Statistical Mechanics - Physics at Oregon State University

5.2. BOSONS IN A BOX. 109
0 < T Tm. Also, in case T → ∞ the sum approaches the integral again, and
there are no convergence problems for the series.
The next question is if the series always converges to the integral ˜ Ω. The
integrant in k-space is again steepest for values of k near zero, and we expect
problems in the first few terms of the series.
Simple example.
Consider the following series:
Define xn = n
L
F (α) = 1
√ L
1
and ∆x = L . We have:
F (α) =
∞
n=0
∞
n=0
n − e L
√
n + αL
and in the limit L → ∞ this approaches the integral:
F (α) =
(5.109)
e−xn √ ∆x (5.110)
xn + α
dx e−x
√ x + α
(5.111)
If we now take the limit α → 0 we get a well defined result. But we did take
the thermodynamic limit L → ∞ first, and that is not correct. If we take the
1
limit α → 0 in the series the very first term, which is equal to √αL blows up!
We can find a way out. Write the series in the following form:
F (α) = 1
L √ 1
+ √
α L
∞
n=1
n − e L
√
n + αL
(5.112)
Now we can replace the series by an integral for all values of α, since the divergent
term is isolated outside the series. Hence we have
L ≫ 1 ⇒ F (α) ≈ 1
L √ α +
dx e−x
√ x + α
(5.113)
where the first term can be ignored only if L ≫ 1
α ! Hence in the limit α → 0
this first term can never be ignored.
The low temperature limit for bosons.
We now return to the question of the magnitude of the error term at low
temperature. The easiest way to discuss the nature of the low temperature limit
is to consider the expansion for N in terms of orbitals: