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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56 CHAPTER 3. VARIABLE NUMBER OF PARTICLES<br />
where in the last case (which we did not discuss here) the volume of the st<strong>at</strong>e<br />
is vs. This last formul<strong>at</strong>ion is only included for completeness, and to show how<br />
other transform<strong>at</strong>ions are made. The argument of the exponent in the sum is<br />
easy to find if we think about the free energies th<strong>at</strong> are used in the situ<strong>at</strong>ions<br />
above. In the first case it is F, in the second case Ω = F − µN, and in the last<br />
case G = F + pV . The rel<strong>at</strong>ions with the free energy follow from:<br />
F (T, V, N) = −kBT log(Z(T, V, N)) (3.65)<br />
Ω(T, V, µ) = −kBT log(Z(T, V, µ)) (3.66)<br />
G(T, p, N) = −kBT log(ζ(T, p, N)) (3.67)<br />
or in exponential form, which resembles the sum in the partition functions,<br />
−βF (T,V,N)<br />
Z(T, V, N) = e<br />
Z(T, V, µ) = e −βΩ(T,V,µ)<br />
−βF (T,p,N)<br />
ζ(T, p, N) = e<br />
(3.68)<br />
(3.69)<br />
(3.70)<br />
Once we have the free energies, we can revert to thermodynamics. We<br />
can also do more calcul<strong>at</strong>ions in st<strong>at</strong>istical mechanics, using the fact th<strong>at</strong> the<br />
probability of finding a st<strong>at</strong>e is always the exponential factor in the sum divided<br />
by the sum. This is a generaliz<strong>at</strong>ion of the Boltzmann factor. We can then<br />
calcul<strong>at</strong>e fluctu<strong>at</strong>ions, and show th<strong>at</strong> certain response functions are rel<strong>at</strong>ed to<br />
fluctu<strong>at</strong>ions. For example, we would find:<br />
(∆V ) 2 <br />
∂V<br />
= kBT<br />
(3.71)<br />
∂p<br />
The example with changing V to p is a bit contrived, but shows the general<br />
principle. A practical example involves magnetic systems, where we can either<br />
use the total magnetiz<strong>at</strong>ion M of the applied field H as an independent variable.<br />
There we would have:<br />
or<br />
Z(T, V, N, M) =<br />
Z(T, V, N, H) = <br />
<br />
s∈S(V,N,M)<br />
s∈S(V,N)<br />
T,N<br />
e −βɛs −βF (T,V,N,M)<br />
= e<br />
e −β(ɛs−Hms) = e −βG(T,V,N,H)<br />
(3.72)<br />
(3.73)<br />
where G is a generalized form of a Gibbs energy. We have used the same symbol<br />
for the partition function (sorry, I could not find a new and better one). This<br />
is also very commonly done in the liter<strong>at</strong>ure. Fluctu<strong>at</strong>ions follow from