gjphw@ihuxm.UUCP (03/28/84)
This overly long item is written in response to the article posted by
D. Gwyn (brl-vgr!gwyn) concerning statistical mechanics and thermodynamics.
It is not intended to be a refutation of the comments made but rather a clari-
fication of the issue. If nothing else, the submissions by D. Gwyn and
J. Stekas have given me an opportunity to clear out my cobwebs. Please excuse
the dust.
The original article that began this discussion concerned the cliche ques-
tion of the air in a room spontaneously clustering into one half. Statistical
mechanics indicates that it is highly unlikely (J. Stekas provided an
estimate) while thermodynamics says that this is impossible. The two items
referenced in the opening paragraph have forced me to greater precision.
Part of the gap between statistical mechanics (based upon dynamics) and
thermodynamics is vocabulary. D. Gwyn states that when the microbehavior of a
system is examined (by unspecified means), the single particle dynamics
(micro-laws?) is seen to be obeyed. Unfortunately, the word microstate, and
the concepts that support this term, do not exist in thermodynamics. I am
inclined to agree with the comment advanced by J. Stekas, that the word
entropy derives its meaning from statistical mechanics. Microstates and
entropy are concepts from statistical mechanics (attributable to
R. Boltzmann?) while the second law of thermodynamics more precisely introdu-
ces the word irreversibility.
Thermodynamics is a strictly macroscopic description derived from empirical
evidence, and usually made axiomatic. There are no microstates to watch, but
the second law describes irreversibility One way to view this property is to
consider that the state where all of the air in a room will arrange itself
into one half of the room is impossible, not merely unlikely. Statistical
mechanics, with its microstates, shows that equilibrium can be defined as the
maximal distribution of these microstates. Any arbitrary distribution, even
if it is not maximal (e.g., air in only one half of a room), is possible,
though most distributions are highly unlikely. The probability of the macro-
states is based upon the number of bodies and the energetics.
Entropy, which is related to the distribution of microstates, is not the
same as irreversibility. This is summarized by the Liouville equation, which
describes the trajectory of the macrostate in phase space. Eventually, all
states are visited. The only expression from statistical mechanics that shows
any irreversibility is the Boltzmann equation, and this is not usually treated
as part of equilibrium statistical mechanics. The Boltzmann equation and the
Liouville equation are at odds, and thermodynamics lends support to the
Boltzmann equation (even at equilibrium).
An irreversible calculation can be obtained in statistical mechanics using a
computer. After operating the model for a while, the round-off and truncation
errors will have eliminated all hope of identifying any long-range correla-
tions. But, this method of obtaining irreversibility is different than the
dynamical foundations for statistical mechanics and its arbitrary precision.
Finally, what this boils down to is whether you believe that physical states
are fundamentally irreversible or merely highly unlikely to be repeated (off
of equilibrium). This same distinction between appearances and actuality was
the basis for the difficulties between A. Einstein's view and W. Heisenberg's
view of quantum mechanics. Prigogine argues that since statistical mechanics
produces the Liouville equation, it cannot adequately describe any macroscopic
system (e.g., a room full of air). He proposes that statistical mechanics,
and perhaps dynamics, needs modifications. While the microstates may or may
not be observable (I am not very clever at examining 10^23 particles), the
resulting macroscopic behavior is the issue.
From an engineering perspective, the differences between *highly unlikely*
and *impossible* are of no consequence. Choose the technique most appropriate
for the system and the questions at hand. From a philosophical perspective,
as employed in science, this difference can be crucial.
--
Patrick Wyant
AT&T Bell Laboratories (Naperville, IL)
*!ihuxm!gjphw