Re: Heat energy, analogies, lecture demos

[William Beaty <billb@ESKIMO.COM>]

On Thu, 9 Sep 1999, Leigh Palmer wrote:

> > This is an analogy for "Heat conduction".  The stored energy in the
> > swinging pendulum is analogous to the "Heat energy" trapped in a single
> > atom.  The moving table is analogous to both the infrared "photon"
> > component of heat transfer, and also the acoustic/mechanical "phonon"
> > component. (Perhaps we should mount magnets on the pendulum bobs, that way
> > the magnets could perform the "photon" transfer, while the wiggling table
> > would illustrate how "phonons" communicate heat between atoms.
>
> I hate it! This "analogy" is incredibly misconceived.

It is a mental model.  It is a puzzle-piece in the construction which I
use to view the world into a coherent interconnected whole.  How can I
know that it's right?  Because once the puzzle-piece is in place, the
yawning gap of my own misunderstandings is almost completely filled.  Once
it's installed, it connects with hundreds of other concepts which are
spread across physics.  It's like a missing gear from a machine which,
when finally restored, allows the entire machine to suddenly start working
for the very first time.  The gear might have a very strange and
nonsensical shape, but once you see the strange and nonsensical function
that it's meant to perform, all will become much clearer.


> In my opinion
> it is exemplary of the misdirection that can be caused by even well
> intentioned but ill informed efforts simplify that which does not
> require simplification.
>
> (calms down a bit)
>
> This well known physical phenomenon is best known because the two
> pendula continue interacting in such a way that the energy of each
> oscillates, ideally, periodically. Phonons are a name we give to a
> mathematical construct which is manifest in vibrations of the
> lattice; phonons *are* the energy; they don't just "transmit it
> between atoms".

Correct, phonons are the quanta of acoustic waves, and the "waves"
normally occupy the entire lattice as if the lattice was an empty cavity
bounded by reflective walls. However, at extremely short acoustic
wavelengths (hypersound), each atom (and its bonds to neighbors)
individually acts as an acoustically resonant cavity, and a single quantum
of acoustic energy can become trapped within the atom (actually within its
mass and within the P.E. of its bonds to its neighbors.)  The pendulum
mentioned before is analogous to an atom which stores a quantum of
hypersonic acoustic energy.  To alter the analogy and force it to act like
sound waves, add some coupling springs between the pendulums.  Each
pendulum in the chain will no longer swing independantly, but will behave
globally as a wave-transmission medium having a particular acoustic
impedance controlled by the spring constants and the mass of each pendulum
bob.

The chain of air-track carts is a 1-D analogy for the lattice.  Move the
end cart slowly back and forth, and you transmit "sound" along the chain.
But give any single cart a sharp "twang", and the cart will vibrate back
and forth at high frequency, and only very slowly will the neighboring
carts aquire the vibration.  This is a direct analogy for the difference
between sound and "heat", at least as far as the acoustic/mechanical side
of "heat vibration" is concerned.  It only lacks quantized states.

Most of this I got from some old multi-volume encyclopedia of physical
science under the entry for "Hypersonics".  I don't recall the name, but
the set of books was sky blue in color.  I think the cart-chain thermal
vibration demo comes from the Meiners book, or perhaps from some old AJP
article I stumbled across.  The path I'm on has many footprints, I did not
create it, I'm only exploring it.


> A single phonon necessarily involves almost every
> atom in the lattice, save only those which might be located at
> nodes of the motion.

As with light, low-frequency phonons are quite wave-like and unlocalized,
while on the other hand high-frequency phonons display their particle
nature. Sound waves are analogous to radio waves, while "Heat" is
analogous to gamma particles.  With "Heat", it's as if the gamma particles
are localized to single atoms and leap from atom to atom.  The acoustic
side of "heat vibrations" could be described as "incoherent hypersound of
wavelength approaching that of the lattice spacing."


> Phonons, like energy, are not localizable to
> a region smaller than the system itself.

Depends on the wavelength.  At very short wavelengths, and where the
boundary conditions form arrays of resonant cavities, a single phonon can
be "stored" in a single cavity, and can roll around from cavity to cavity
like some sort of marble.

Electromagnetism has the same characteristics: at extremely high
frequencies, we imagine that single photons are being emitted by atoms and
being absorbed by their neighbors.  The photons in the sun take centuries
to wiggle their way out to the surface.  But at extremely low frequencies,
a single photon spans across billions of lattice spacings, and behaves
like an extended wave, not like a silver ball in a pinball machine.


> Why on Earth would a teacher want to introduce an erroneous concept
> using a name already associated with a perfectly respectable
> physical entity!? In what way will this help a student to grasp the
> workings of Nature?


I think my understanding of Quantum Acoustics does not map onto yours.
One of our mental models needs repair (or perhaps both of us are wrong,
and the real experts on Phys-L will leap out and hold up a mirror so we
can see our flaws clearly.)


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