Statistical Mechanics - Physics at Oregon State University

6.4. EQUIVALENCE OF ENTROPY DEFINITIONS FOR CANONICAL ENSEMBLE.135
results for the entropy and density matrix should also be the same in this case,
independent of which definition of the entropy we use. In this section we illustrate
this equivalence directly. This direct method also shows how to use the
maximum entropy principle.
In the canonical ensemble we specify only the average energy U of the system,
the volume V and the number of particles N. If other extensive state variables
are known, they are also included in this list. The trace in the expressions for
the entropy is taken over all states with volume V, number of particles N, and
other extensive state variables. These variables are not allowed to fluctuate.
The energy is allowed to vary, however, and we sum over states with all possible
energies. The additional requirement we have to impose is U = Tr Hρ for the
average energy. This requirement is taken into account via a Lagrange multiplier
β, and we have to maximize
X = −kBTr ρ log ρ + λkB (Tr ρ − 1) − βkB (Tr ρH − U) (6.95)
The choice of sign in front of β is conventional. Clearly we see that
∂X
∂λ =
0 ⇒ Tr ρ = 1 and = 0 ⇒ Tr ρH = U. Maximization with respect to ρ
leads to
∂ρnm
∂X
∂β
∂X
0 = = kBλδnm − kB(log ρ)mn − kBδnm − βkBHmn
(6.96)
where we have used the results derived in the previous section. The addition of
one term did not make the derivative much more complicated. This is valid in
general. Most constraints we want to add are linear, and give very easy terms
in the Lagrange equation!
The minimization leads to
and hence
kB(log ρ)mn = kB(λ − 1)δnm − βkBHmn
ρ = e λ−1 e −βH = Ce −βH
(6.97)
(6.98)
The normalization constant C (or λ) follows again from the requirement that
Tr ρ = 1. The constant β is obtained by demanding Tr ρH = U. This condition
can be compared with thermodynamical expressions via (6.97). If we multiply
(6.97) by ρnm and sum over n and m, we obtain
Tr ρ log ρ = (λ − 1)Tr ρ − βTr ρH (6.99)
and using the definition of the entropy and the constraints on the density matrix:
S = −kB(λ − 1) + kBβU (6.100)
The constant λ − 1 is equal to log C. Since Tr ρ = 1 we have C −1 = Tr e −βH .
1
The last expression is the partition function Z at a temperature . The
βkB