27.4 Classical Statistical Mechanics 1133
at a pressure of 1.000 bar. Compare your result with the
result of Problem 27.10.27.13a.Show that〈(E−〈E〉)^2 〉〈E^2 〉−〈E〉^2
b.Show thatCV(1/kBT)〈(E−〈E〉)^2 〉so thatCVis
always positive.
c.Show that asT→0,〈E^2 〉〈E〉^2 so thatCV→0as
T→0.27.4 Classical Statistical Mechanics
We now give a brief introduction to equilibrium statistical mechanics based on classical
mechanics. Classical mechanics is a good approximation for the translational motion of
atoms and molecules near room temperature, and appears to be a usable approximation
for the rotational motion of most molecules near room temperature. It works very poorly
for vibrations, and fails completely for electronic motion. However, we have seen that
in many systems the vibrational and electronic energies are numerically unimportant,
and classical statistical mechanics can be used with good results in these systems.
Our rather modest goal is to obtain some general formulas and to show how clas-
sical and quantum statistical mechanics are related to each other. We will make some
elementary comments on the application to nonideal gases and liquids. More detailed
discussions are found in the statistical mechanics textbooks listed at the end of this
book. Many research articles in classical statistical mechanics appear in journals such
asThe Journal of Chemical Physics,The Journal of Physical Chemistry,The Physical
Review, andPhysica.Phase Space
We first consider a system containing a single monatomic substance such as neon or
argon. Ignoring electronic motion, the classical state of an atom can be specified by
its location and velocity, given by three coordinates and three velocity components.
We can also use momentum components instead of velocity components. In Cartesian
coordinates thexcomponent of the momentum is denoted bypxand is equal tomvx.
Theyandzcomponents are similar.
The coordinates correspond to a point in ordinary 3-dimensional space, and the
momentum components correspond to the location of a point in a three-dimensional
momentum space (a mathematical space withpxplotted on one axis,pyon another axis,
andpzplotted on a third axis). These two 3-dimensional spaces combine to constitute
a 6-dimensional mathematical space and the state of the particle is specified by a single
point in this space. A mathematical space with axes that represent time-dependent
variables is called aphase space. In addition to the 6-dimensional phase space of a
single atom we define a 6N-dimensional phase space forNatoms, corresponding to
3 Ncoordinates and 3Nmomentum components.The Classical Canonical Ensemble
We assume that our real system is closed and maintained at constant temperature, and
represent the system by a canonical ensemble of replicas of the system, all in thermal
contact with their neighboring systems. We plot all of the phase points for the many
systems of the ensemble in a single 6N-dimensional phase space so that there is a
swarm of very many phase points in this phase space, one point for each system in the
ensemble. This swarm of points moves about in phase space in a way that is analogous