in­trin­sic en­ergy ${\vphantom' E}^{\rm p}_{\rm {i}}$ is 1$\raisebox{.5pt}{$/$}$​$e^{({\vphantom' E}^{\rm p}_{\rm {i}}+mgh-\mu)/{k_{\rm B}}T}$. That de­creases with height pro­por­tional to $e^{-mgh/{k_{\rm B}}T}$, just like clas­si­cal analy­sis pre­dicts.

Now sup­pose that you make the par­ti­cles in one of the boxes hot­ter. There will then be a flow of heat out of that box to the neigh­bor­ing boxes un­til a sin­gle tem­per­a­ture has been reestab­lished. On the other hand, as­sume that you keep the tem­per­a­ture un­changed, but in­crease the chem­i­cal po­ten­tial in one of the boxes. That means that you must put more par­ti­cles in the box, be­cause the Maxwell-Boltz­mann dis­tri­b­u­tion has the num­ber of par­ti­cles per state equal to $e^{\mu/{k_{\rm B}}T}$. The ex­cess par­ti­cles will slowly leak out through the slightly porous walls un­til a sin­gle chem­i­cal po­ten­tial has been reestab­lished. Ap­par­ently, then, too high a chem­i­cal po­ten­tial pro­motes par­ti­cle dif­fu­sion away from a site, just like too high a tem­per­a­ture pro­motes ther­mal en­ergy dif­fu­sion away from a site.

While the Maxwell-Boltz­mann dis­tri­b­u­tion was al­ready known clas­si­cally, quan­tum me­chan­ics adds the no­tion of dis­crete en­ergy states. If there are more en­ergy states at a given en­ergy, there are go­ing to be more par­ti­cles at that en­ergy, be­cause (6.21) is per state. For ex­am­ple, con­sider the num­ber of ther­mally ex­cited atoms in a thin gas of hy­dro­gen atoms. The num­ber $I_2$ of atoms that are ther­mally ex­cited to en­ergy $E_2$ is in terms of the num­ber $I_1$ with the ground state en­ergy $E_1$:

$$\frac{I_2}{I_1}= \frac{8}{2}e^{-(E_2-E_1)/{k_{\rm B}}T}$$

The fi­nal ex­po­nen­tial is due to the Maxwell-Boltz­mann dis­tri­b­u­tion. The lead­ing fac­tor arises be­cause there are eight elec­tron states at en­ergy $E_2$ and only two at en­ergy $E_1$ in a hy­dro­gen atom. At room tem­per­a­ture ${k_{\rm B}}T$ is about 0.025 eV, while $E_2-E_1$ is 10.2 eV, so there are not go­ing to be any ther­mally ex­cited atoms at room tem­per­a­ture.

Key Points

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The Maxwell-Boltz­mann dis­tri­b­u­tion gives the num­ber of par­ti­cles per sin­gle-par­ti­cle state for a macro­scopic sys­tem at a nonzero tem­per­a­ture.

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It as­sumes that the par­ti­cle den­sity is low enough, and the tem­per­a­ture high enough, that (anti) sym­metriza­tion re­quire­ments can be ig­nored.

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In par­tic­u­lar, the av­er­age num­ber of par­ti­cles per sin­gle-par­ti­cle state should be much less than one.

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Ac­cord­ing to the dis­tri­b­u­tion, the av­er­age num­ber of par­ti­cles in a state de­creases ex­po­nen­tially with its en­ergy.

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Sys­tems for which the dis­tri­b­u­tion ap­plies can of­ten be de­scribed well by clas­si­cal physics.

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Dif­fer­ences in chem­i­cal po­ten­tial pro­mote par­ti­cle dif­fu­sion.