re:Flow of energy

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

Brian Whatcott wrote:
> ....
> I understand your point that heat is a statistical or emergent property,
> but I hope you will not demur at the use of average particle or molecular
> momentum as the underlying basis of 'temperature' in this simple case,
> where I do not need to press you to define 'microstate'.

Sorry, but I do demur a little here.  The average particle momentum is not
the underlying basis of temperature.  Rather, it is the underlying basis of
macroscopic *translation*.  The average particle momentum divided by the
mass of the particles (assuming they are non-relativistic as this point) is
the average *drift velocity* of the system of particles as a whole.  The 
temperature of a (classical equilibrated) system of particles is more related
to (actually, proportional to) the *variance* of the particle's momenta (or
velocities) than to the mean of the momentum distribution.  When quantum
mechanics and relativity become important then even this connection to the
temperature is broken and the variance of the distribution of the momenta is
no longer directly proportional to the (absolute) temperature.

> The reason I am unreasonably insisting on inspecting individual particles
> where statistical quantities are really called for, is because a few years
> ago, I attended an interesting talk given by a research engineer or
> physicist from one of the big commercial labs - perhaps 'son of Bell labs'.
>
> He set out to describe his laser-cooling method of reducing macroscopic
> particles to millikelvin temperatures, perhaps much lower, I cannot recall.
>
> The details grow dim, but I recall his lasers in several orthogonal axes
> and in each direction were able to couple energy into particles moving at
> particular speeds, by means of tuning the laser frequency to interact with
> the doppler-shifted frequency corresponding to particular speeds.

This laser optical cooling method has been given the descriptive moniker 
'optical molasses' and has been successfully used in cooling ensembles of
particles to such a low temperature that Bose-Einstein condensation occurs.  
The lowest temperatures reached are *sub* microkelvin.  The technique works
sort of like an old-fashioned method of demodulating FM signals (before the
days of phase-locked loops and, even, ratio detectors) that used a 'frequency
discriminator'.  Here the laser frequency is tuned to be a little off-
resonance for an absorption line for the particles.  Ideally the laser
frequency is at the inflection point of maximum slope on the low frequency
side of the absorption line curve. Because of this a particle is more likely
to absorb a photon if the particle encounters it "head-on" due to the Doppler
blue-shift, and it is less likely to absorb a photon if it is trying to run
away from the photon due to the Doppler red-shift.  If an on-coming particle
absorbs a photon the magnitude ot the particle's momentum is reduced.  If
a receding particle absorbs a photon the particle's momentum magnitude is 
increased.  Thus a particle is more likely to be slowed down by the photons
than to be sped up by them by absorption.  After the particle absorbs a
photon it eventually de-excites by emission.  The direction of this emitted
photon is essentially isotropically distributed, so, on average, there is no
net slowing or speeding of the emitting particles.  Once the particles de-
excite they are ready to absorb another laser photon.  As time goes on the
particle motions get ever-slower effectively lowering the temperature of the
ensemble.  The bath of laser photons effectively acts like a viscous fluid in
slowing the particles down.

> He was able to segregate particles all of a closely controlled speed,
> and he claimed that by definition, where internal random motion was
> suppressed, this corresponded to reducing the internal temperature.

It is true that such a suppression will reduce the temperature, but it is
not "by definition".

> This naturally raised the possibility that photons can provide a momentum
> carrying mechanism in my mind.

Photons certainly *do* carry momentum.  Being massless particles their energy
is related to the magnitude of their momentum by E = p*c.

> ....
> I take exception to the idea that a working process can change the
> Hamiltonian, thereby disturbing the system's energy levels.
> I expect this is a mathematical shorthand ( but it seems quite on a level
> with 'energy flowing' to me) because a Hamiltonian is simply a math construct.
> A math construct only ever describes some change - it never causes the
> change, to my view...

