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