4.4 Interconnections Between Ensembles
203
V = −k B T
∂ ln
∂P
N,T
.
(4.3.19b)
The enthalpy H is given by H = T S + G, so that
H = k B T
2
∂ ln
∂T
N,P
.
(4.3.20)
Finally, we obtain a formal expression for the Helmholtz energy A in the
isothermal–isobaric ensemble via the relation A = G − P V , so that
A = G − P V = −k B T ln + k B T P
∂ ln
∂T
N,T
.
(4.3.21)
Note that a formal expression for the internal energy U for this ensemble may be
obtained from the relation U = H − P V as
U = k B T
2
∂ ln
∂T
N,P
+ P k B T
∂ ln
∂P
N,T
.
(4.3.22)
We thus see that the constant-volume canonical ensemble has simpler formal
expressions for U and A (corresponding to the ‘natural’ thermodynamic variables
for these two thermodynamic state functions), while the constant- pressure ensemble
has simpler expressions for H and G (corresponding to the ‘natural’ thermodynamic
variables for these two thermodynamic state functions).
4.4 Interconnections Between Ensembles
The symbol was chosen to designate the microcanonical partition function
specifically because the degeneracy factor i for discrete energy levels or, for
continuous energies, the density of (energy) states (E, V ) represents the number
of microstates accessible to a system having fixed energy E. For a continuous
distribution of energy states, we found that the canonical partition function z(β, V )
for a single structureless particle is related to the microcanonical partition function
(E, V ) via Eq. (3.2.20), namely,
z(β, V ) =
∞
0
V )e
−βE dE ,
(4.4.1)
from which we may conclude that the canonical partition function z(β, V ) is a
Laplace transform of the microcanonical partition function (E, V ). The microcanonical partition function for a system of N structureless particles is given [2]
by