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Re: Entropy /correction/



At 01:42 PM 2/5/00 -0500, David Bowman wrote:

It's true that the entropy of a system is not quite extensive for
composite systems made of nearly identical subsystems. Actually, in your
latter case the entropy would only be practically independent of the
size of the pile *if* you could by some means guarantee that the
microscopic state of each of the identical decks of cards was exactly the
same one.

First answer: I was taking the information-processing point of view, in
which the entropy of the symbols on the cards is taken to be relevant, and
the entropy of the atoms that carry the symbols is taken to be
irrelevant. This is often a perfectly fine approximation, because in
typical signal-processing situations the two pools of entropy are
thoroughly decoupled.

Sorry for not being clear about this decoupling.

This would be a very difficult feat to try to pull off in
practice.

That leads to a second answer: Once upon a time I earned my living doing
low temperature physics. Suppose we make the following substitutions:
*) card ==> atom
*) symbol on card ==> spin state of atom
*) gross thermal properties of card material ==> thermal agitation of
atom's center-of-mass motion (which is decoupled from the spin degrees of
freedom)

... then it *is* operationally possible to cool the atoms to the point
where the center-of-mass motion has zero entropy (crystal or superfluid)
and the discrete "symbols" carried in the spin state represent the entire
entropy.

In such a system, a locally-complicated pattern of spins, repeated
indefinitely, has an entropy that is essentially constant, not
extensive. To say the same thing in different words: The entropy depends
on the spin-wave occupation numbers, not on the number of cycles of wave
that fit within the boundaries of the sample. Increasing the number of
atoms, while keeping the spin-wave occupation numbers constant, does not
increase the entropy.

=======

General pedagogical point: spin systems such as this are often very useful
when thinking about (or explaining) the relationship between Carnot entropy
and Shannon entropy.