Statistical Mechanics - Physics at Oregon State University

Chapter 2
The canonical ensemble:
a practical way for
microscopic calculations.
2.1 Introduction.
Which state variables?
The state of a system depends on a number of parameters. In most measurements
it is a good idea to change only one of those parameters and to observe
the response of the system. Hence one needs a way to keep the other parameters
at fixed, prescribed values. The same is true in theory. So far we have only
considered the entropy S as a function of the internal energy U, the volume V,
and the number of particles N. If we specify U we can find T. Of course, this
is not what is happening in most measurements. It is easier to specify T and
to measure U (or rather the specific heat in calorimetry). The same is true in
our theoretical formalism. It is often easier to calculate all quantities at a given
temperature.
The formal development in the previous chapter was necessary to establish
that a thermodynamical and a statistical mechanical description of a large system
are the same. We argued that the entropy as defined in thermodynamics
and in statistical mechanics lead to the same equations describing the physics
of a system, but only if we work in the thermodynamic limit. What the differences
can be for small systems is not clear. The formal definition of entropy is,
however, in many cases not the best way to attempt to obtain a quantitative
description of a system. In this chapter we develop a more practical approach
for statistical mechanical calculations.
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