4 The canonical ensemble
4.1 Introduction: A different set of experimental conditions
The microcanonical ensemble is composed of a collection of systems isolated from any
surroundings. Each system in the ensemble is characterized by fixed values of the par-
ticle numberN, volumeV, and total energyE. Moreover, since all members of the
ensemble have the same underlying HamiltonianH(x), the phase space distribution
of the system is uniform over the constant energy hypersurfaceH(x) =Eand zero
off the hypersurface. Therefore, the entire ensemble can be generated by a dynami-
cal system evolving according to Hamilton’s equations of motion ̇qα=∂H/∂pαand
p ̇α=−∂H/∂qαunder the assumption that the dynamical system is ergodic, i.e., that
in an infinite time, it visits all points on the constant energy hypersurface. Under this
assumption, a molecular dynamics calculation can be used to generate a microcanon-
ical distribution.
The main disadvantage of the microcanonical ensemble is that conditions of con-
stant total energy are not those under which experiments are performed. It is, there-
fore, important to develop ensembles that have different sets of thermodynamic control
variables in order to reflect more common experimental setups. Thecanonical ensem-
bleis an example. Its thermodynamic control variables are particle numberN, volume
V, and temperatureT, which characterize a system in thermal contact with an infi-
nite heat source. Although experiments are more commonly performed at conditions of
constant pressureP, rather than constant volume, or they may fix the chemical poten-
tialμrather than constant particle number, the canonical ensemble nevertheless forms
the basis for theNPT(isothermal-isobaric) andμV T(grand canonical) ensembles,
which will be discussed in the subsequent two chapters. Moreover,for large systems,
the canonical distribution is often a good approximation to the isothermal-isobaric
and grand canonical distributions, and when this is true, results from the canonical
ensemble will not deviate much from results of the other ensembles.
In this chapter, we will formulate the basic thermodynamics of the canonical en-
semble. Recall that thermodynamics always divides the universe intoa system and
its surroundings. When a system is in thermal contact with an infiniteexternal heat
source, its energy will fluctuate in such a way that its temperatureremains fixed, lead-
ing to the conditions of the canonical ensemble. This thermodynamicparadigm will
be used in a microcanonical formulation of the universe (system + surroundings) to
derive the partition function and phase space distribution of the system under these
conditions. It will be shown that the HamiltonianH(x) of the system, which is not
conserved, obeys aBoltzmann distributionexp[−βH(x)]. Once we have laid out the
the underlying statistical mechanics, we will work through a numberof examples em-