Page 25 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

Foundations of equilibrium statistical mechanics 11

conditions,theonlywayinwhichtheseaveragescandependontheinitialconditions
is through the values of the energy, linear momentum and angular momentum of the
entire system. The general study of the dependence of averages of the form (1.16)
on the initial conditions is part of ergodic theory. An ergodic system is (loosely
from a mathematical point of view) defined as an energetically isolated system for
which the phase point eventually passes through every point on the surface in phase
¯
space consistent with its energy. It is not hard to prove that the averages φ t in such
an ergodic system depend only on the energy of the system. It is worth pointing out
that the existence of ergodic systems in phase space of more than two dimensions
is quite surprising. The trajectory of the system in phase space is a topologically
one dimensional object (a path, parametrized by one variable, the time) yet we
want this trajectory to fill the 6N − 1 dimensional surface defined by the energy.
The possibility of space filling curves is known mathematically (for a semipopular
account see reference 1). However, for a large system, the requirement is extreme:
23
thetrajectorymustfillanenormouslyopenspaceoftheorderof10 dimensions!By
contrast the path of a random walk has dimension 2 (in any embedding dimension)!
(Very briefly, the (fractal or Hausdorff–Besicovitch) dimension of a random walk
can be understood to be 2 as follows. The dimension of an object in this sense
is determined as D H defined so that when one covers the object in question with
spheres of radius η a minimum of N(η) spheres is required and

L H = lim N(η)η D H
η→0

2
is finite and nonzero. For a random walk of mean square radius R
, N(η) =
2
2
R
/η and D H = 2. See reference 1 for details.) Nevertheless something like
ergodicity is required for statistical mechanics to work, and so the paths in phase
space of large systems must in fact achieve this enormous convolution in order to
account for the known facts from experiment and simulation. It is not true that every
system consisting of small numbers of particles is ergodic. Some of the problems at
the end of this section illustrate this point. For example, a one dimensional harmonic
oscillator is ergodic, but a billiard ball on a two dimensional table is not (Figure 1.1).
On the other hand, in the latter case, the set of initial conditions for which it is not
ergodic is in some sense “small.” Another instructive example is a two dimensional
harmonic oscillator (Problem 1.1).
There are several rationally equivalent ways of talking about equation (1.10).
These occur in textbooks and other discussions and reflect the history of the subject
as well as useful approaches to its extension to nonequilibrium systems. What we
have discussed so far may be termed the Boltzmann interpretation of ρ (in which ρ
is related to the time which the system phase point spends in each region of phase
space). This is closely related to the probability interpretation of ρ because the