Lecture #2a-- Statistical mechanics and irreversibility
1) Irreversibility
That the universe exhibits irreversible phenomena on large scales is
surprising at first glance, when one considers that the elementary
processes governing individual particles are typically time-reversible.
So what is the source of irreversibility? It comes from the fact that
large systems have so vastly many different ways of being actualized,
different possible outcomes, that we necessarily have to lump together
vast subsets of those outcomes and label them "effectively indistinguishable".
Then we can make progress simply by counting the number of different
potential outcomes that are lumped together in each of these subsets.
If each of those specific outcomes is equally likely (a common assumption
in statistical mechanics and thermodynamics), then the overall likelihood
of the entire subset of possible outcomes is simply proportional to the
number of outcomes included in that subset.
Irreversibility is then not a time evolution from one particular state to
another (which we generally cannot control well enough to test the
reversibility that the equations predict), it is a time evolution from
one subset of outcomes (or "states") to another, which we term irreversible
if the latter subset is so much more vastly numerous in the outcomes it
includes, that it would simply be vastly unlikely for an outcome in that
subset to ever evolve into an outcome from the former subset.
Hence statistical physics is all about what is more likely by virtue of
being allowed to happen in more ways.
It is only in systems that are so large and have such complexity
that some types of evolution are spectacularly unlikely
to proceed backwards, that we may
call those processes "irreversible".
2) Equilibrium
Once we decide what kinds of distinguishing aspects of a system we are going
to recognize as "relevant", we can define gross system properties like
"temperature" and "density" as the factors of relevance.
Then, large systems will tend to have so many different outcomes that
are consistent with certain overall configurations of those parameters,
such as temperature and density gradients, that those particular
configurations become vastly the most likely.
Once a
large system enters a state that is consistent with the most likely set of
these gross parameters, it is unlikely that the system will depart from
it very much or for very long, so we can say the system has reached
"equilibrium" once it enters that most likely subset of states.
3) Rate balance
A system in equilibrium achieves a kind of "marginal utility", in
economics terms, for making changes.
That is, it has reached an extremum in the number of ways it can
happen and be indistinguishable for what we have chosen is relevant, so
any small redistribution of those
parameters (conforming to any necessary conservation laws and
constraint equations)
should not
appreciably change the number of ways that particular subset of outcomes
can be actualized.
This way of thinking about things has no concept of "history" and no
concept of "time", it is just a means of finding the final equilibrium.
But if we want to know how much time it takes to reach equilibrium, and
what path is taken, we need explicit time dependence, and the concept
of "rate equations".
In this latter view, the equilibrium is the place where the rates are
in net balance.
Rate equations require time constants, which stem from details of
the physics that we often can ignore in determining the final equilibrium.
We simply imagine a particular configuration of the allowable parameters
for the system, and separate all the processes that increase a particular
parameter, with their appropriate timescales, from all the processes
that reduce that parameter, and their associated timescales.
In equilibrium, those processes must balance, so that provides a useful
means for finding the equilibrium.
This is particularly convenient when the rates depend linearly on the
parameters themselves, leading to a concept of a "rate coefficient matrix".
In that case, the ratios of the equilibrium parameters can be determined
in terms of the ratios of the rate coefficients, while the timescale to
achieve the equilibrium depends on the magnitudes of the rates themselves
(via the eigenvalues of the rate coefficient matrix).