To the extent that all natural law is purely descriptive rather than 
prescriptive of nature it is true that *actual* causes are never identified
for *any* natural phenomena.  In order to do science at all we must assume
(for merely operational purposes) that such a thing as natural laws exist,
and they (in whatever form they eventually are found to be) do accurately
describe the physical make-up and behavior of things.  With this assumption
and the assumption that our current knowledge of such laws is sufficient for
our purposes at hand, we can use the shorthand language that the laws of
nature that we know *govern* the phenomena that they really merely describe.
I certainly would not forbid God from performing some miracle because it
happens to violate some law of nature, but we cannot come to a scientific 
understanding of such miracles if they, by definition, violate the laws that
we use to come to a scientific understanding of the phenomena around us.  So
even though we really don't know the real ultimate causes of phenomena or
existence, we can pretend that we do as long as we confine our discussion to
phenomena that obey the laws of nature that we know about.

When I mentioned that a macroscopic working process changed the system's
Hamiltonian and that *caused* a change in the system's energy levels, I took
it for granted that such a use of the word 'cause' would be interpreted as
loosely as is customary in scientific discussions.  The Hamiltonian is a
mathematical construction, but it actually describes the system and its
dynamical behavior.  If some macroscopic work is performed on the system then
the system is changed in some way.  This change appears in a change in the
Hamiltonian that describes the system's subsequent behavior.

Consider the simple case of a macroscopic collection of monatomic ideal gas
particles in a cubic box of side L = V^3.  The Hamiltonian is the sum of the
kinetic energy of the particles and the potential energy function for the
particles' positions.  This potential energy function is 0 when all the
particles are in the box, and is infinite otherwise.  If this system is
subjected to a process that expands the volume of the box to V' = L'^3 > V
then the potential energy function is changed to a new function because some
regions that formerly were outside the box are now inside it (so some former
infinite P.E. configurations are now zero P.E. configurations).  The energy
levels of a single particle in a cubic box of side L are given by
E(n_x,n_y,n_z) = (1/(2*m))*((h_bar*[pi]/L)^2)*(n_x^2 + n_y^2 + n_z^2) where
the quantum numbers (n_x, n_y, n_z) each belong to the natural numbers.  The
energy levels for the entire collection of particles are the sum of the
single-particle levels for each of the system's particles.  A given
microstate involves a specification of a collection of {(n_x, n_y, n_z)_1,
(n_x, n_y, n_z)_2, ..., (n_x, n_y, n_z)_N, m_1, m_2, ..., m_N} where m_i
represents the spin component of the ith particle along some fiducial axis.
If the particles are all identical then the microstates are given by an
(anti)symmetrized superposition of states involving the permutation of the
particle labels 1,2, .. , i, ... N among themselves in the above collection.
If the particles are fermions then no one set{(n_x, n_y, n_z)_i, m_i} for any
particle i can agree in detail with another corresponding set for any other
particle j present in the system.  Notice that each of the energy levels
(no matter what the complicated collection of quantum numbers may be) is
proportional to the common factor of 1/L^2 = 1/V^(2/3).  Thus, when L --> L'
and V --> V' we see that all of the energy levels are scaled by the same
factor.  When averaging these energy levels over the distribution of
possible microstates we see that the resulting average in E_total is
proportional to V^(-2/3) under a process that changes V (but adiabatically
keeps the microstate distribution the same in terms of the quantum number
labels).  The amount by which this total average energy changes is the
macroscopic work done on the system by the process that changed its volume.

The pressure is the negative of the derivative of the total energy wrt the
volume change (under adiabatic conditions).  The pressure for a given
microscopic state (i.e. complete collection of quantum numbers) is the
negative of the derivative of the energy level for that state wrt a change
in the system's volume.  The macroscopic average pressure for the system is
the average over the distribution of microstates of the pressures for each of
those microstates.  Since microstate-by-microstate we see that
p = -dE/dV =(2/3)*E/V,  after averaging this relationship still must hold.
We thus see that an ideal gas must obey the equation
(p_avg)*V = (2/3)*(E_tot.avg).

> So finally Thanks again. And is it possible to say a photon is involved
> with transfering momentum between gas molecules?

Certainly this trivially is the case when one molecule emits a photon and
later another one absorbs that photon.  In general, it would be a real pain
to treat things as complicated as molecules and their behavior of
intermolecular collisions in terms of all the Feynman diagrams which account
for all of the virtual processes involving virtual photon exchange for those
collisions.  Since the electromagnetic interaction is the only one of
relevance for the behavior and structure of molecules such collisions are
ultimately mediated by virtual photon exchange at the ultramicroscopic level. 

David Bowman
dbowman@gtc.georgetown.ky.us