Statistical Mechanics - Physics at Oregon State University

5.1. FERMIONS IN A BOX. 93
T ′ m T Tm we can choose a range of volumes so that the series converges
uniformly and is arbitrarily close to the integral, with an error independent of
the values of µ and T . Therefore, we can replace the series by the integral for
all such values of µ and T , and if the volume is arbitrarily large, the error is
arbitrarily small in a uniform manner. As a result, the thermodynamic limit,
which we need to take at the end, will not change the results from what we
obtained by using the integral.
There are still three problem areas. What if µ → ∞? What if T → 0? What
if T → ∞? The last case is the easiest. If the temperature is very large, the
series is very close to the integral for all volumes that are larger than a certain
value and all values of µ in the range −∞ µ µm. The latter is true, since
the values of the second order derivative which determine the error are largest
at µm, and hence this value can be used for uniform convergence criteria. The
limit T → 0 is very important and will be discussed after the next paragraph.
What about µ → ∞? It turns out that this limit is equal to the limit of
infinite density. Although we often do calculations at large densities, this limit
is not of physical interest since the behavior of the system will be very different
from what we are interested in.
There is, however, one more detail. If we replace the summation by an
integral, the error is also proportional to the value of the second order derivative
of the function somewhere in the interval. If this second order derivative is
bounded, there is no problem. If it can go to infinity, there is a problem. As we
will see, for fermions there are no problems. But it is good to address this issue
for fermions, because after this we will study bosons where there are problems
with replacing the sum by an integral. The resulting errors do show up in
physics, and are the cause of Bose-Einstein condensation!
The function in the integrant is
log(1 + λe − 2 k 2
2Mk B T ) (5.16)
and is smoothly decaying. The largest order derivatives are when k → 0 and are
inversely proportional to the temperature. So the only problem area is when
T → 0. In this limit the series has no convergence properties, but we need to
investigate whether the series converges to the value of the integral. In this
limit the first few terms are so important, that they determine the behavior of
the series.
The argument of the summation drops from a value of log(1 + e µ
k B T ) at the
origin of k-space to 0 at infinity. When T is very small, though, the value of the
logarithm near the origin becomes very large, and it seems that we need a very
dense mesh in k-space to be able to convert the series to an integral with a small
error. Hence one should take the limit V → ∞ before the limit T → 0, which
is not the correct procedure. For fermions this turns out not to be necessary.
Because the series converges uniformly we are able to interchange summation
and limit and find