6.7 Bose-Ein­stein Dis­tri­b­u­tion

As the pre­vi­ous sec­tion ex­plained, the en­ergy dis­tri­b­u­tion of a macro­scopic sys­tem of par­ti­cles can be found by merely count­ing sys­tem en­ergy eigen­func­tions.

The de­tails of do­ing so are messy but the re­sults are sim­ple. For a sys­tem of iden­ti­cal bosons, it gives the so-called:

$$\fbox{$\displaystyle \mbox{Bose-Einstein distribution:}\qu... ...c{1}{e^{({\vphantom' E}^{\rm p}- \mu)/{k_{\rm B}}T} - 1} $} %$$ (6.9)

Here $\iota^{\rm {b}}$ is the av­er­age num­ber of bosons in a sin­gle-par­ti­cle state with sin­gle-par­ti­cle en­ergy ${\vphantom' E}^{\rm p}$. Fur­ther $T$ is the ab­solute tem­per­a­ture, and $k_{\rm B}$ is the Boltz­mann con­stant, equal to 1.380 65 10$\POW9,{-23}$ J/K.

Fi­nally, $\mu$ is known as the chem­i­cal po­ten­tial and is a func­tion of the tem­per­a­ture and par­ti­cle den­sity. The chem­i­cal po­ten­tial is an im­por­tant phys­i­cal quan­tity, re­lated to such di­verse ar­eas as par­ti­cle dif­fu­sion, the work that a de­vice can pro­duce, and to chem­i­cal and phase equi­lib­ria. It equals the so-called Gibbs free en­ergy on a mo­lar ba­sis. It is dis­cussed in more de­tail in chap­ter 11.12.

The Bose-Ein­stein dis­tri­b­u­tion is de­rived in chap­ter 11. In fact, for var­i­ous rea­sons that chap­ter gives three dif­fer­ent de­riva­tions of the dis­tri­b­u­tion. For­tu­nately they all give the same an­swer. Keep in mind that what­ever this book tells you thrice is ab­solutely true.

The Bose-Ein­stein dis­tri­b­u­tion may be used to bet­ter un­der­stand Bose-Ein­stein con­den­sa­tion us­ing a bit of sim­ple al­ge­bra. First note that the chem­i­cal po­ten­tial for bosons must al­ways be less than the low­est sin­gle-par­ti­cle en­ergy ${\vphantom' E}^{\rm p}_{\rm {gs}}$. Just check it out us­ing the for­mula above: if $\mu$ would be greater than ${\vphantom' E}^{\rm p}_{\rm {gs}}$, then the num­ber of par­ti­cles $\iota^{\rm {b}}$ in the low­est sin­gle-par­ti­cle state would be neg­a­tive. Neg­a­tive num­bers of par­ti­cles do not ex­ist. Sim­i­larly, if $\mu$ would equal ${\vphantom' E}^{\rm p}_{\rm {gs}}$ then the num­ber of par­ti­cles in the low­est sin­gle-par­ti­cle state would be in­fi­nite.

The fact that $\mu$ must stay less than ${\vphantom' E}^{\rm p}_{\rm {gs}}$ means that the num­ber of par­ti­cles in any­thing but the low­est sin­gle-par­ti­cle state has a limit. It can­not be­come greater than

$$\iota^{\rm {b}}_{\rm max} =\frac{1}{e^{({\vphantom' E}^{\rm p}- {\vphantom' E}^{\rm p}_{\rm gs})/{k_{\rm B}}T} - 1}$$

Now as­sume that you keep the box size and tem­per­a­ture both fixed and start putting more and more par­ti­cles in the box. Then even­tu­ally, all the sin­gle-par­ti­cle states ex­cept the ground state hit their limit. Any fur­ther par­ti­cles have nowhere else to go than into the ground state. That is when Bose-Ein­stein con­den­sa­tion starts.

The above ar­gu­ment also il­lus­trates that there are two main ways to pro­duce Bose-Ein­stein con­den­sa­tion: you can keep the box and num­ber of par­ti­cles con­stant and lower the tem­per­a­ture, or you can keep the tem­per­a­ture and box con­stant and push more par­ti­cles in the box. Or a suit­able com­bi­na­tion of these two, of course.

If you keep the box and num­ber of par­ti­cles con­stant and lower the tem­per­a­ture, the math­e­mat­ics is more sub­tle. By it­self, low­er­ing the tem­per­a­ture low­ers the num­ber of par­ti­cles $\iota^{\rm {b}}$ in all states. How­ever, that would lower the to­tal num­ber of par­ti­cles, which is kept con­stant. To com­pen­sate, $\mu$ inches closer to ${\vphantom' E}^{\rm p}_{\rm {gs}}$. This even­tu­ally causes all states ex­cept the ground state to hit their limit, and be­yond that stage the left-over par­ti­cles must then go into the ground state.

You may re­call that Bose-Ein­stein con­den­sa­tion is only Bose-Ein­stein con­den­sa­tion if it does not dis­ap­pear with in­creas­ing sys­tem size. That too can be ver­i­fied from the Bose-Ein­stein dis­tri­b­u­tion un­der fairly gen­eral con­di­tions that in­clude non­in­ter­act­ing par­ti­cles in a box. How­ever, the de­tails are messy and will be left to chap­ter 11.14.1.

Key Points

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The Bose-Ein­stein dis­tri­b­u­tion gives the num­ber of bosons per sin­gle-par­ti­cle state for a macro­scopic sys­tem at a nonzero tem­per­a­ture.

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It also in­volves the Boltz­mann con­stant and the chem­i­cal po­ten­tial.

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It can be used to ex­plain Bose-Ein­stein con­den­sa­tion.