Re: [Phys-L] heat content

[John Denker <jsd@av8n.com>]

On 02/12/2014 07:23 AM, Folkerts, Timothy J wrote:

> So, do we think about these differently because we are comparing 
> * large vs small? (if so, at what size should we switch? What about
> nanotechnology?)
> * many vs few? (if so, how many is enough? What if I set up 10^4
> pucks on an air table all connected by spring?)
> * interacting vs non-interacting? (suppose I set up 1000 airtracks
> with pairs of carts; does it matter conceptually if they are weakly
> connected by springs vs each track moving on its own?)
>
> Perhaps "thermal energy" is simply "potential energy and kinetic
> energy of objects that I am too lazy to deal with on an individual
> basis". :-/

That last suggestion is pretty close to the mark.  I would
suggest only a slight refinement:  Thermodynamics makes statistical
predictions about the ensemble.  This is /sometimes/ the only way
to deal with the situation ... but not necessarily.  There are
some very interesting situations where you can do the molecular
dynamics -- dealing with things on an individual basis -- *and*
also do the statistical analysis, leveraging each approach against
the other. 

In particular:  It is /not/ safe to assume that the "internal"
degrees of freedom of a given particle are "thermal" while
the "external" degrees of freedom are "non-thermal".  If you 
had thought of this 110 years ago, you could have written a 
really nifty paper:

   "Über die von der molekularkinetischen Theorie der Wärme 
    geforderte Bewegung von in ruhenden 
    Flüssigkeiten suspendierten Teilchen"

    Annalen der Physik 322 (8): 549–560.
    http://www.physik.uni-augsburg.de/theo1/hanggi/History/Einstein1906BMII.pdf




On 02/11/2014 02:17 PM, I wrote:
> .... this may help explain why what I've been 
> saying is imperfect but not entirely crazy:

Let's continue down that road.  I'm doing a bit of "thinking 
out loud" here.

We have been considering the contrast between a cold,
fast-moving bullet and a hot, non-moving bullet.  We
arrange that they have the same total energy;  the 
difference is in the entropy.

I would like to shift the focus a tiny, tiny bit.  Rather
than a bullet or other free particle, let's consider a
mechanical oscillator such as a pendulum or a tuning fork.
We can analyze it in terms of its normal modes.  The 
macroscopic oscillation is one mode.  Let's call that 
subsystem A.  In addition, there is a huge number of 
microscopic modes, the phonons within the structural
materials.  Let's call that subsystem B.

We assume it is a very high-Q oscillator, so the macroscopic
mode is only very loosely coupled to the microscopic modes.

Under a wide (but not unlimited) range of "normal" operating
conditions, for a pendulum clock or a tuning fork, the energy 
in the macroscopic mode is enormous compared to kT.  In such
a case, the temperature of subsystem A is essentially infinite.
You can understand that from the fact that it has macroscopic
amounts of energy but only microscopic amounts of entropy.

It is conventional to explain the heat capacity of the 
structural materials in terms of "thermal phonons".  Similarly 
the main oscillatory motion of the oscillator is often called 
"non-thermal".  I'm not saying that's wrong, but I will say 
it is not particularly fundamental.

The reason I say that is rooted in my experience.  In my
line of work, it is likely that somebody will come to me
and ask what happens if we make a super-accurate measurement
of the pendulum, in some not-very-clock-like situation.  At
some point, the uncertainty in the measurement will be dominated 
by *thermal* noise in subsystem A.  At this point *both*
subsystems are "thermal".

So ... from my point of view, it is sometimes convenient
but somewhat risky to label some of the modes as "thermal"
and some of them as "non-thermal".

To repeat:  It is /not/ safe to assume that the "internal"
degrees of freedom of a given particle are "thermal" while
the "external" degrees of freedom are "non-thermal".  If you 
had thought of this 110 years ago, you could have written a 
really nifty paper:

   "Über die von der molekularkinetischen Theorie der Wärme 
    geforderte Bewegung von in ruhenden 
    Flüssigkeiten suspendierten Teilchen"

    Annalen der Physik 322 (8): 549–560.
    http://www.physik.uni-augsburg.de/theo1/hanggi/History/Einstein1906BMII.pdf


===========

At this point, the astute reader might say "The oscillator
has only one exceptional mode.  The 'thermal' properties 
such as heat capacity are determined by the 'thermal' modes,
and it doesn't matter whether you include the heat capacity
of the exceptional mode or not."

Well, that's true for the oscillator example, but once
again I say it's not particularly fundamental.  It's risky.
Somebody is going to come along with something like an 
adiabatic demagnetization refrigerator, where subsystem A 
is the spin system and subsystem B is the lattice ... and 
depending on the applied magnetic field and other details,
the heat capacity of A might be much bigger than B, or much 
less, or comparable.

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

Note: In both the tuning-fork example and the spin/lattice
example, subsystem A and subsystem B occupy the same
physical space.  This stands in contrast to (say) fluid
dynamics, where typically different parcels occupy different
locations in space.

In the spin/lattice example especially, you have basically
no choice but to recognize the two subsystems as separate.
Then, having learned the technique, you might as well
apply it to the oscillator and innumerable other situations.

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

We could go on like this all day.  I can think of many
many examples where dividing the system into subsystems
makes sense.

In /some/ of these systems it is /sometimes/ the case 
that one of the subsystems will be non-thermal i.e. not 
in thermal equilibrium with itself.  However, I do not
recommend that as the main way of analyzing things.  Other
tools are simpler and better.