Classical virial theorem 83〈
xi∂H
∂xj〉
=
MNδij
Ω(N,V,E)∫
H(x)<Edx=
δij
Ω(N,V,E)MN
∫
dxθ(E−H(x)).=
E 0 δij
Ω(N,V,E)CN
∫
dxθ(E−H(x)), (3.3.8)whereCN = 1/(N!h^3 N). The phase space integral appearing in eqn. (3.3.8) is the
partition function of an ensemble that closely resembles the microcanonical ensemble,
known as theuniform ensemble:
Σ(N,V,E) =CN∫
dxθ(E−H(x)). (3.3.9)The partition function of this ensemble is related to the microcanonical partition
function by
Ω(N,V,E) =E 0∂Σ(N,V,E)
∂E
. (3.3.10)
As noted previously, the uniform ensemble requires that the phasespace integral be
performed over a phase space volume for whichH(x)< E, which is the volume enclosed
by the constant-energy hypersurface. While the dimensionality ofthe constant energy
hypersurface is 6N−1, the volume enclosed has dimension 6N. However, in the ther-
modynamic limit, whereN→ ∞, the difference between the number of microstates
associated with the uniform and microcanonical ensembles becomesvanishingly small
since 6N≈ 6 N−1 forNvery large. Thus, the entropyS ̃(N,V,E) =kln Σ(N,V,E)
derived from the uniform ensemble and that derived from the microcanonical ensemble
S(N,V,E) =kln Ω(N,V,E) become very nearly equal as the thermodynamic limit is
approached. Substituting eqn. (3.3.10) into eqn. (3.3.8) gives
〈
xi
∂H
∂xj〉
=δijΣ(E)
∂Σ(N,V,E)/∂E
=δij(
∂ln Σ(E)
∂E)− 1
=kδij(
∂S ̃
∂E
)− 1
≈kδij(
∂S
∂E
)− 1
=kTδij, (3.3.11)which proves the theorem. The virial theorem allows for the construction of microscopic
phase space functions whose ensemble averages yield macroscopicthermodynamic
observables.