Re: [Phys-L] heat content

[John Denker <jsd@av8n.com>]

On 02/12/2014 01:37 PM, Bill Nettles wrote:

> It would probably do us all much good to go (re)read what Francis
> Sears has written in his text, An Introduction to Thermodynamics and

OK, thanks for that.

What Sears says is exactly correct in context.  Sears is 
expressing one of the big ideas I've been trying to get 
across.

BE THAT AS IT MAY ......  The bane of all teaching, of all
learning -- indeed of all thinking -- is trying to keep 
track of the context.  Things that are true and useful in 
one context might be not-so-true and not-so-useful in another 
context.

I'm pretty sure Sears was working in a context where we
assume
    E is known as a function of V and S           [0]
in which case (subject to very mild provisos) we can write
        dE = - P dV + T dS                        [1]

ON THE OTHER HAND .......  Several people have pointed out,
quite correctly, that there are other contexts, other situations
that we absolutely must deal with.  Assumption [0] is not
always true.  As a consequence, equation [1] is restricted to
situations where everything in sight is in equilibrium at the 
temperature T.  A theory of thermodynamics restricted to 
completely isothermal situations would be pretty much useless.
You could not build a heat engine under such restrictions.

I still think the right way to generalize equation [1] is 
to divide the system into subsystems.  This is conceptually
simple yet very powerful.  The ith subsystem has its own
value of E_i, T_i, S_i, et cetera.

I hope nothing I've said so far is controversial.


The place where we get into trouble is when only one of the 
subsystems is "thermal" in the sense that it is "thermalized" 
i.e. in equilibrium with itself.  Then the other subsystems 
can be called "non-thermal".

The idea shows up in a lot of places:

    A cold, rapidly-moving bullet.
    A cold, rapidly-spinning flywheel.
    A cold spring with a lot of stored mechanical energy.
    A cold capacitor with a lot of stored electrical energy.
    A cold tuning fork undergoing large-amplitude high-Q oscillations.
    A cold ocean with large-amplitude waves on its surface,
     plus a well-organized Gulf Stream.

This idea is so common and so important that one can understand 
why people like to have some shorthand terminology for dealing 
with it.  I'm not sure the "thermal" terminology is wise, but 
I'm also not sure it is wrong.  I'm conflicted.

In particular:  Terminology is important only insofar as it
helps us formulate and communicate the ideas.  I think I
have a handle on the ideas -- subsystem A versus subsystem
B -- but that is ugly, non-descriptive terminology.  Meanwhile,
the "thermal" terminology is colorful and descriptive, but
misleading.  I feel I ought to come up with some better
terminology, but I haven't yet done so.  This means I am
at risk of breaking one of my deeply held rules:  Don't 
complain about the imperfections in a thing unless you've
got something better to offer.

The hazard is that non-experts will hear us talk about
"thermal" this and "non-thermal" that and take it as an
excuse to go against Sears's warning. to do what the math
and the physics forbids.  Let's be clear:  within any particular
*subsystem* where assumption [0] applies or where equation [1]
applies, you /cannot/ distinguish thermal energy from plain
old energy.  There is no state-function Q such that dQ = T dS
(except in trivial cases).

This is a huge problem, because everybody who has ever tried
to study thermodynamics has tried to define such a Q-function.
Years of experience with trivial cases, such as the heating 
and cooling of baby bottles, makes this misconception virtually
inevitable.

So let me leave it as a question:  Can anybody suggest some
better terminology to capture the idea?  For (say) a tuning 
fork, how do we name the exceptional modes, the ones that are
not in equilibrium with the "main" part of the system?

To get the ball rolling, here is the best I can do at the
moment:

-- We could fall back on ye olde thermal / non-thermal terminology.
 This has several serious disadvantages, but it's better than 
 nothing.

-- In the physics business, the idea of "collective modes" is
 pretty well established.  One fly in this ointment is that I
 have not been able to come up with a good antonym or any clever
 way of describing the non-collective modes.  Maybe something
 like "disorganized rabble" modes.

 The thesaurus lists "distributed" as an antonym for collective,
 but that's guaranteed to convey the wrong idea, because 

-- All of this is a sideshow, because it doesn't cover the
 general case.  There remain lots of situations where we
 simply must divide the system in to parcels, into subsystems,
 and there is no reason to expect that there will be anything
 resembling a "thermal" / "non-thermal" distinction.  Examples
 include:
   NMR spin/lattice distinction
   Styrofoam box containing a hot potato and a cold potato
   Brownian motion
   etc. etc. etc.

 This approach is not colorful, but it is very powerful, and
 AFAICT necessary.  Gack.  I'm conflicted about these compound
 systems.

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

One thing I'm not conflicted about:  There is no Q-function
such that dQ = T dS.  If you think "heat content" (as in
the Subject: of this thread) refers to Q, I guarantee there
is no such thing (except maybe in trivial cases).

This is not a problem with the terminology.  Given that it
doesn't exist, it doesn't matter what you call it.