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Re: entropy





On Fri, 22 Nov 1996, Leigh Palmer wrote:

The baby's gone out with the bathwater! The idea of instilling a concept
involves getting the student to relate something he doesn't know about
to something he does. You have utterly abandoned this goal when you take
the concept with which he is familiar, disorder, and relate it to yet
another definition. You presented him with a tautology, not an analogy!

Most students, and most people, have a notion of order/disorder which is
hardly useful for a scientific discussion, and can be downright misleading
when applied to the real world. So first we must get across a useful
defintition, and some understanding of it, before proceding to
applications and conclusions.

I have a very clear idea of what disorder is. Related to a pack of cards
I can tell you that a new pack is rarely as disordered straight out of
the box as it is after it has been shuffled. I shall demonstrate* that
the entropy of the pack of cards has nothing whatever to do with its
entropy, and that should thorougly discredit the analogy.

Does your clear idea of disorder...related to a pack of cards...allow you
to predict the result of the following little experiment? Arrange a deck
with cyclic alternation of suits, Heart Club Diamond Spade, for example.
Have someone do a fair cut of the deck, turn one of the two portions of
the deck upside down and do a fair riffle shuffle with them. Now you have
a deck with mixed upside down and right side up cards. Not much order to
that, a nice mix of face up and face down cards, which to most naiver
persons would appear 'random'. But of course even that isn't 'random' by a
strict mathematical definition. Random is another word most people have no
useful understanding of.

Now deal four cards from the top of this 'mixed deck' They will have all
four suits present. So will the next four dealt, and so on throughout the
deck. Yet the groups of four will not have any apparent consistent
ordering of the four suits present. So in spite of some mixing, and
destroying of some of the order, other aspects of order still remain.

This is just one of many 'tricks' based on the Gilbreath principle, and
these have been discussed in Martin Gardner's books and other places.

The explanation of this isn't difficult, but my point is that most
people's understanding of 'order' is useless to deal with it, and most are
surprised at the result. It is just one example of the fact that a riffle
shuffle doesn't destroy all order, but simply interleaves two subsets of
the original ordered deck. It takes something like 9 such shuffles to
ensure that all remnants of the original order are destroyed.

-- Donald