4.2 Grand Ensemble: Open Systems
187
= −
1
β
N,r N
ln p N,r N dp N,r N −
γ
β
N,r N
N dp N,r N ,
(4.2.13)
as the ln term vanishes because
N,r N
dp N,r N = 0, while λ and γ are related by
λ = e −γ .
Now, since N is always integer, and the average number of particles N is given
by
N ≡
N
p N N ,
(4.2.14)
any infinitesimal change in N must come as a result of changes in the values of p N ,
so that 2
dN =
N
N dp N .
(4.2.15)
As a consequence of this result, Eq. (4.2.13) becomes
N,r N
r N dp N,r N = −
1
β
d
⎛
⎝
N,r N
p N,r N ln p N,r N
⎞
⎠ −
γ
β
dN .
(4.2.16)
From Eqs. (4.2.16) and (4.2.11), we see that
dU
stats
=
−
1
β
d
⎛
⎝
N,r N
p N,r N ln p N,r N
⎞
⎠ + δW −
γ
β
dN
(4.2.17)
thermo
=
δQ rev + δW + μ dN ,
(4.2.18)
where we have now associated the thermodynamic variable N and its differential dN
with the statistical mechanical quantities N, dN, respectively. From the comparison
between Eqs. (4.2.17) and (4.2.18), we obtain
δQ rev = T dS = −
1
β
d
⎛
⎝
N,r N
p N,r N ln p N,r N
⎞
⎠ = −k B T d
⎛
⎝
N,r N
p N,r N ln p N,r N
⎞
⎠ ,
or
2 Note also that even were N not an integer, its value is fixed for the individual canonical ensembles
that make up the grand ensemble.