Statistical Mechanics - Physics at Oregon State University

7.3. ERGODIC THEOREM. 145
The trajectory X(t) is well-described. For an isolated system the energy is
conserved, and X(t) is confined on the surface in phase space given by H( X) =
U. In the micro-canonical ensemble ρ( X) = Cδ(H( X) − U) and all states with
H( X) = U are included in the average. In the time average, only a small subset
of these states is sampled. The requirement that both averages are the same
leads to the requirement that in the time τ, which has to be short compared
with the time scale of changes in < A > (t), the trajectory X(t) in phase
space samples all parts of the accessible phase space equally well. If this is
true, and the time-average is equal to the ensemble average, the system is called
ergodic. Most systems with many degrees of freedom seem to obey this ergodic
hypothesis.
If a system is ergodic the orbit X(t) should approach any point in the accessible
region in phase space arbitrarily close. Originally it was thought that this
statement could be related to Poincaré’s theorem. This theorem states that a
system having a finite energy and a finite volume will, after a sufficiently long
time, return to an arbitrary small neighborhood of almost any given initial state.
The time involved, however, is prohibitively long. It is on the order of e N , where
N is the number of degrees of freedom.
Chaos.
A more useful approach combines the classical trajectories with small external
perturbations, like in the quantum-mechanical treatment. An orbit X(t) is
called chaotic if an arbitrary small change in the initial conditions in general
leads to a very large change in the state of the system at a later time t. Classical
statistical mechanics therefore requires that most of the orbits of the system are
chaotic. The time τ still has to be large in order to obtain a good sampling of
phase space. For meta-stable systems like glass the time τ can be longer than a
human life-span. This means that the ergodic theorem in itself is not sufficient
for the time-average and ensemble-average to be the same in a real experiment.
If we want to measure properties of a system, we have to connect the system
to some outside instrument. If we want to specify any intensive state variable
of a system, we have to connect it to a reservoir. In all these cases there will
be small perturbations on the system and the description given in the previous
paragraph is valid. What happens in a truly isolated system though? The
answer to this question is impossible to derive from experiments, since any
experiment involves a perturbation. Hence questions pertaining to a statistical
mechanical description of a truly isolated system in terms of ensembles are
purely academic. Nevertheless scientists have tried to answer this question and
two opposing points of view have been defended. Landau claims that it is not
possible to give such a description and that you need outside perturbations,
while Khinchin tells us that it is possible to give such a description.