For many, one of the most perplexing questions of the last century has
been the origin of irreversibility. The microscopic equations of motion
are time-reversal invariant and solvable, at least in principle, given
appropriate microscopic initial conditions. Not only are these initial
conditions never available, but by the very definition of the problem we
never have any control over the microscopic variables. It should
be clear immediately, then, that this microscopic invariance has little
to do with the irreversibility we see on the macroscopic scale, for we
cannot deal with those equations of motion directly in any event. A great
many of the microscopic states and initial conditions will serve to define
the corresponding macroscopic quantities, however, and we certainly possess
initial conditions on the latter if we are studying experimentally reproducible
phenomena, say. In turn, the microscopic equations of motion are manifested
in those for the statistical operator 
,
or phase-space distribution 
in
the classical case. These are exact and also irreversible, as we have already
noted above. From whence, then, comes the macroscopic irreversibility we
observe in nature?
The operator is
essentially a collection of probabilities, through its expectation values,
so that one can already find the difficulties at that level. There is no
harm in contemplating a time-dependent probability and studying 
,
say, but there is absolutely nothing in probability theory itself to tell
us what the equations of motion for P(t) should be. [Indeed,
we wonder how P can change at all in the absence of any change in
the hypotheses upon which it was constructed.] Only the physical
equations of motion can guide us in the present scenario and, as we have
seen, they are time-reversal invariant. The traditional response to this
impasse has been to approximate the equations of motion for or
in
some way, thereby immediately and artificially introducing irreversibility
into the description by discarding relevant information. The most common
approximation is to truncate the hierarchy and close the resulting equations
for the lower-order distribution functions 
by
some means. There is simply no way to tell if this `irreversibility' is
merely mathematical or not, a point that continues to be missed [13].
This brings us to the question of time-dependent entropy, and one can
now see the difficulty with defining 
.
There are no definite criteria for determining how P(t) itself
should develop, other than from the physical equations of motion. And if
S is defined as in Eq.(1) it is a constant of the motion. It may
be possible, of course, to define S(t) in a number of other
ways, but there do not seem to be any criteria available to tell us what
is unique and correct.
How, then, are we to understand irreversibility and the second law of thermodynamics? Because much of the conventional wisdom in this area appears flawed, as noted, another alternative might be to follow a line advocated by Prigogine and his collaborators [19], and others [20], who suggest that there is a heretofore unrecognized time irreversibility in all the microscopic laws of physics, even at the quantum level. They envision a fundamental irreversibility throughout the microscopic world, despite the absence of any empirical evidence to support those beliefs. Rather than move further into this twilight zone of irrationality, we shall consider a simpler and more logical resolution of these questions arising from reflection on the above discussion.