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Re: Entropy /correction/



Regarding John D.'s corrected scenario:

!! Oops. That last sentence should be replaced by something like this:

I take a _shuffled_ deck of cards and make 10^20 Xerox copies, then pile up
all the decks. I don't tell you the order of any of the decks, but I tell
you they're all the same. The entropy of the whole pile is much less than
what you'd get by summing the entropy of the constituent parts.

I have my own "oops" to correct. I seem to have misread your original
scenario anyway. I mistakenly thought you had 10^20 cards not 10^20
*decks* of cards. But little matter (conceptually speaking, if not
necessarily quantitatively speaking). In *any* of these situations, your
first one, the one I misinferred, and your later corrected one, the
disorder (by my definition) of all of the relevant arrangements is
positive and measures the 'right' thing such a measure ought to measure.

In fact the entropy is not even an extensive quantity; it's practically
independent of the size of the pile.

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. This would be a very difficult feat to try to pull off in
practice. In real life the microscopic state of each of the decks would
be nearly independent of the microscopic state of all the other decks.
This means that in real life the stack's entropy would be quite
proportional to the size of the stack because the information represented
by the entropy of any one deck could not really be used to determine
the miscoscopic state of any of the others, and to an excellent
approximation the entropy of the whole stack would be the sum the entropies
of the individual decks. But this is whole line of thinking is irrelevant
for my definition of disorder. Similarly, the complexity is also not
quite extensive. But this, too, is irrelevant, and the phenomenon of
non-detailed-extensivity for both entropy and for complexity does not
constitute a bug in my definition of disorder.

This is easy to see using a
Chaitin-style "description length" argument.

True. But so what?

!! My apologies for any confusion this caused...

& I'm sorry to have misread your gedanken situation.

David Bowman
David Bowman@georgetowncollege.edu