Re: Entropy (was very long) ERRATUM

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

It seems I need to correct a statement/explanation I made in response to
John D. about the entropy difference between an ordered deck and a
shuffled deck of cards.  After a private correspondence between Leigh
myself I have become convinced that my explanation was wrong.  Thanks
Leigh!  Leigh and I still disagree about a point or two about philosophy
(related to the 'in principle' possibility--not practical possibility--of
macroscopic tunnelling and to just what is objective and what is
subjective in considerations of entropy), but we both agree that both
decks of cards have exactly the same entropy assuming everything else
about them, other than their sequence order, is the same.

I had written:
DB:
>   For instance, a thoroughly
> shuffled deck of cards has essentially the same thermodynamic entropy
> as a well-ordered deck (assuming, of course, both decks have the same
> temperature and are subject to the same external macroscopic
> environment).  OTOH, a cold deck of cards has less thermodynamic entropy
> than a warm one regardless of how the decks were shuffled or not.

John responded:
JD:
> Shuffling the deck increases its entropy by a few bits.  Cooling the deck
> (in the usual way) decreases its entropy by something like 10^23 bits.

I answered:
> That is why I hedged my comment by saying the shuffled deck had
> "essentially the same thermodynamic entropy as a well-ordered deck.
> ^^^^^^^^^^^
>
> The weasel word was to account for a *possible* unobservable
> contribution to the ~21st-23rd significant digit or so.  In fact, whether
> or not the shuffled deck's entropy is greater or not than the unshuffled
> one by a few bits depends on the details of the macroscopic
> specification of the two decks.  For instance, if the actual
> card sequence in the shuffled desk was included in its macro-level
> description then the few bits needed to specify the sequence would
> not contribute to the entropy.

I now believe that there are *not any* extra few bits of thermodynamic
entropy associated with the statistical entropy of the macroscopic
sequence of cards for the disordered deck *even* in the case where the
actual card sequence of this shuffled deck was not specified as part of
the macroscopic description of the deck.  Thus, I wish to completely
eliminate the weasel word "essentially" from my prior statement.

In my view any such deck, whether shuffled or not, whether the actual
card sequence is specified as part of the macroscopic description or not,
has the same entropy as any of the others as long as everything else
about the decks' macroscopic state is scrupulously held the same.  The
reason (Leigh disagrees somewhat with the reason but agrees with the
conclusion) is that in all cases the actual order of the cards remains
fixed (over realizable time-scales) whether or not the actual card
sequence is specified as part of the macroscopic description.  Thus all
the microscopic states of the deck involving all the other sequences
than the initial sequence are dynamically inaccessible, and even the deck
with an unspecified sequence is not free to sample those other
microstates associated with a different macroscopic card sequence.  This
is a case of quenched disorder, not all that unlike that found in
amorphous materials.  In the later case the thermodynamic entropy does
not include any contribution from the microscopic states involving major
rearrangements of the effectively frozen lattice whose precise disordered
structure is dynamically 'stuck'--even though such rearrangements do not
entail a difference in the description of the macroscopic state of the
system.  The main difference in the two cases is that in the case of
amorphous materials the frozen disorder involves very many bits at the
nearly microscopic scale, and in the case of the cards the frozen
disorder involves a few bits at the macroscopic scale.

See my next post for an amended definition of thermodynamic entropy
that includes the possibility of quenched disorder.

David Bowman
David_Bowman@georgetowncollege.edu