[Fis] definition(s) of disorder/chaos

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

Dear FISers,

I would like to outline the excellent remarks of Jerry LR Chandler:
> the terms "order" and "disorder" are not opposites in the
> mathematical sense
and:
> Set theory is one foundation of mathematics. An abstract concept of order
> enters set theory via the "well - ordered axiom." One expression of this
> axiom ( Stoll, Set Theory and Logic, 1961) states:
>
>"a well ordered set is a partially ordered set such that each non-empty
> subset has a least or first element."
> ...

Partial ordering and total ordering are well known
concepts in set theory. In fact the transitive (but non symmetric)
relation of order between elements is a basic concept.
So, in this sense, there is no ambiguity about what is order.
Now, except negating order, disorder is not defined.
But it seems to me that many authors working with concepts of disorder
or chaos, are not working with the well known definition issued
from set theory.
In the context of the FIS session about Information/Entropy,
my question would have been rather:
which definitions for disorder/chaos ?
Pertinent replies were posted by Devin, Gyorgy, Robert, Loet,
and Stan.
An other way to discuss about order/disorder/chaos is to look
to what could be their maximum/minimum, if any. Coming back to
set theory, we could see that "well ordered" and "totally ordered"
are different criteria: the experimentalist has to clarify in his
mind what are his needs. If the actual concepts of order in set
theory suffice, it is OK. If not, we have to elaborate definitions
from properties, these latter depending on the local area in which
we are working.
I notice that the order concept in set theory falls in what I called
the "static" case (no time) in a previous post, although it
appeared from other postings that the time has to play a role
in order/disorder/chaos. We move forward.
Some minor questions:
- are the measure(s) of order/disorder/chaos changed when the space unit is changed ?
- are the measure(s) of order/disorder/chaos changed when the time unit is changed ?
Other questions:
- Processes: what relations between periodicity, aperiodicity, and
order/disorder/chaos ? And what about ergodicity ? Equilibrium ? Randomness ?

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 Tue May 25 17:47:22 2004