1549380323-Statistical Mechanics Theory and Molecular Simulation

220 Isobaric ensembles

Eqns. (5.2.5) constitute the basic thermodynamic relations in the isoenthalpic-isobaric
ensemble. The reason that enthalpy is designated as a control variable rather than the
entropy is the same as for the microcanonical ensemble: It is not possible to “dial
up” a desired entropy, whereas, in principle, the enthalpy can be set by the external
conditions, even if it is never done in practice (except in computer simulations).
The isothermal-isobaric ensemble results from performing the sameLegendre trans-
form on the canonical ensemble. The volume in the Helmholtz free energyA(N,V,T) is
transformed into the external pressurePyielding a new free energy denotedG(N,P,T):

G(N,P,T) =A(N,V(P),T)−V(P)

∂A

∂V

. (5.2.6)

Using the fact thatP=−∂A/∂V, we obtain

G(N,P,T) =A(N,V(P),T) +PV(P). (5.2.7)

The functionG(N,P,T) is known as theGibbs free energy. SinceGis a function of
N,P, andT, a small change in each of these control variables yields a change inG
given by

dG=

(

∂G

∂N

)

P,T

dN+

(

∂G

∂P

)

N,T

dP+

(

∂G

∂T

)

N,P

dT. (5.2.8)

However, sinceG=A+PV, the differential changedGcan also be expressed as

dG= dA+PdV+VdP

=−PdV+μdN−SdT+PdV+VdP

=μdN+VdP−SdT, (5.2.9)

where the second line follows from eqn. (4.2.5). Thus, equating eqn.(5.2.9) with eqn.
(5.2.8), the thermodynamic relations of the isothermal-isobaric ensemble follow:

μ=

(

∂G

∂N

)

P,T

, 〈V〉=

(

∂G

∂P

)

N,T

, S=−

(

∂G

∂T

)

N,P

. (5.2.10)

As before, the volume in eqn. (5.2.10) must be regarded as an average over instanta-
neous volume fluctuations.

5.3 Isobaric phase space distributions and partition functions

The relationship between the isoenthalpic-isobaric and isothermal-isobaric ensembles is
similar to that between the microcanonical and canonical ensembles. In the isoenthalpic-
isobaric ensemble, theinstantaneousenthalpy is given byH(x) +PV, whereVis the
instantaneous volume andH(x) is the Hamiltonian. Note thatH(x) +PVis strictly
conserved under isoenthalpic conditions. Thus, the ensemble is defined by a collection
of systems evolving according to Hamilton’s equations in a containing volume; in turn,