144 Canonical ensemble
Ebecomes negligible. The implication of this result is that in the thermodynamic limit,
the canonical ensemble becomes equivalent to the microcanonical ensemble, where, in
the latter, the Hamiltonian is explicitly fixed. In the next two chapters, we will analyze
fluctuations associated with other ensembles, and we will see that the tendency of these
fluctuations to become negligible in the thermodynamic limit is a generalresult. The
consequence of this fact is that all ensembles become equivalent in the thermodynamic
limit. Thus, we are always at liberty to choose the ensemble that is most convenient
for a particular problem and still obtain the same macroscopic observables. It must
be stressed, however, that this freedom only exists in the thermodynamic limit. In
numerical simulations, for example, systems are finite, and fluctuations, which decay
slowly as 1/
√
N, might be large, depending on the system size chosen. Thus, the
choice of ensemble can influence the results of the calculation, and one should choose
the ensemble that best reflects the experimental conditions of the problem.
Now that we have the fundamental principles of the classical canonical ensemble
at hand, we proceed next to consider a few simple analytical solvableexamples of this
ensemble in order to demonstrate how it is used.
4.5 Simple examples in the canonical ensemble
4.5.1 The free particle and the ideal gas
Consider a free particle of massmmoving in a one-dimensional “box” of lengthL.
The Hamiltonian is simplyH=p^2 / 2 m. The partition function for an ensemble of such
systems at temperatureTis
Q(L,T) =
1
h∫L
0dx∫∞
−∞dpe−βp(^2) / 2 m
. (4.5.1)
The positionxcan be integrated trivially, yielding a factor ofL. The momentum
integral is an example of aGaussian integral, for which the general formula is
∫∞−∞dye−αy2
=√
π
α(4.5.2)
(see Section 3.8.3, where a method for performing Gaussian integrals is discussed).
Applying eqn. (4.5.2) to the partition function gives the final result
Q(L,T) =L
√
2 πm
βh^2. (4.5.3)
The quantity
√
βh^2 / 2 πmappearing in eqn. (4.5.3) can be easily seen to have units
of a length. For reasons that will become clear when we consider quantum statistical
mechanics in Chapter 10, this quantity, denotedλ, is often referred to as thethermal
wavelengthof the particle. Thus, the partition function is simply the ratio of thebox
lengthLto the thermal wavelength of the particle:
Q(L,T) =
L
λ