Statistical Mechanics - Physics at Oregon State University

34 CHAPTER 2. THE CANONICAL ENSEMBLE
F = U + T
∂F
∂T V,N
which leads to the following differential equation for F
∂ F
= −
∂T T
U
T 2
(2.41)
(2.42)
Combined with the relation between U and log(Z) this leads to the simple
equation
F
∂ T
∂ log(Z)
= −kB
(2.43)
∂T
∂T
V,N
V,N
or
F = −kBT log(Z) + c(V, N)T (2.44)
The constant of integration c(V,N) is determined by the state of the system
at low temperatures. Suppose the ground state has energy ɛ0 and degeneracy
g0. The only terms playing a role in the partition function at low temperature
correspond to the ground state and hence we have
and hence in lowest order in T near T=0:
lim
T →0 Z = g0e − ɛ0 kB T (2.45)
F (T, V, N) = −kBT log(g0) + ɛ0 + c(V, N)T (2.46)
which gives for the entropy
∂F
S = −
∂T V,N
≈ kB log(g0) − c(V, N) (2.47)
and since we know that at T = 0 the entropy is given by the first term, we
need to have c(V, N) = 0, and
or
F (T, V, N) = −kBT log(Z(T, V, N)) (2.48)
F (T,V,N)
− k Z(T, V, N) = e B T (2.49)
In other words, we not only know that F can be expressed as a function of
T and V (and N) , we also have an explicit construction of this function. First
we specify the volume V, then calculate all the quantum states ɛs(V ) of the
system. Next we specify the temperature T and evaluate the partition function
Z(T, V ), the Helmholtz free energy F(T,V) and all other quantities of interest.