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

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

Dear Devin,

Devin wrote:
> I was a bit surprised by the lack of direct discussion by this group
> concerning the intriguing issue of opposition between grouping and
> symmetry, in part because I assume most here are aware of Shu-Kun's
> correlation between symmetry and entropy.

There are various ways to handle relations between symmetry and entropy.
Here is my own, extracted from my email of the 22 April 2004:
Subj: [Fis] Re: miscellanea / temperature / symmetry
I wrote:
> About symmetry and its relations to entropy: symmetry theory and
> entropy (information theory) are both related to distributions.
> But, symmetry theory has to work for distributions, discarding whether
> they are finite discrete, or infinite, or continuous. Informational
> entropy is mainly connected to finite discrete distributions, despite
> some extensions in the continuous case.
> Much more problematic is that the entropy associated to a discrete
> distribution does not care of the numerical values to which non null
> probabilities are attached, and in fact, entropy exists even if the
> probabilities are defined on a non numerical space: e.g.:
> P(red)=1/3, P(green)=1/3, P(blue)=1/3 is a distribution for which
> entropy can be calculated. This is false for symmetry calculations,
> which deals mainly with euclidean multivariate distributions.
> So, entropy and symmetry are quite different. An other question
> is their relations with order and disorder...

But of course, there are other points of view.

Devin wrote:
> The most simple example I gave was
> of dots along a straight line. There is no need to limit the frame of
> reference. The dots can only either move toward or move apart. Break any
> dot into smaller pieces, symmetry has increased, not disorder.

It is an excellent idea to think about the 1-dimensional case before
thinking to the 2 or 3-dimensional case. There are measures of symmetry
in the d-dimensional case, including when colors are considered
(see my review in Entropy 2003,5[3],271-312 http://www.mdpi.net/entropy)
but which measures have we for the disorder of n points on the real line ?
Examples are welcome, but a mathematical expression would be better.

Devin propose an interesting property for what should be a measure of chaos:
> All could at least appreciate here that disorder (if there was such a
> property) could only exist between extremes of perfect grouping and
> perfect symmetry.
> ... rest of email not quoted, but quite interesting, too...

These are the first steps toward a mathematical expression.
It is easier to ask questions that to elaborate consistent theories,
but questions help.

About disorder/chaos, the following remarks of Shu-Kun are very good:
> A small suggestion for discussion:
> When order and other concepts are defined in a quantitative way (how
> much is it?) and relative way (which one has more chaos?), very simple
> examples should be given.

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 May 27 10:30:23 2004