Statistical Mechanics - Physics at Oregon State University

3.6. A SIMPLE EXAMPLE. 57
(∆M) 2 = kBT
∂M
∂H T,V,N
(3.74)
Finally, the different partition sums are related by Laplace transformations.
For example, we have
Z(T, V, µ) =
e βµN Z(T, V, N) (3.75)
N
We can also do the back transformation, but that involves an integration in the
complex µ plane. Although we can relate complex temperature T to time, I
have no idea what the physical meaning of complex µ would be.
3.6 A simple example.
A very simple example of the use of the grand canonical ensemble is a study of a
one-dimensional photon gas where all the photons have frequency ω. The energy
ɛn of the quantum state n is nω, while the number of particles in that state is
Nn = n. The values for n are 0,1,2,.... This example is oversimplified, since one
quantum number controls two quantum states. Also, there are only two sets of
thermodynamic variables. We should therefore only use it as an illustration of
how we sometimes can do calculations, and not draw strong conclusions from
the final results. The grand partition function follows from
Z(T, µ) =
∞
e β(µ−ω)n
n=0
This series can only be summed for µ < ω, leading to
Z(T, µ) =
1
1 − e β(µ−ω)
(3.76)
(3.77)
All thermodynamic quantities can now be evaluated from the grand potential
and we find
Ω(T, µ) = kBT log(1 − e β(µ−ω) ) (3.78)
S(T, µ) = −kB log(1 − e β(µ−ω) ) + 1 ω − µ
T 1 − eβ(µ−ω) (3.79)
N(T, µ) = frac1e β(ω−µ) − 1 (3.80)
U(T, µ) = Ω + T S + µN = Nω (3.81)
We have followed all steps in the statistical mechanical prescription for the
grand canonical ensemble, and we obtained reasonable answers. Nevertheless,
thermodynamics tells us that for a system with only T, S, µ, and N as state