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4 Mean Values and Thermodynamics
dS = −k B d
⎛
⎝
N,r N
p N,r N ln p N,r N
⎞
⎠ ⇒ S = −k B
N,r N
p N,r N ln p N,r N .
(4.2.19)
Notice that this statistical expression for the entropy has the same form as that
obtained earlier for the canonical ensemble. In fact, this remains the case for every
ensemble, including those describing nonequilibrium situations.
We also see from Eqs. (4.2.17) and (4.2.18) that the thermodynamic chemical
potential μ is related to γ by
μ = −
γ
β
= −k B T γ
or
γ = −βμ ≡ −μ(k B T )
−1 ,
(4.2.20)
so that the absolute activity λ becomes 3
λ = e
−γ
= e
βμ .
(4.2.21)
Now, from expression (4.2.19) for the entropy, together with Eq. (4.2.12), we find
that
S = −k B
N,r N
p N,r N [−ββ r N − γ N − ln ]
=
U
T
+ k B γ N + k B ln ,
or
T S = U − μN + k B T ln .
(4.2.22)
Thus, we may express k B T ln as
k B T ln = T S − U + μN = μN − A .
(4.2.23)
If we now consider the Gibbs energy G, defined in thermodynamics by
G = A + P V ,
(4.2.24)
then its differential is
dG = dA + P dV + V dP ,
3 It is a common practice in quantum mechanical treatments to define a dimensionless parameter
α ≡ βμ = −γ so that the absolute activity is written as λ = e α .