Statistical Mechanics - Physics at Oregon State University

144 CHAPTER 7. CLASSICAL STATISTICAL MECHANICS.
7.3 Ergodic theorem.
Connection with experiment.
Most experiments involve systems that involve many degrees of freedom. In
all those cases we are only interested in average quantities. All average quantities
follow from the quantum mechanical formula, and can be expressed as classical
averages if we change sums to integrals:
< A >=
1
N!h 3N
d Xρ( X)A( X) (7.21)
The factor N! is needed for indistinguishable particles, and should be omitted
for distinguishable particles.
The previous formula does, of course, not describe the way we measure
quantities! In a measurement we often consider time averaged quantities or
space averaged quantities. In the first case, we start the system at an initial
state
∂pi
∂H
Xin at t = 0 and use Hamilton’s equations ∂t = − and ∂xi
∂xi
∂t =
+ to calculate X(t). We then evaluate
∂H
∂pi
< A >τ (t) = 1
t+τ
A(
τ t
X(t ′ ))dt ′
(7.22)
where X(t) is the state of the system at time t. The value of this average
depends on the measurement time τ and on the initial state. Birkhoff showed
that in general limτ→∞ < A >τ (t) is independent of the initial conditions. So
if we take τ large enough it does not matter where we start. Note that in cases
where we have external potentials that vary with time we cannot make τ too
large, it has to be small compared to the time scale of the external potential. In
different words, the effects of external potentials that vary too rapidly are not
described by thermodynamic equilibrium processes anymore.
Equivalency of averages.
Now we ask the more important question. Under what conditions is this
measured quantity equal to the thermodynamical ensemble average? In the
quantum-mechanical case we argued that all accessible quantum states are
equally probable if the total energy U of the system is fixed. We needed to
invoke some random external small perturbation to allow the system to switch
quantum states. In the classical case we need a similar argument.
The time evolution of a classical system is determined by Hamilton’s equations:
∂pi
∂t
= −∂H
∂xi
, ∂xi
∂t
= ∂H
∂pi
(7.23)