Re: macroscopic vs microscopic degrees of freedom

[John Denker <jsd@MONMOUTH.COM>]

At 03:49 PM 10/30/99 -0700, Leigh Palmer wrote:
> I agree with all John Denker says about time scale;

Good!

> it is a point I
> emphasize when I teach thermodynamics, but not classical mechanics.

If classical mechanics is going to include discussions of the heat and work
done by blocks sliding on tables, then the syllabus will need revision in
more than one place.

> Since he really only disagreed with my answer to his reformulated
> version of my question we really have no disagreement.

OK.

> My point is
> that there is no value to worrying about work *per se* done in such
> a system, that's all.

I find value in it.  Lots of other people find value in it.  I'm sorry to
hear that you've been unable to find value in it.

> If the definition of work in mechanics depends
> on time scale then I think it is not terribly useful. Do you feel
> that the amount of work varies *continuously* in time, John?

1) I was careful to say what I wanted to say in terms of "timescale" not
"time".

2) The very notion of heat itself is predicated on *not* watching the
ultramicroscopic motions.  If you are asking for an ultramicroscopic
description of the heat-generating process, the request is self-defeating.

3) It is a common debating tactic to argue that anything that is not
completely white should be called black.  The point has been made that
there exist timescales on which the distinction between heat and work is
not clear.  OK.  Fine.  So what?  My point remains:  there exist timescales
on which the distinction *is* clear and useful.  Some of us like to use
physics to solve problems.

Just because there are potholes in the road doesn't mean I can't get to my
destination.  I'm not required to hit the potholes.  In this case it is
trivial to drive around them.

4) In many cases of interest, a qualitative separation of timescales
suffices.  Quantifying the timescale is unnecessary.

In particular, consider the case of the ordinary block sliding on the
ordinary table, which is where I came into this thread.  The thermalization
timescale is fantastically short compared to the natural, conventional
timescale in the problem (i.e. the duration of the sliding motion).

A sliding block is reasonably well described by a handful of variables:
center-of-mass motion, overall rotation, and temperature.

Hydrodynamics is a more difficult case because certain flow patterns (such
as eddies) take a remarkably long time to thermalize.  A large-scale
hydrodynamic system is obviously not well described by a handful of
variables, unless and until the eddies thermalize.

So what's the point here?

  Positive statement: We have simple tools that tell us useful things about
simple systems (and even moderately complicated systems) on relevant
timescales.
  Negative statement:  The simple tools do not suffice to give a detailed
description of a fantastically complicated system on all timescales.

"The light shines and the darkness cannot overcome it."

5) To give a clear, definite non-answer to the question about
time-dependence of work:  The question may or may not be answerable.
During the time-period when macroscopic motions are being degraded to
microscopic motions, the notion of temperature may or may not be well
defined, and it may or may not be useful to attribute whatever heat and
work there is to the paddle (or whatever) that set events in motion.

> Can you
> really say with a straight face that after the eddies have died out,
> the canoeist has done *no mechanical work at all* on the water?

Read my lips:

In a finite pond, if the water is macroscopically stationary before
paddling and also macroscopically stationary afterward, after the eddies
have died out, then I can paddle for hours with no effect other than
heating the water.  On my specified timescale, paddling has done no
mechanical work at all on the water.  No work.  Just heat.

I can do long-term work against water by pumping it to a higher reservoir.
But doing long-term work by paddling around in a pond?  No way.

The distinction between work and heat rests on the notion of entropy.
Entropy is, alas, somewhat subjective.  Suppose the water contains a
certain amount of energy.  If the energy is in the form of eddies, and you
*know* it's in that form, and you know where the eddies are, and you are
prepared with special equipment, then you can go harvest that energy by
nonthermal means.  On the other hand, in a typical canoeing situation, you
don't have a ghost of a chance of harvesting the eddy-energy before it is
thermalized.

Therefore, even on moderately-short timescales (before the eddies have
fully thermalized) I would be pretty uncomfortable calling the eddies
"work".  Certainly they are not "useful work".  I am also uncomfortable
calling them "heat" on this timescale.  Perhaps we need a third term.  I
might go for the term "dissipated energy" which includes out-and-out heat
plus eddies that are destined to turn into heat.

James Prescott Joule operated his paddle-wheel and called the long-term
result heat.  Lots of other people do the same.

(The question of whether the process did work on the canoe is another
question.)

(In the case of paddling against a flowing river, all bets are off.)

> In a
> mechanics course I would never say that; I would come down firmly on
> the side that says the canoeist does an amount of work equal to the
> force he exerts on the paddle time the length of the stroke in his
> frame.

Various possibilities:

1a) When you say the canoeist do you really mean the canoeist?  In that
case this statement is irrelevant and distractive.  We are talking about
the work and heat in the water, not in the canoiest.

1b) Or are you implicitly invoking some conservation-of-work notion to
argue that the work in the water is equal to the work in the canoeist?

2) In any case, were you perhaps trying to say that the paddle does work on
the water, in the amount equal to force times length of stroke?  Really?
So this says you can paddle a canoe with no dissipation at all?  The
lift-to-drag ratio of your paddle is infinite?  Wow!  Patent it!

______________________________________________________________
copyright (C) 1999 John S. Denker             jsd@monmouth.com