Statistical Physics, Second Revised and Enlarged Edition

60 Gases: the distributions

whichrearranges togive

n∗i=gi/[exp(−α−βεi)− 1 ]

Intermsofthedistributionfunctionthis becomes

fffi=ni∗/gi= 1 /[exp(−α−βεi)− 1 ] (5.11a)

or

fffBE(ε)= 1 /[exp(−α−βε)− 1 ] (5.11b)

Thisisthe Bose–Einsteindistribution.

5.4. 3 Maxwell–Boltzmannstatistics

For the dilute (fermion or boson)gas, the procedure maybe followed for the third
time, starting now with the expression ( 5 .6) for the numbertttMBof microstates.
Weobtain
lntttMB=

∑

i

{nilngi−nilnni+ni}

which using(5.9) with (5.8) and (5.7)gives

∑

i

{ln[gi/ni]+α+βεi}dni= 0

Removingthe summation for the equilibrium distribution and rearrangingnowgives

n∗i =giexp(α+βεi)

The final result for the Maxwell–Boltzmann distribution is therefore

fffi=n∗i/gi=exp(α+βεi) ( 5 .12a)

or

fffMB(ε)=exp(α+βε) (5.12b)

One maynotein passingthat thisdistributionbears a markedsimilarityto theBoltz-
mann distribution for localized particles derived in Chapter 2. This greatly simplifies
thediscussion of dilute gases, since wehave unknowinglyalready coveredmuchof
theground, as we shall see in Chapter6.