Classical virial theorem 81The form of the microcanonical partition function shows that the phase space vari-
ables are not all independent in this ensemble. In particular, the conditionH(x) =E
specifies a condition or constraint placed on the total number of degrees of freedom.
AnN-particle system in the microcanonical ensemble therefore has 6N−1 indepen-
dent phase-space degrees of freedom (or 2dN−1 inddimensions). In the limit where
N→∞andV→∞such thatN/V= const, a limit referred to as thethermodynamic
limit, we may approximate, 6N− 1 ≈ 6 Nso that the system behaves approximately
as if it had 6Nphase-space degrees of freedom.
Eqn. (3.2.22) now allows us to understand Boltzmann’s relationS(N,V,E) =
kln Ω(N,V,E) between the entropy and the partition function. From eqn. (3.2.10),
it is clear that quantities such as 1/T,P/T, andμ/T, which are themselves macro-
scopic observables, must be expressible as ensemble averages of phase space functions,
which we can denote asaT(x),aP(x) andaμ(x). Consider, for example, 1/T, which
can be expressed as
1
T=
MN
Ω(N,V,E)
∫
dxaT(x)δ(H(x)−E) =(
∂S
∂E
)
N,V. (3.2.23)
We seek to relate these two expressions for 1/Tby postulating thatS(N,V,E) =
CG(Ω(N,V,E)), whereGis an arbitrary function andCis an arbitrary constant, so
that (
∂S
∂E
)
N,V=CG′(Ω(N,V,E))
(
∂Ω
∂E
)
N,V. (3.2.24)
Now,
(
∂Ω
∂E
)
N,V=MN
∫
dx
∂δ(H(x)−E)
∂E=MN
∫
dxδ(H(x)−E)
∂lnδ(H(x)−E)
∂E. (3.2.25)
Thus,
(
∂S
∂E
)
N,V=CG′(Ω(N,V,E))MN
∫
dxδ(H(x)−E)
∂lnδ(H(x)−E)
∂E. (3.2.26)
Eqn. (3.2.26) is in the form of a phase space average as in eqn. (3.2.23). If we iden-
tifyaT(x) = (1/k)∂[lnδ(H(x)−E)]/∂E, wherekis an arbitrary constant, then it is
clear thatG′(Ω(N,V,E)) =k/Ω(N,V,E), which is only satisfied ifG(Ω(N,V,E)) =
kln Ω(N,V,E), with the arbitrary constant identified as Boltzmann’s constant.In the
next few sections, we shall see how to use the microcanonical and related ensembles to
derive the thermodynamics for several example problems and to prove an important
theorem known as thevirial theorem.
3.3 The classical virial theorem
In this section, we present a microcanonical ensemble proof of theclassical virial theo-
rem. Consider a system with HamiltonianH(x). Letxiandxjbe specific components
of the phase space vector x. The classical virial theorem states that