Re: Ludwig Boltzmann, entropy


Paul J Gans wrote:

> The site you refer to above is the creation of Rob Swenson of
> the Center for the Ecological Study of Perception and Action
> University of Connecticut.
>
> I did a quick read of the site. He's got a point of view
> which isn't quite right, but not totally wrong either.

[Very nice discussion brutally snipped - I'm just adding a few things
and re-expressing some of your points in less polite language.]

My reaction is significantly more negative - I think he's got things
hopelessly garbled. As you say, the key distinction, which he
completely blew, is that between equilibrium and nonequilibrium
thermodynamics. What seems to have confused Swenson is the fact that
Boltzmann did seminal work in both areas, and in both his major
contribution was providing a microscopic interpretation of entropy -
but these were distinct pieces of work and have very different limits
of application.

As you said, Boltzmann's definition of the entropy of an equilibrium
system is pretty nearly universally valid. It is expressed in terms of
an isolated system, but one can readily derive from it the appropriate
expressions for closed and open systems, so that is not a limitation. I
believe that, with a proper interpretation of "number of states", it
even applies to such exotic phenomena as black holes - IIRC Steve
Carlip had a paper on that a few years ago.

However, equilibrium thermo is tricky - you are not really supposed to
say that "in an isolated system entropy increases in a spontaneous
process", although everybody does, because you don't have an expression
for the entropy in between the initial and final states. You are
supposed to say something like "when an internal constraint on an
isolated system is removed, the entropy of the unconstrained system is
higher than the entropy of the constrained system" - i.e. to delete
language that refers to anything changing in time. In practice this is
too cumbersome, but we're supposed to keep it in the back of our minds.

Boltzmann was not satisfied with having a formula for the entropy of a
system at equilibrium - he wanted to be able to say something about how
the entropy (and other thermodynamic variables) changed in a real time
process. Specifically, he wanted to prove, from mechanical
considerations alone, that there existed some function of a
many-particle system that always increased with time. As a start, he
tried to prove this for a hard-sphere gas. The result was the
celebrated "H theorem" - which turned out to be not entirely true
because of Lochmidt's and Zermelo's paradoxes, which then led Boltzmann
to reformulate his theorem in terms like "increases almost everywhere",
which - but this goes on and on and on and nobody is paying attention
except the folks who already know this stuff so I'll stop here.

My point (which I think is Paul's too, but he's being more polite about
it) is that Swenson appears to have gotten Boltzmann's nonequilibrium
formula, which applies to a particular model system, mixed up with his
equilibrium formula, which is very general.

By the way, "Entropy Law" is one of those phrases that sets my alarm
bells ringing - whenever I see it I am 90 percent sure I'm going to
read some pretentious bombast. I've seen much worse than this. One of
my favorites was a claim by George Monbiot that the 2nd Law proves that
Capitalism is impossible.

-----
Robert

.

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