80 Microcanonical ensemble
small compared to all of phase space, eqn. (3.2.17) can be very wellapproximated by
an integral
Ω(N,V,E) =1
h^3 N∫
E<H(x)<E+E 0dx. (3.2.18)Finally, since eqn. (3.2.18) is measuring the volume of a very thin 6N-dimensional
shell in phase space, we can approximate this volume by the (6N−1)-dimensional
area of the surface defined byH(x) =Etimes the thicknessE 0 of the shell, leading to
Ω(N,V,E) =
E 0
h^3 N∫
dxδ(H(x)−E), (3.2.19)which is eqn. (3.2.16) withM=E 0 /h^3 N. In principle, eqn. (3.2.19) should be sufficient
to define the microcanonical partition function. However, remember we assumed at
the outset that all particles are identical, so that exchanging any two particles does
not yield a uniquely different microstate. Unfortunately, classical mechanics is not
equipped to handle this situation, as all classical particles carry an imaginary “tag”
that allows them to be distinguished from other particles. Thus, in order to avoid
overcounting, we need to include a factor of 1/N! inMfor the number of possible
particle exchanges that can be performed. This factor can only beproperly derived
using the laws of quantum mechanics, which we will discuss in Chapter 9. Adding the
1 /N! in by hand yields the normalization factor
M≡MN=
E 0
N!h^3 N. (3.2.20)
Note that the constantE 0 is irrelevant and will not affect any thermodynamic or
equilibrium properties. However, this normalization constant renders Ω(N,V,E) di-
mensionless according to eqn. (3.2.16).^2
Since Ω(N,V,E) counts the number of microscopic states available to a system with
given values ofN,V, andE, the thermodynamics of the microcanonical ensemble can
now be expressed directly in terms of the partition function via Boltzmann’s relation:
1
kT=
(
∂ln Ω
∂E)
N,V,
P
kT=
(
∂ln Ω
∂V)
N,E,
μ
kT=−
(
∂ln Ω
∂N)
V,E. (3.2.21)
Moreover, an observableAis obtained from the ensemble average of a phase space
functiona(x) according to
A=〈a〉=MN
Ω(N,V,E)
∫
dxa(x)δ(H(x)−E) =∫
dxa(x)δ(H(x)−E)
∫
dxδ(H(x)−E). (3.2.22)
(^2) Of course, in a multicomponent system, if the system containsNAparticles of species A,NB
particles of species B,...., andNtotal particles, then the normalization factor becomesM{N}, where
M{N}=
E 0
h^3 N[NA!NB!···]
.
Throughout the book, we will not complicate the expressionswith these general normalization factors
and simply use the one-component system factors. However, the reader should always keep in mind
when the factorsMN,CN(canonical ensemble), andIN(isothermal-isobaric ensemble) need to be
generalized.