10.5 Statistical Definition of Entropy
433
10.5.2 Thermodynamics from Statistics
The Helmholtz free energy, A ≡ E − T S, is simply related to the partition function
Z by
A = E − T S
= −Nk B T ln (Z/N ) − Nk B T
= −Nk B T ln(eZ/N) .
(10.24)
From Eq. (9.110), we know that the entropy, pressure, and chemical potentials in
such a system are determined by the Helmholtz free energy function according to
S = −
∂A
∂T
V ,N s
,
(10.25)
p = −
∂A
∂V
T ,N s
,
(10.26)
μ k =
∂A
∂N k
T ,V
.
(10.27)
It follows from Eq. (10.24) that all the thermodynamic variables can be expressed in
terms of the partition function. For examples,
E = −N
∂
∂β
ln (eZ/N) ,
(10.28)
c v =
∂E
∂T
V ,N s
= Nk B β
2 ∂ 2
∂β 2 ln (eZ/N)
(10.29)
κ T =
∂V
∂p
T ,N s
=
∂p
∂V
−1
=
Nk B T
∂ 2
∂V 2 ln (eZ/N)
−1
. (10.30)
where c v is the heat capacity at constant volume and κ T is the compressibility at
constant temperature. The ‘e’ in the above three equations is the base of the natural
logs.
10.5.3 Balance of Intensive Quantities
Suppose we isolate a pair of systems, allowing the exchange of heat, work, and
particles between them. We will assume that each of the systems contains a large