78 Microcanonical ensemble
That logarithmic dependence of the entropy on Ω(N,V,E) will be explained shortly.
Assuming we can determine Ω(N,V,E) from a microscopic description of the system,
eqn. (3.2.11) then provides a connection between this microscopic description and
macroscopic thermodynamic observables.
In the last chapter, we saw that the most general solution to the equilibrium Li-
ouville equation,{f(x),H(x)}= 0, for the ensemble distribution functionf(x) is any
function of the Hamiltonian:f(x) =F(H(x)), where x is the phase space vector.
The specific choice ofF(H(x)) is determined by the conditions of the ensemble. The
microcanonical ensemble pertains to a collection of systems in isolation obeying Hamil-
ton’s equations of motion. Recall, however, from Section 1.6, that asystem obeying
Hamilton’s equations conserves the total Hamiltonian
H(x) =E (3.2.12)withEbeing the total energy of the system. Conservation ofH(x) was demonstrated
explicitly in eqn. (1.6.15). Moreover, the ensemble distribution functionf(x) is static
in the sense that∂f/∂t= 0. Therefore, each member of an equilibrium ensemble is in
a single unique microscopic state. For the microcanonical ensemble,each unique state
is described by a unique phase space vector x that satisfies eqn. (3.2.12). It follows that
the choice ofF(H(x)) must be consistent with eqn. (3.2.12). That is,F(H(x)) must
restrict x to those microscopic states for whichH(x) =E. A function that achieves
this is the Diracδ-function
F(H(x)) =Nδ(H(x)−E) (3.2.13)expressing the conservation of energy condition. Here,Nis an overall normalization
constant. For readers not familiar with the properties of the Diracδ-function, a detailed
discussion is provided in Appendix A. Since eqn. (3.2.12) defines the constant-energy
hypersurface in phase space, eqn. (3.2.13) expresses the fact that, in the microcanon-
ical ensemble, all phase space points must lie on this hypersurface and that all such
points are equally probable; all points not on this surface have zeroprobability. The
notion that Ω(N,V,E) can be computed from an ensemble in which all accessible
microscopic states are equally probable is an assumption that is consistent with clas-
sical mechanics, as the preceding discussion makes clear. More generally, we assume
that for an isolated system in equilibrium, all accessible microscopic states are equally
probable, which is known as theassumption of equal a priori probability. The quantity
1 /Ω(N,V,E) is a measure of the probability of randomly selecting a microstate in any
small neighborhood of phase space anywhere on the constant-energy hypersurface.
The number Ω(N,V,E) is a measure of the amount of phase space available to
the system. It must, therefore, be proportional to the fraction of phase space consis-
tent with eqn. (3.2.12), which is proportional to the (6N−1)-dimensional “area” of
the constant-energy hypersurface. This number can be obtained by integrating eqn.
(3.2.13) over the phase space, as indicated by eqn. (2.6.5).^1 An integration over the
(^1) If we imagined discretizing the constant-energy hypersurface such that each discrete patch con-
tained a single microstate, then the integral would revert to a sum that would represent a literal
counting of the number of microstates contained on the surface.