Molecular dynamics 239πi=V^1 /^3 piπ ̇i=V^1 /^3 p ̇i+1
3
V−^2 /^3 V ̇pi. (5.8.5)Substituting eqns. (5.8.5) into eqns. (5.8.4) yields
r ̇i=pi
mi+
1
3
V ̇
V
rip ̇i=−∂U
∂ri−
1
3
V ̇
V
piV ̇=pV
Wp ̇V=1
3 V
∑
i[
p^2 i
mi−
∂U
∂ri·ri]
−P. (5.8.6)
Note that the right side of the equation of motion forpV is simply the difference
between the instantaneous pressure estimator of eqn. (4.6.57) or (4.6.58) and the ex-
ternal pressureP. Although eqns. (5.8.6) cannot be derived from a Hamiltonian, they
nevertheless possess the important conservation law
H′=
∑N
i=1p^2 i
2 mi+U(r 1 ,...,rN) +p^2 V
2 W+PV
=H(r,p) +p^2 V
2 W+PV, (5.8.7)
and they are incompressible. Here,His the physical Hamiltonian of the system. Eqns.
(5.8.6) therefore generate a partition function of the form
ΩP=
∫
dpV∫∞
0dV∫
dNp∫
D(V)dNrδ(
H(r,p) +p^2 V
2 W+PV−H
)
(5.8.8)
at a pressureP.^2 Eqn. (5.8.8) is not precisely equivalent to the true isoenthalpic-
isobaric partition function given in eqn. (5.3.3) because the conserved energy in eqn.
(5.8.7) differs from the true enthalpy byp^2 V/ 2 W. However, when the system is equipar-
titioned, then according to the classical virial theorem,〈p^2 V/W〉=kT, and forNvery
large, this constitutes only a small deviation from the true enthalpy. In fact, thiskT
is related to the extrakTappearing in the work-virial theorem of eqn. (5.4.8). If the
fluctuations inp^2 V/ 2 W are small, then the instantaneous enthalpyH(r,p) +PV is
confined to a thin shell betweenHandH+ ∆.
(^2) If∑
iFi = −
∑
i∂U/∂ri = 0, then an additional conservation law of the formK =
Pexp[(1/3) lnV] exists, and the equations will not generate eqn. (5.8.8). Note that the equations
of motion in scaled variables, eqns. (5.8.4), do not suffer from this pathology.