Re: Entropy: quenched disorder is entropy

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Barlow Newbolt's comment/complaint:

> You fellows are confusing me mightily with this thread.

I'm sorry.

>  I can't even
> figure out if you are talking about thermodynamic entropy, informational
> entropy, or the twilight zone between.

This might be part of the source of our disagreements.  We might be
using the same word for subtlely different concepts.  I hope to be able
to tell further when John responds again.  It seems that John may think
that many/most kinds of entropy are really the same, but I'm not sure.

>  Statistical mechanics was always
> difficult for me so I am sure part of the problem is just my
> understanding, but can you make your posts clearer to the "slow
> learners"?                   WBN

Here is a brief summary of our positions as I see them:

John believes a shuffled deck of cards has a few bits more of
thermodynamic entropy than a well sorted one by virtue of the disorder
in its sequence.  He also (apparently) claims that quenched disorder is
thermodynamic entropy.  He gave an argument in support of this latter
claim (that I cannot follow) having to do with the thermodynamic cost of
computations.

Originally, I claimed that the shuffled deck would *only* have a few
bits more entropy *if* its actual macroscopic sequence was *not
specified* (this would allow possible actual sequences that are both
'ordered' and 'disordered'; but the 'disordered' ones would strongly
predominate in number) as part of its macroscopic description.  This is
because there are a few more bits needed to identify the exact
microscopic state when all the microscopic states for all the possible
sequences become part of the mix of all the microstates consistent with
this less restrictive macroscopic description.

However, upon further consideration (stimulated by a private
correspondence with Leigh) I changed my mind and now claim that both a
deck of cards whose sequence is specified as part of its macrostate
(such as a deck specified to be in perfect order by suits and number)
*and* one whose sequence is left completely unspecified (so its order is
effectively random and most likely is quite disordered) *both* have the
exact same thermodynamic entropy.  This is because even though the
unspecified deck has more microstates consistent with it than a deck with
a specified order has, it ends up that the microscopic dynamics for an
unspecified deck is incapable of spontaneously permuting its inital
order (over any realizable timescale) and this restricts the actual
number of microscopic states dynamically accessible to the deck of
unspecified order to be the same as for a deck whose order is specified.
I (& the consensus of stat. mech. types) believe that the thermodynamic
entropy for a system should only count those microstates that are
consistent with the macrostate that *also* happen to be accessible from
the initial microstate by execution of the underlying dynamics (assuming
the dynamics is allowed an infinitesimal noise).  This means we should
not count the microstates for the impossible card sequences for the
unspecified deck, and only count those for the actual sequence that the
unspecified deck actually happened to have when the deck was initially
prepared.  This means that all decks have the same thermodynamic entropy
whether or not their card sequences are specified or not.

This restriction to a proper *subset* of the microstates consistent with
a specified macrostate (i.e. the subset that are mutually dymanically
accessible) is what happens whenever a thermodynamic system has quenched
disorder such as in glassy system, where the randomness in the lattice
structure is frozen in place and the system's microscopic dynamics is
incapable of exploring all the microstates corresponding to the different
frozen structures that are all macroscopically identical.

The conclusion of my latter position agrees with that of Leigh, who all
along, contended that all card sequences had the same thermodynamic
entropy.  But Leigh disputes that the case of a deck of unspecified card
sequence is one of quenched disorder.  He believes that all sequences
have the same entropy whether or not they are macroscopically specified,
and no matter how disordered a given sequence might be.  He seems to have
never cared whether or not a deck of unspecified order had more
microstates consistent with its macroscopic description than a deck whose
card order was specified.

I hope this recap helps (somewhat).

David Bowman
David_Bowman@georgetowncollege.edu