Statistical Mechanics - Physics at Oregon State University

1.4. AVERAGES. 11
A different way of stating this basic assumption is that if we wait long
enough, a system will go through all accessible quantum states. This seems to
be in contradiction with ordinary quantum mechanics, which specifies that once
a system is prepared in a given quantum state, it will remain in that quantum
state unless there are external perturbations (i.e. a measurement). In statistical
mechanics we therefore assume that our system is always in contact with a large
outside world. This makes sense, since it is hard to shield a sample from all
external influences. For example, try to counteract gravity! The atoms in a gas
do collide with the walls of the container of that gas, and since the atoms in
this wall move in a random way (even at zero temperature due to zero point
motion), the wall adds a small random component to the motion of the atoms
in the gas.
This scenario also requires that the system we are considering is chaotic. In
a chaotic system a small difference in initial conditions growth exponentially
fast, and the random, small perturbations by the walls of the container are very
efficient in causing the system to cycle through all accessible states. An interesting
question is if all systems are chaotic when they are very large (remember, we
need to take the thermodynamic limit!). In addition, we do not know for sure
what chaos means in quantum mechanics. In practice, however, all macroscopic
systems seem to obey the laws of thermodynamics.
You run into a number of paradoxes when you want to calculate the entropy
of the universe in the standard thermodynamical way. The discussion above
shows you, however, that the entropy of the universe is not a conceptually
sound quantity. There is no outside reservoir which can cause the universe to
cycle through all possible states, and one can argue that the universe is in a
given quantum state for all eternity.
Take your time!
The argument made in the previous paragraphs allows for a statistical mechanical
description of the solar system. It is an almost closed system, but
there are weak interactions with the external world. Such a description is, of
course, not very useful. The keywords here are long enough. A system can be
trapped in a metastable state. Glass is a standard example of such a metastable
state and in order to give a meaningful statistical mechanical description of the
behavior of glass we have to exclude all crystalline states. If the duration of
the measurement would be millions of years we have, of course, to include all
those states. A solar system is actually a chaotic system and if we wait long
enough all kinds of weird things can happen. A planet could leave and go off
to infinity and beyond. Fortunately, the time scale of these phenomena is much
larger than the lifetime of a typical star. Therefore, we do not worry about this
happening. Should we?
Truly isolated?