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Re: Entropy



At 4:13 PM -0800 2/9/00, David Bowman wrote:
Regarding Leigh's airing of our remaining disagreement:

As David mentioned, I have only small disagreement with his picture.
That disagreement turns on the topic of "quenched disorder".

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.

Nope. I know nothing about that.

While
I understand what that means for a glass, where a recognizable
breech of degeneracy exists, I will point out that no such breech
of degeneracy occurs in the cese of a deck of cards.

I'm not sure what you mean by "breech of degeneracy"

A "breech of degeneracy" could be called a "breakage of degeneracy".
I just consider the latter an uglier locution, that's all.

but I suspect it
may mean the idea that a proper subset of degenerate possibilities
allowed by the macrostate are actually capable of being realized (because
of a bottleneck or restriction in the dynamics), rather than all of them.
If this is the case, then there *is* a "breach of degeneracy" in this
problem. There is a degeneracy of 52! card sequences that all have the
equivalent collections of microstates in that they all are consistent
with the defined macrostate. When a deck of *unspecified* sequence is
prepared only 1 of these 52! sequences is actually realized. Such a
deck's dynamics is unable to allow the microscopic states of the other
sequences that all belong to the same unspecified macroscopic description
from being sampled. However, any deck whose sequence *is* specified as
part of its defined macroscopic state, then has an extra imposed
constraint that forces only the specified sequence to be originally
allowed. None of the other sequences are, in this instance, even in the
conceptual universe of conceivable microscopic states from which the
microscopic dynamics of the deck is allowed to consider. They have been
ruled 'out of court' a priori. In such a case there is no such
"breeched degeneracy" since there is no such degeneracy to breech.

I'm not sure I followed that. 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. 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.

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.

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.

It is only a deck that *doesn't* have its particular sequence
specified as part of its macroscopic description that has a few bits
of quenched disorder because all sequences are equally consistent with
its macroscopic description, but only the microstates for one such
sequence are mutually dynamically accessible because the microscopic
dynamics effectively prevents the system from getting to any of the
microscopic states corresponding to a different sequence. 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 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 think the idea of teaching that entropy is a measure of the
disorder of a system is likely to confuse the student who seeks to
understand entropy as we do.

I agree. But I wouldn't be adverse to mentioning that thermo (and,
indeed, all kinds of) entropy *is/(are)* a measure of 'uncertainty'
even if is not actually a measure of 'disorder' (as I conceive of the
notion).

I've never said that entropy is not a measure of the disorder of a
system. My quarrels are with the notion that an undefined concept of
order leaves the concept sufficiently well constrained to make it
useful, and that even well defined the concept is useless at the
high school level (except, perhaps, at AP - I don't know). In either
event The statement the "entropy is a measure of disorder" is
perfectly true if one simply defines disorder quantitatively - as
being identical with entropy (or, perhaps, entropy/k).

Uncertainty is ok as long as it is made clear to the
student that the kind of uncertainty meant is *not* a subjective
notion, but is an objective measure determined by the macrostate
description (with the proviso that the only microstates whose
uncertainty is to be considered are those that are dynamically
accessible). In general (possibly non-thermodynamic contexts) the
entropy of a generic probability distribution is *a particular*
nonparametric measure (there are other inequivalent such measures) of
the average uncertainty associated with an otherwise unspecified
outcome drawn from that distribution given that the only known
information about that outcome is present in the probability assignments
of the distribution itself.

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.

The association of entropy with disorder should not be introduced to
students until a quantitative measure of disorder can be defined
and associated with the entropy, and that can't be done at the
high school level.

I agree here. In fact, even *after* the notion of disorder is carefully
nailed down any association it has with entropy is mostly one of being
a cousin-level info-theoretic *relative* notion to entropy. It is
certainly not an association of complete identity anyway. (This being
said, I still didn't find anything objectionable in the AIP news update,
since the context made clear what was meant by the term disorder.)

Yes, and that is the crucial requirement.

I think our differences are only about the employment of the term
"quenched entropy", onto which I admit I have projected my own
meaning, since I'd not seen it before you mentioned it. We both
know that the entropy of a deck of cards does not depend on the
order of the cards. It turns out that it doesn't even depend on
whether they are faced properly in the deck or not, and it is the
same for a card house built from the deck as it is for the stacked
deck. The notion of entropy is very powerful if applied correctly.

Leigh