[Fis] definition(s) of disorder/chaos: an example

From: Michel Petitjean <[email protected]>
Date: Thu 03 Jun 2004 - 14:21:23 CEST

To: <fis@listas.unizar.es>
Subject: [Fis] definition(s) of disorder/chaos: an example

Dear Rafael,

You are right when you underline that "indistiguishable points" is not
a trivial concept:
> your question on "indistiguishable points" depends on what you understand by
> "point."

There are two replies.
(a) short reply: (x1,x2,x3)=(1,2,3) and (x1,x2,x3)=(1,3,2) constitutes
either the same data ("indistiguishable points") or different data
("distiguishable points").
(b) long reply: points are allowed to have the same coordinates, and
in fact, they may be weighted, and in fact we have to work with
distributions rather with sets. But this does not suffice.
The full model involves a mixture (in the probabilistic sense) of
distributions, each distribution being flagged with a color, and having
its own weight. A rigorous presentation is too long, but may be
found in my papers: J.Math.Chem. 2004,35[3],147-158, and also
in J.Math.Phys. 2002,43,4147-4157. The model involves a measurable
"space of colors". A single point with a color is just a marginal
distribution in the euclidean space (the marginal r.v. is almost
surely constant). Two points with a color and a third one with an
other color are, after projecting in the euclidean space, a finite
discrete distribution of two equally probable points and a discrete
distribution of one point with prob.=1, both distributions having
each their own weight, respectively 2/3 and 1/3 here. There are
more complex examples.

Defining disorder/chaos in a general way needs to clarify the
nature of the input data. Your remarks were thus highly useful.

Michel Petitjean Email: petitjean@itodys.jussieu.fr
Editor-in-Chief of Entropy entropy@mdpi.org
ITODYS (CNRS, UMR 7086) ptitjean@ccr.jussieu.fr
1 rue Guy de la Brosse Phone: +33 (0)1 44 27 48 57
75005 Paris, France. FAX : +33 (0)1 44 27 68 14
http://www.mdpi.net http://www.mdpi.org
http://petitjeanmichel.free.fr/itoweb.petitjean.html
http://petitjeanmichel.free.fr/itoweb.petitjean.freeware.html
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Received on Thu Jun 3 14:23:17 2004

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