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Re: Entropy



At 6:04 PM -0800 2/16/00, David Bowman wrote

Regarding Leigh's comments on one of my long entropy posts (sorry about
the delay in responding; I've not had the time to respond until now):

Well, mine will set a new standard for delay in responding!

I had thought it turned on the 'in principle' conceptual possibility
of macroscopic tunnelling and on which aspects of the info-theoretic
formulation of thermo entropy are subjective and which are objective.

[Leigh] Nope. I know nothing about that.

I thought you had commented (privately) about the impossibility of
macroscopic tunnelling and accused me of having a subjective/solipsistic
view of thermo entropy. Did I misunderstand your comments?

It is my opinion that tunneling of playing cards from one position in
a deck to another can't occur, even in principle. Such an occurence,
if it were possible, would be very much rarer than a multitude of more
probable occurences, for example the tunneling of half a card. Once
the tunnelings of single cards become terms in the partition function
(or rather the final states of those processes) very many other terms
of greater significance must be included as well, far too many of them
to count.

I recall (though I have lost the note by now) that I did comment on
the notion that the mere possession by an individual of information
regarding the state of a system affects its entropy; it does not. To
hold otherwise *is* solipsistic, but I don't believe I accused you of
solipsism. The entropy of a system (or, more properly, the change in
the entropy of a system during some process) is an objective quantity.
It is entirely analogous to the state of Schrödinger's cat, alive or
dead, but never in an indeterminate state, regardless of who can see
it.

The crystalline form of quartz has a
lower internal energy than the glass, I believe. In any event the
internal energies (and the free energies) are different.

The crystalline form of quartz has a lower internal energy than the glass
at a common *fixed temperature* (which is maintained fixed by an external
thermal reservoir).

Yes, at room temperature, and also at a common pressure, atmospheric.
It is, more correctly, the Gibbs free energies of the phases that
differ. That is also true anywhere below the melting point of the
crystalline phase.

The Hamiltonian for both systems is the same so the
energy levels and their degeneracies also coincide. The difference in the
different macroscopic forms is in differences in the distribution of which
of the microsopic states are likely to be observed under various
macroscopic conditions. For instance, the crystal (when fully
equlibrated and isolated from its surroundings) for a given amount of
energy has a greater entropy than the isolated glass has when it is given
the same energy.



The two forms have a different entropy vs. internal energy functional
relationship (for the same fixed mass & volume) since at a given energy
the glass (not being fully yet equilibrated) has a lower entropy than
the crystal. Since both curves merge to coincidence at a high enough
energy/temperature (i.e. in the melt) this means that the slope of the
S vs. E curve for the glass is steeper than for the crystal at a given
energy. This means the same energy has a lower temperature for the
glass (relative to the crystal), or equivalently, the same temperature
has a higher energy for the glass. Calling this difference a
breach of degeneracy is quite idiosyncratic.

That is
what I meant by a broken degeneracy. There is no such difference in
the internal energies of two ordered states in a deck of cards.

True, but what makes the deck of unspecified order quenched has to do
with a constriction on its dynamics at a given fixed internal energy, not
with a different value of a mean internal energy at a fixed temperature
(fixed by the environment).

There is a tendency (on a very long timescale) for viteous quartz to
"devitrify", or return to a crystalline phase. This tendency is
accelerated by the presence of some foreign materials on the surface
of hot fused quartz, whence the warning that quartz light bulbs not
be handled with the bare hand.

This demonstrates that the glass is not an equilibrium phase and that
the time scale for full equilibration can sometimes be sped up enough for
to it be experimentally measurable. Admittedly the activation energy
for having the cards switch their order is much higher than for the
glassy structure to find the appropriate crystal structure, and a
suitable analogical catalyst is missing from the card example.

The ordered
pack has exactly the same entropy as the disordered pack, and the
ordered pack is clearly not in a state of "quenched disorder".

Actually, neither deck is in a state of quenched disorder *if* they
both have their sequences specified as part of their macroscopic
state.

