Basic thermodynamics 79entire phase space is an integration over the momentumpiand positionriof each
particle in the system and is, therefore, a 6N-dimensional integration. Moreover, while
the range of integration of each momentum variable is infinite, integration over each
position variable is restricted to that region of space defined by thecontaining vol-
ume. We denote this region asD(V), i.e., the spatial domain defined by the containing
volume. For example, if the container is a cube of side lengthL, lying in the positive
octant of Cartesian space with a corner at the origin, thenD(V) would be defined
byx∈[0,L],y∈[0,L],z∈[0,L] for each Cartesian vectorr= (x,y,z). Therefore,
Ω(N,V,E) is given by the integral
Ω(N,V,E) =M
∫
dp 1 ···∫
dpN∫
D(V)dr 1 ···∫
D(V)drNδ(H(r,p)−E), (3.2.14)whereMis an overall constant whose value we will discuss shortly. Eqn. (3.2.14) defines
the partition function of the microcanonical ensemble. For notational simplicity, we
often write eqn. (3.2.14) in a briefer notation as
Ω(N,V,E) =M
∫
dNp∫
D(V)dNrδ(H(r,p)−E), (3.2.15)or more simply as
Ω(N,V,E) =M∫
dxδ(H(x)−E), (3.2.16)using the phase space vector. However, it should be remembered that these shorter
versions refer to the explicit form of eqn. (3.2.14).
In order to understand eqn. (3.2.14) somewhat better and definethe normalization
constantM, let us consider determining Ω(N,V,E) in a somewhat different way. We
perform a thought experiment in which we “count” the number of microstates via a
“device” capable of determining a position component, sayx, to a precision ∆xand
a momentum componentpto a precision ∆p. Since quantum mechanics places an
actual limit on the product ∆x∆p, namely Planck’s constanth(this is Heisenberg’s
uncertainty relation to be discussed in Chapter 9),his a natural choice for our thought
experiment. Thus, we can imagine dividing phase space up into small hypercubes of
volume ∆x = (∆x)^3 N(∆p)^3 N =h^3 N, such that each hypercube contains a single
measurable microstate. Let us denote this phase space volume simply as ∆x. We will
also assume that we can only determine the energy of each microstate to be within
EandE+E 0 , whereE 0 defines a very thin energy shell above the constant-energy
hypersurface. For each phase space hypercube, if the energy of the corresponding
microstate lies within this shell, we increment our counting by 1, which we represent
identically as ∆x/h^3 N. We can therefore write Ω as
Ω(N,V,E) =
∑
hypercubes
E<H(x)<E+E 0∆x
h^3 N, (3.2.17)
where the summand ∆x/h^3 Nis added only if the phase space vector x in a given hy-
percube lies in the energy shell. Since the hypercube volume ∆x is certainly extremely