1549380323-Statistical Mechanics Theory and Molecular Simulation

140 Canonical ensemble

f(x)∝e−H(x)/kT. (4.3.12)

The overall normalization of eqn. (4.3.12) must be proportional to

∫

dx e−H(x)/kT.

As was the case for the microcanonical ensemble, the integral is accompanied by an
N-dependent factor that accounts for the identical nature of the particles and yields
an overall dimensionless quantity. This factor is denotedCNand is given by

CN=

1

N!h^3 N

, (4.3.13)

so that the phase space distribution function becomes

f(x) =

CNe−βH(x)

Q(N,V,T)

. (4.3.14)

(As we noted in Section 3.2, in a multicomponent system withNAparticles of type
A,NBparticles of type B,..., andNtotal particles,CNwould be replaced byC{N}=
1 /[h^3 N(NA!NB!···)].) The parameterβ= 1/kThas been introduced, and the denom-
inator in eqn. (4.3.14) is

Q(N,V,T) =CN

∫

dx e−βH(x). (4.3.15)

The quantityQ(N,V,T) (or, equivalently,Q(N,V,β)) is the partition function of the
canonical ensemble, and, as with the microcanonical ensemble, it is ameasure of
the total number of accessible microscopic states. In contrast to the microcanonical
ensemble, however, the Hamiltonian is not conserved. Rather, it obeys the Boltzmann
distribution as a consequence of the fact that the system can exchange energy with
its surroundings. This energy exchange changes the number of accessible microscopic
states. Note that the canonical partition functionQ(N,V,T) can be directly related
to the microcanonical partition function Ω(N,V,E) as follows:

Q(N,V,T) =

1

E 0

∫∞

0

dEe−βEMN

∫

dxδ(H(x)−E)

=

1

E 0

∫∞

0

dEe−βEΩ(N,V,E). (4.3.16)

In the first line, if the integration over energyEis performed first, then theδ-function
allowsEto be replaced by the HamiltonianH(x) in the exponential, leading to eqn.
(4.3.15). The second line shows that the canonical partition function is simply the
Laplace transform of the microcanonical partition function. Readers unfamiliar with
Laplace transforms are referred to Appendix D.