Statistical Mechanics - Physics at Oregon State University

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