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