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Re: Entropy: sorted=0 unsorted=237



John wrote:
It shouldn't matter (in my view) whether the cards
are all arranged in order or not--they are in some order and that is all
that matters.

Well, it depends on what you are told about the deck.
--) If you are told it is sorted (in the standard order), then there is
only one ordering that is consistent with that description. That's zero
entropy.
--) In contrast, if you are told that the deck is unsorted, then there are
something like 54 factorial different orderings that are consistent with
that description. That's 237 bits of entropy.

Does that help?

Perhaps it clarifies matters if we refer to the "standard order" of a
deck of cards as a "reference state", in the same way that a temperature
scale needs a reference temperature. One of the definitions of
thermodynamic temperature makes use of entropy: the reference
temperature (T = 0 K) corresponds to s = 0. Then it's easy to see,
as per John's example, that if the state of a system (whether it
be an ideal gas or a deck of cards) differs from its reference state,
the entropy must be nonzero.

But now I wonder: the reference state of a deck of cards seems
arbitrary; as has been pointed out, the "standard order" is equally
likely to exist as any other order. But the same cannot be said
of an ideal gas: its reference state (s=0) is NOT as likely
as the many other states it might occupy (equilibrium or not).

Yet perhaps this does not matter; if shuffling a deck of cards
corresponds to doing work on this system, then after time passes
from when we last knew it was in its reference state, the likelihood
of again observing it in its reference state is quite small, regardless
of which ordering we chose as its reference state. Thus, we would
characterize the many-shuffled deck as having high entropy.

This may be stretching the ideal gas / deck of cards analogy, and
I must also confess that I am unfamiliar with what John refers to
as Shannon entropy.

Andy




~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


Dr. Andrew A. Piacsek voice: 509-963-2723
Assistant Professor
Department of Physics fax: 509-963-2728
Central Washington University
Ellensburg, WA 98926 piacsek@tahoma.cwu.edu