190
4 Mean Values and Thermodynamics
S =
∂(P V )
∂T
μ,V
= k B ln + k B T
∂ ln
∂T
μ,V
,
(4.2.30b)
P =
∂(P V )
∂V
T ,μ
= k B T
∂ ln
∂V
T ,μ
,
(4.2.30c)
U = −
∂ ln
∂β
λ,V
= k B T
2
∂ ln
∂T
λ,V
.
(4.2.30d)
Some texts choose to define a ‘grand potential’ , V , μ) via
, V , μ) ≡ −k B T ln (T , V , μ) ,
(4.2.31)
so that P V = −, and N, S, and P are obtained as
N = −
∂∂
∂μ
T ,V
,
S = −
∂∂
∂T
V ,μ
,
P = −
∂∂
∂V
T ,μ
.
(4.2.32)
This grand potential thus plays the same role for the grand ensemble that the
Helmholtz energy A plays for the canonical ensemble. The internal energy, U , is
then given in terms of by
U = − T
∂∂
∂T
V ,μ
.
(4.2.33)
Finally, from the association N ↔ N =
N,r N
p N,r N N, we have
N =
N,r N
Ne
−ββ r N λ N
=
λ
N,r N
Ne
−ββ r N λ N −1
.
This expression relates N to via
N =
1
λ
∂
∂λ
⎛
⎝
N,r N
e
−ββ r N λ
N
⎞
⎠ =
λ
∂∂
∂λ
T ,V
,
thereby allowing us to associate the thermodynamic N (which is equivalent to our
ensemble average number N) to the grand partition function through the relation
N thermo = λ
∂ ln
∂λ
V ,T
.
(4.2.34)