Virial theorems 227=P
1
∆
∫∞
0dVe−βPVQ(N,V,T) =P. (5.4.3)The boundary term in the first line of eqn. (5.4.3) vanishes at both endpoints: At
V = 0, the configurational integrals inQ(N,V,T) over a box of zero volume must
vanish, and atV =∞, the exponential exp(−βPV) decays faster thanQ(N,V,T)
increases withV.^1 Recognizing that the integral in the last line of eqn. (5.4.3) is just
the partition function ∆(N,P,T), it follows that
〈P(int)〉=P. (5.4.4)Eqn. (5.4.4) expresses the expected result that the volume-averaged internal pressure
is equal to the external pressure. This result is known as thepressure virial theorem.
Any computational approach that seeks to generate the isothermal-isobaric ensemble
must obey this theorem.
We next consider the average of the pressure–volume product〈P(int)V〉. At a fixed
volumeV, the productP(int)Vis given in terms of the canonical partition function by
P(int)V=kTV
∂lnQ
∂V=
kTV
Q∂Q
∂V
. (5.4.5)
Averaging eqn. (5.4.5) over an isothermal-isobaric ensemble yields
〈P(int)V〉=1
∆
∫∞
0dVe−βPVkTV∂
∂V
Q(N,V,T). (5.4.6)
As was done for eqn. (5.4.3), we integrate eqn. (5.4.6) by parts, which gives
〈P(int)V〉=1
∆
[
e−βPVkTV Q(N,V,T)]
∣
∣
∣
∣
∞0−
1
∆
∫∞
0dV kT[
∂
∂V
Ve−βPV]
Q(N,V,T)
=
1
∆
[
−kT∫∞
0dVe−βPVQ(V) +P∫∞
0dVe−βPVV Q(V)]
=−kT+P〈V〉, (5.4.7)or
〈P(int)V〉+kT=P〈V〉. (5.4.8)
Eqn. (5.4.8) is known as thework virial theorem. Note the presence of the extrakT
term on the left side. SinceP〈V〉and〈P(int)V〉are both extensive quantities and hence
proportional toN, the extrakT term can be neglected in the thermodynamic limit,
and eqn. (5.4.8) becomes〈P(int)V〉 ≈P〈V〉. Nevertheless, eqn. (5.4.8) is rigorously
correct, and it is interesting to consider the origin of the extrakTterm, since it will
arise again in Section 5.9, where we discuss molecular dynamics algorithms for the
isothermal-isobaric ensemble.
(^1) Recall that asV→∞,Q(N,V,T) approaches the ideal-gas and grows asVN.