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Re: Entropy



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.

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", 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.

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 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.

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 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). 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.

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.)

David Bowman
David_Bowman@georgetowncollege.edu