Page 23 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

Foundations of equilibrium statistical mechanics 9

in time and what is measured is a time average:
1 t+τ/2
¯
φ t = φ(q i (t ), p i (t )) dt (1.8)
τ t−τ/2
τ is an averaging time determined by the apparatus and the measurement made
(or chosen for analysis by the simulator). Experience has shown that for many
¯
systems, an experimental situation can be achieved in which measurements of φ t
are independent of τ for all τ> τ 0 for some finite τ 0 . It is easy to show that, in
¯
such a case, φ t is also independent of t. If this is observed to be the case for the
macroscopic observables of interest, then the system is said to be in equilibrium. A
similar operational definition of equilibrium is applied to simulations. In practice
it is never possible to test this equilibrium condition for arbitrarily long times, in
eitherexperimentorsimulation.Thusexceptintherarecasesinwhichmathematical
proofs exist for relatively simple models, the existence and nature of equilibrium
states are hypothesized on the basis of partial empirical evidence. Furthermore, in
experimental situations, we do not expect any system to satisfy the equilibrium
condition for arbitrarily long times, because interactions with the surroundings
will inevitably change the values of macroscopic variables eventually. Making the
system considered ever larger and the time scales longer and longer does not help
here, because there is no empirical evidence that the universe itself is in equilibrium
in this sense. Nevertheless, the concept of equilibrium turns out to be an extremely
useful idealization because of the strong evidence that many systems do satisfy
the relevant conditions over a very wide range of averaging times τ and that,
under sufficiently isolated conditions, many systems spontaneously evolve rapidly
toward an approximately equilibrium state whose characteristics are not sensitive
to the details of the initial microscopic conditions. These empirical statements
lack mathematical proofs for most systems of experimental or engineering interest,
though mathematicians have made progress in proving them for simple models.
For systems in equilibrium defined in this way we are concerned with the calcu-
lation of averages of the type
1 τ
¯

φ t = lim φ({q i (t )}, {p i (t )})dt (1.9)
τ→∞ τ
0
We will show that it is always possible in principle to write this average in the form

¯ 3N 3N
φ t = ρ({q i }, {p i })φ({q i }, {p i })d q d p (1.10)
inwhichρ({q i }, {p i })iscalledtheclassicaldistributionfunction.Thedemonstration
provides useful insight into the meaning of ρ({q i }, {p i }). We consider the 6N dimen-
sional space of the variables {q i }, {p i }, called phase space. In this space the time