Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
5.2. BOSONS IN A BOX. 107<br />
with<br />
Znx,ny,nz(T, µ, V ) =<br />
<br />
1 − e<br />
−1 µ−ɛ(nx,ny ,nz )<br />
kB T<br />
(5.98)<br />
The same trick applies here as we used for fermions, and we replace the series<br />
by an integral:<br />
and<br />
˜Ω(T,<br />
1<br />
µ, V ) = V kBT (2S + 1)<br />
(2π) 3<br />
<br />
d 3 µ−<br />
k log(1 − e<br />
2k 2<br />
2M<br />
kB T ) (5.99)<br />
Ω(T, µ, V ) = ˜ Ω(T, µ, V ) + error(T, µ, V ) (5.100)<br />
where the error is defined by making this equ<strong>at</strong>ion exact. One can again show<br />
th<strong>at</strong> also for bosons we have limV →∞ error(T, µ, V ) = 0, but the big difference<br />
with a system of fermions is th<strong>at</strong> it is not possible to give an upper-bound to<br />
the error which is only a function of V. The value of V we need in order to get<br />
for example a 1% error becomes infinitely large when T → 0.<br />
Note th<strong>at</strong> µ < 0 for free particles, and hence 0 < λ < 1. Problems also occur<br />
in the limit µ → 0, where the integrant diverges <strong>at</strong> the lower limit of integr<strong>at</strong>ion.<br />
These can be dealt with. The integral is of the form:<br />
<br />
1<br />
d 3 k log(<br />
1 − λe − 2 k 2<br />
2Mk B T<br />
) (5.101)<br />
and when k is small the argument of the logarithm is approxim<strong>at</strong>ely log( 2MkBT<br />
2k2 ).<br />
Therefore, near the origin we need to evalu<strong>at</strong>e d3k log(k), which behaves well<br />
even though the logarithm diverges, since k2 log(k) goes to zero. Hence the<br />
function ˜ Ω is well defined for 0 < λ 1, or for µ 0, and there are no problems<br />
in this integral with the limit µ → 0. As we will see, this limit corresponds to<br />
the limit T → 0.<br />
Special functions for bosons.<br />
In a manner similar to wh<strong>at</strong> we found for fermions, we define a set of special<br />
functions by<br />
gα(z) =<br />
∞<br />
n=1<br />
zn , |z| < 1 (5.102)<br />
nα and use analytic continu<strong>at</strong>ion to define these functions everywhere. The formula<br />
for ˜ Ω is now manipul<strong>at</strong>ed in the same way we did for fermions, and we find<br />
˜Ω(T, µ, V ) = −(2S + 1)V kBT nQ(T )g 5 (λ) (5.103)<br />
2