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# Re: [Phys-l] Temp & Energy density

• From: John Denker <jsd@av8n.com>
• Date: Wed, 05 Aug 2009 18:09:00 -0700

On 08/04/2009 09:03 AM, Donald Smith wrote:
.... Ultimately, I agree with the other posters -- you're applying
macroscopic concepts like pressure, temperature and volume to a
system where they don't apply. Hence, the paradoxes.

Excellent. This is what they call "taking the game to the next level".

The levels here are:

1) Expert level: mastery of the subject matter:
2) Instructor level: All of level 1, plus mastery of the pedagogical
and psychological issues.

When we start asking where the paradoxes come from, we are clearly
at the instructional level. After all, there are no paradoxes in
the real laws of physics; paradoxes only arise if/when we
misunderstand the laws of physics.

As previously mentioned, there is a magnificent discussion of
fluctuations in Feynman volume I chapter 46 ("Ratchet and Pawl").

I reckon there are several ideas that need to be clarified:
-- Temperature is not the same as energy density. Suppose
we have a 500 psi bottle of compressed air and also a 1000
psi bottle, both at 300K, both standard size. The latter
has quite a bit more energy (and energy density).
-- Any finite system will exhibit fluctuations, even at
equilibrium. The laws of thermodynamics apply just fine to
finite system.
-- A Maxwell demon is a device that purports to violate the
2nd law of thermodynamics by looking for fluctuations. Such
devices never work as advertised, but they sometimes do other
interesting things; see Feynman, op.cit.
-- Other issues not yet identified.
-- Combinations of the above.

... applying macroscopic concepts like pressure, temperature and
volume to a system where they don't apply.

I'm not sure that's the whole story, or even the main part of the
story.

1) Actually the standard notions of pressure and volume apply just
fine to systems of all sizes (microscopic, moderately large, and
infinite). Temperature is not always well defined, but certainly
there are some systems, even microscopic systems, where it is well
defined.

2) Secondly, "macroscopic" is not quite the right word.

Any _finite_ system (whether macroscopic or microscopic) will exhibit
thermal fluctuations, and I think fluctuations are a big part of the
story here.

-- For microscopic systems, fluctuations dominate.
-- For medium-large systems, fluctuations lead to smallish
correction terms.
-- For infinite systems, fluctuations are negligible.

As far as I can tell, the issue here is not that the system is less
than macroscopic, but rather that it is less than infinite.

This is a pedagogical problem, because students typically start by
studying situations where fluctuations are negligible ... and they
may never get around to really understanding fluctuations.

This is quite a fundamental issue, affecting even our basic notion
of equilibrium. For finite systems, there will be fluctuations
/even at equilibrium/.

Also keep in mind that fluctuations are related to dissipation via
the _fluctuation / dissipation theorem_ ... which is a corollary
of the second law of thermodynamics. So understanding fluctuations
is part and parcel of any attempt to understand dissipation.

All this falls under the heading of "finite size effects". I just
now added a section on this to my thermodynamics document:
http://www.av8n.com/physics/thermo-laws.htm#sec-finite

As discussed there, finite size effects can be categorized as:
* Boundary effects;
* Quantization effects;
* Mean free path effects;
* Transport and dissipation effects; and
* Fluctuations.

==================

Remark: This stuff is not trivial, and it is not old hat. Remember
that Einstein published the first real explanation of Brownian motion.
A hundred years ago, this was considered very advanced physics.

Even in 2009, you can still publish papers on Maxwell's demon.
http://arxiv.org/abs/0707.3400
Authors: Koji Maruyama, Franco Nori, Vlatko Vedral

Abstract:
Maxwell's demon was born in 1867 and still thrives in modern physics.
He plays important roles in clarifying the connections between two
theories: thermodynamics and information. Here, we present the
history of the demon and a variety of interesting consequences of the
second law of thermodynamics, mainly in quantum mechanics, but also
in the theory of gravity. We also highlight some of the recent work
that explores the role of information, illuminated by Maxwell's
demon, in the arena of quantum information theory.