It matters not one bit whether a particular sequence is "specified"
or not! A physical deck of cards *has* a perfectly well-determined
sequence whether it is known to or specified by anyone, or not.

Specification of the macrostate does not pertain to anyone's arcane
knowledge, but to the objective definition of the precise meaning of
the system under study. If the dynamics of the deck of unspecified
sequence allowed sequence switching over reasonable realizable time
scales then a system whose sequence was specified to be fixed to
some particular order would have slightly different equilibrium physical
properties (its entropy would be a few bits less, for instance) than a
deck whose external physical constraints were loose enough to allow it
to keep switching sequences.

It is true that *in this case* it "matters not one bit whether a
particular sequence is 'specified' or not" because the internal dynamics
for the microscopic states do not allow for sequence switching whether
or not the sequence is externally specified and fixed as part of the
macrostate. It is this constriction on the dynamics that allows both the
specified sequence deck and the unspecified sequence deck to have the
same entropy (and other thermodynamic properties).

.... This *is* a
quenched system. It is prevented from sampling the microstates of the
other sequences, i.e. annealing, by a dynamical constraint/bottleneck.

Methinks this smacks of magic! A deck of cards doesn't anneal if
you don't watch it any more than a cat can be in a mixed state of
vitality if it remains unobserved. The idea of quenched entropy
surely can't apply here.

The concept applies. It's just that the relevant needed timescale is
far too astronomical for it to be actually observed 'in the field'.

This exchange shows we *do* seem to disagree on "the 'in principle'
conceptual possibility of macroscopic tunnelling" (not the realizable
possibility) as I stated in my previous post quoted at the top of this
one.

The problem here arises from the common perception of order. If a
pack of cards is arranged according to suits and pips we say it is
ordered, but that is a cultural, not a physical distinction. The
physical idea of order is quite different in this case. All
possible arrangements of the deck are equally orderly from a
physical perspective.

True, but some card sequences require fewer bits of algorithmic
information (i.e. complexity) to precisely describe, and, by my
definition of disorder, those such sequences *are* more ordered than
those sequences which require more information to completely
characterize. But, I agree each of the sequences are physically
equivalent as far as their thermodynamics is concerned.

I guess I must apologize to John Denker; I thought he had made
the assertion above. (I'm responding more slowly to David Bowman
because I know he's smarter than I am.) Please accept the
challenge to John in my last note in this long series.

I'm sorry, what challenge was that? Remember, I'm not defending any
notion of the entropy of different sequences being different. I merely
mentioned that not all sequences have the same algorithmic complexity.
If they did then all sequences would be equally ordered/disordered (by
my definition of disorder). The info for the algorithm for specifying
an ordered deck only needs 2 bits to specify the order of the 4
suits, and needs 4 bits to hold the repeat count of 13 for iteratively
counting up the cards in order for each suit. The info for the algorithm
for specifying a disordered sequence needs nearly 52 (or 54) separate
instructions for which card is to come next, and each one needs about
6 bits to identify each appropriate card color/number.

...

Moreover it gives aid and comfort to
religious fanatics who do not seek physical understanding.

I do not understand what is meant here.>

You haven't seen Wayne Gish perform. The "Scientific Creationists"
invoke the second law of thermodynamics in its popular form as a
neo-Aquinian argument from design. The existence of any order in
the universe is antithermodynamic because in the beginning there
was chaos. Order cannot arise spontaneously from chaos. Therefor
there must have been a creator.

I think you must be referring to one *Duane* Gish. It's true, that I
haven't seen him "perform". It's also true that many (not all)
Creationists terribly abuse thermodynamics and the 2nd law in their
polemics. Of course a flawed argument only invalidates the process by
which a conclusion is derived. It doesn't necessarily mean the
conclusion itself is wrong. The conclusion could be correct for some
other reason(s). I don't think we can invoke thermodynamics to prove the
existence of a creator. This doesn't mean that I don't think the
universe has a creator/sustainer.

David Bowman
David_Bowman@georgetowncollege.edu