Statistical Mechanics - Physics at Oregon State University

3.7. IDEAL GAS IN FIRST APPROXIMATION. 63
Our new condition is therefore that the second term is small compared to the
first term, or
n
| log(1 + ε)| ≪ | log( ) − 1| (3.112)
nQ(T )
We now take the same requirement as in the one particle case, and assume
that
n ≪ nQ(T ) (3.113)
which also implies V nQ(T ) ≫ N or ε ≪ 1. This means that the left hand side
of 3.112 is much smaller than one. Since the right hand side is much larger
than one (note the absolute sign, the logarithm itself is large and negative!), the
Helmholtz free energy is indeed approximated by the formula we gave.
The inequality n ≪ nQ(T ) inspired the name quantum concentration. If
the concentration is comparable to or larger than the quantum concentration,
we have to use the series. Also, at that point quantum effects like the Pauli
principle will start to play a role. If the condition above is fulfilled, we are in
the classical limit
Hence everything seems OK. But there might be one last objection. We
need to take derivatives of the free energy, and is the derivative of the error
term small? The two terms in the free energy are:
n
Fideal = −NkBT log( ) − 1
(3.114)
nQ(T )
and
1 −
Ftrunc = −NkBT ε = −NkBT f(V, T )(V nQ(T )) 3 (3.115)
where f(V, T ) is some function of order one. As an example, we take the derivative
with respect to V, in order to get the pressure. The ideal gas result is
NkBT
V , and the derivative of the truncation error in the free energy has two
terms. One part comes from the last factor, and gives
1 NkBT
1 −
f(V, T )(V nQ(T )) 3 (3.116)
3 V
which is again small if V nQ(T ) ≫ N. The second part gives
− NkBT
∂f
V (V, T )
V ∂V
T
1 −
(V nQ(T )) 3 (3.117)
and could be large if f(V, T ) is an oscillating function. This is fortunately not
the case. Formula 3.89 tells us that f(V, T ) is equal to − 1
2 plus terms that are
exponentially small. The first part gives zero derivatives, and the second part
gives derivatives which are exponentially small.
The error analysis above is typical for fundamental questions in statistical
mechanics. In many cases it is ignored, and often without consequences. In