Re: maximum entropy and the seeking of lowest PE

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

Joel Rauber wrote:
> I'm not quite sure how to respond, because I either agree or disagree. 
> Depending on what is meant by Mechanical Energy.  If by "mechanical energy" 
> one means the sum total of all the kinetic and potential energies of all 
> particles in the mechanical system in question; then I disagree, as I would 
> maintain that this is a conserved quantity.
>
> On the other hand (auf die andere Seite), if by "mechanical energy" you mean 
> only that part of the energy described above that David referred to as 
> "macroscopic degrees of freedom" then I think I agree and at 8:00am in the 
> morning on the great plains think that it is a more or less equivalent 
> statement for the 2nd law.  My personal taste is too find this to be a 
> rather restrictive statement applicable only to mechanical systems and only 
> if you divvy up the total energy in the specified fashion.    E.g. does the 
> statement help with understanding why excited atoms tend to decay to their 
> ground state emitting a photon.  I guess I'm stating a preference for 
> statements of the 2nd law that refer to entropy, as being the basic or 
> fundamental principal.  Other specialized statements coming from the 
> entropic version.
>
> Leigh, thanks for responding, I made the post hoping that some of you more 
> experienced folks would give us (me) the benefit of your experience.
>
> David, I hope I haven't misrepresented your opinions or statements you've. 
> Do you have any comments or opinions on what I've said?
>
> Joel

I don't think you have misrepresented me.  I think that maybe you may have
missed Leigh's main point though.  I don't think that Leigh was so much
arguing for an energy minimization principle (although it did read that
way) as much as he was arguing against a *potential* energy minimization
principle.  I think his point was that dissipation will reduce the whole
mechanical energy (or as I would prefer to phrase it the energy in the
macroscopic degrees of freedom) as much as possible consistent with any
other constraints (such as other conservation laws) on the system.  What is
being lowered is not just potential energy but kinetic energy as well
(since their average values above their respective minima are typically
proportional to each other via the Virial theorem).  The second law
dissipation mechanism that reduces the energy is only concerned with the
overall energy (i.e. energy surplus in the macroscopic modes) though -- not
so much on the particular partition of it into potential and kinetic
components.

In the perverse case of gravitating systems we have an exception.  In such
systems the average kinetic energy is proportional to the *negative* of the
average potential energy by the Virial theorem.  In this case as a
bound gravitating system dissipates energy its average potential energy
decreases while its average kinetic energy increases.  This explains why
the collapse of a interstellar gas cloud resulting in the formation of a
protostar is an unstable process.  The system effectively has a negative heat
capacity.  As the system adiabatically shrinks loosing gravitational
potential energy it gains kinetic energy which raises the temperature of the
constituent gas particles.  The higher temperature and higher gas density
translates into an increased internal gas pressure which brakes the
gravitational collapse.  But the increase in temperature increases the
temperature difference between the temp. in the cloud and the external
interstellar ambient.  This turns on a radiant dissipation mechanism which
saps energy out of the cloud.  As this happens the internal gas temperature
and pressure tends to drop causing further gravitational shrinkage.  This
causes further heating (er, warming, or temperature increases) which causes
still more intense radiant dissipation which causes further gravitational
shrinkage.  This vicious cycle has the overall effect of the system
dissipating energy at an ever higher rate as its internal temperature
*increases* away from the cooler background ambient.  The system heats up
(its temperature increases) as it cools (radiates away heat).  LeChatelier's
principle does not tend to apply here (at least in its usual form).

Another example of the weird incompatibility of the ususal approach to
equilibrium with gravitationally dominated systems is in the case of the
decay of a near-Earth satellite's orbit due to atmospheric friction.  In
this case dissipative frictional forces result in the satellite's speed
*increasing* rather than decreasing as is the usual case for dissipative
friction.

David Bowman
dbowman@gtc.georgetown.ky.us