Thermodynamics 219for molecular dynamics calculations in these ensembles. We will show how the volume
and density distributions can be generated by treating the volume as an additional
dynamical variable with a corresponding momentum, the latter serving as a barostatic
control of the fluctuations in the internal pressure. This idea will be extended to
anisotropic volume shape-changes by treating the cell vectors asdynamical variables.
5.2 Thermodynamics of isobaric ensembles
We begin by considering the isoenthalpic-isobaric ensemble, which derives from a Leg-
endre transformation performed on the microcanonical ensemble. In the microcanon-
ical ensemble, the energyEis constant and is expressed as a function of the number
of particlesN, the volumeV, and the entropyS:E=E(N,V,S). Since we seek to
use an external applied pressureP as the control variable in place of the volumeV,
it is necessary to perform a Legendre transform ofEwith respect to the volumeV.
Denoting the new energy asE ̃, we find
E ̃(N,P,S) =E(N,V(P),S)−∂E
∂V
V(P). (5.2.1)
However, sinceP=−∂E/∂V, the new energy is justE ̃=E+PV, which we recognize
as the enthalpyH:
H(N,P,S) =E(N,V(P),S) +PV(P). (5.2.2)The enthalpy is naturally a function ofN,P, andS. Thus, for a process in which
these variables change by small amounts, dN, dP, and dS, respectively, the change in
the enthalpy is
dH=(
∂H
∂N
)
P,SdN+(
∂H
∂P
)
N,SdP+(
∂H
∂S
)
N,PdS. (5.2.3)SinceH=E+PV, it also follows that
dH= dE+PdV+VdP=TdS−PdV+μdN+PdV+VdP=TdS+VdP+μdN, (5.2.4)where the second line follows from the first law of thermodynamics. Comparing eqns.
(5.2.3) and (5.2.4) leads to the thermodynamic relations
μ=(
∂H
∂N
)
P,S, 〈V〉=
(
∂H
∂P
)
N,S, T=
(
∂H
∂S
)
N,P. (5.2.5)
The notation〈V〉for the volume appearing in eqn. (5.2.5) serves to remind us that
the observable volume results from a sampling of instantaneous volume fluctuations.