RE: [Fis] Re: miscellanea / temperature / symmetry

Dear Shu-Kun,

Thank you for the explanation. I now understand the confusion.

Probablistic entropy can be considered as a measure of the uncertainty
in a distribution. Which label we attach to this uncertainty depends on
the theoretical perspective that we use. For example, from an
evolutionary perspective we can call it variation as opposed to
selection. From a dynamic perspective change versus stability. Selection
is taking place at each moment in time; stabilization can only be
evaluated over a time axis.

Shannon's H --I am almost sorry for using it-- was defined for the
measurement of the entropy at a specific moment in time. One can further
derive from it a measure which is sometimes indicated as I (e.g., Theil
1972) which measures the "dissipation":

I = Sigma q(i) 2log q(i)/p(i)

In this formula Sigma q(i) represents the a posteriori probability
distribution and Sigma p(i) the a priori one. I then measures the change
in terms of bits of information. It can be shown that I >= 0. This
accords also with the second law in thermodynamic entropy, but it can be
considered as a probabilistic (formal) equivalent.

The measure I can easily be extended to the multidimensional case and
then be used as a measure for the stability. It remains to be proven
that symmetrical configurations are more stable than non-symmetrical
ones because this depends on the systems properties. For example, in
almost all evolutionary selection processes highly skewed distributions
are produced which can be extremely well buffered against change.

A convenient extension of this reasoning about measuring change in
distributions is provided by the Markov property. The Markov property:
the best prediction of the next state of a system is its current state.
If a distribution (e.g., a system) is stable one expect it to exhibit
the Markov property (because variations in one part of the system can be
compensated in another part). For example, one can test the amount of
change on the assumption of systemness in a distribution versus the
amount of change in the sum of the composing elements and then make an
evaluation of the stability of the system. (This can also be extended to
higher-dimensional arrays!)

I have elaborated this into a test on systemness. For example, in:

Loet Leydesdorff & Nienke Oomes, Is the European Monetary System
Converging to Integration? Social Science Information 38 (1999) 57-86;
preprint version at
<http://users.fmg.uva.nl/lleydesdorff/avril/ems.pdf>
http://users.fmg.uva.nl/lleydesdorff/avril/ems.pdf

Are EU Networks Anticipatory Systems? An empirical and analytical
approach, in: Daniel M. Dubois (Ed.), Computing Anticipatory Systems --
CASYS'99 (Woodbury, NY: American Physics Institute, 2000), pp. 171-181;
Preprint version at
<http://users.fmg.uva.nl/lleydesdorff/casys99/index.htm>
http://users.fmg.uva.nl/lleydesdorff/casys99/index.htm

I don't exclude that symmetrical systems may have other entropical
properties than non-symmetrical ones, but I think that this needs to be
proven. I don't expect this to be the case for living and
meaning-processing systems except perhaps for lower-order ones like
amoebae. These systems tend to produce highly skewed distributions
because of ongoing selection and hyperselection processes.

With kind regards,

Loet

  _____

Loet Leydesdorff
Amsterdam School of Communications Research (ASCoR)
Kloveniersburgwal 48, 1012 CX Amsterdam
Tel.: +31-20- 525 6598; fax: +31-20- 525 3681
 <mailto:loet@leydesdorff.net> loet@leydesdorff.net ;
<http://www.leydesdorff.net/> http://www.leydesdorff.net/

 <http://www.upublish.com/books/leydesdorff-sci.htm> The Challenge of
Scientometrics ; <http://www.upublish.com/books/leydesdorff.htm> The
Self-Organization of the Knowledge-Based Society
Received on Fri Apr 23 08:44:46 2004