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

From: Devin Harris <[email protected]>
Date: Thu 27 May 2004 - 01:41:49 CEST

>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.
Shu-Kun

Michel wrote:
>An other way to discuss about order/disorder/chaos is to look to what
could be their maximum/minimum, if any.

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. Can entropy simultaneously
correlate with disorder as prescribed by Boltzmann (the second law). I
can explain that the disintegration of grouping is invariably toward
symmetry, not disorder. One response took notice but then it seemed like
responses were toward silencing further discussion.

Perhaps the specific issue of exclusivity, the either/or of grouping
(order) and symmetry (order) could be analyzed properly and discussed. I
pointed out the simple principle that a set of objects can either group
together or spread out more evenly. 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.
Disintegration of grouping invariably leads to integration with the
reference frame, in this case fusing together to become a continuous line.

The dots on a line analogy can be improved by using red and blue dots,
to suggest positive and negative, and simplified by closing the line
into a circle (a closed system). The dots are still restricted to the
line. The diagram above (or click here)
<http://macrocosmicsymmetry.com/transition.gif> shows a range of
possible patterns between two extremes where the colors are polarized on
opposite sides (grouped) shown in pattern A versus the two colors
fused/blended/disintegrated/mixed into the neutrality of perfect
symmetry in this model in pattern H. Note that each pair of dots in F
then G repeats this process of disintegration and reintegration which in
extreme becomes H. Note the grouping in E compared to F. Which has
greater symmetry order? The grouping in E draws the pattern away from
increased symmetry.

In my original post <http://fis.iguw.tuwien.ac.at/mailings/1523.html> I
explained the same either/or principle also with checkers, as it is more
applicable to three dimensions, showing how the intermediary stage
between two orders merely appears to be generally disordered. Using game
pieces on a checkerboard, moving from the original grouping of the
checkers toward matching the symmetry of the board, any move either
increases grouping or increases symmetry. Simply imagine switching any
two adjacent squares of a checkerboard. The symmetry of the board is
obviously decreased yet there are four like colored squares then
grouped. Any move in which grouping disintegrates is invariably toward
the symmetry of the checkerboard pattern.

Maximum symmetry order is perfect balance, integration, combination,
uniformity, homogeneity, singularity, formlessness, symmetry, and unity
in reference to the whole system or any limited frame. While grouping
order involves division, separation, distinction, individuality,
density, pronunciation, opposition, and conflict also in the whole
system or limited frame. The definition of chaos is a whole other issue. :)

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. That is the first step. But notice pattern D in the
image, necessary in the transition from C to E. Pattern D is meant here
to represent the myriad of irregular (disordered) patterns which exist
at this stage of the transition between two orders. Observing the
transition, one is led to question whether pattern D is disordered or is
it actually a blend of grouping and symmetry? What is disorder in
reference to grouping in pattern A? What is disorder in reference to
symmetry in patterns FG and H? Perhaps this is why order and disorder
are not considered opposites in some corners of mathematics or why
different measures of order and disorder emerge.

Note most importantly that what is being proposed is simply a logical
extension of Shu-Kun's correlation between symmetry and entropy to
include in contrast to symmetry increasing with time, a precise model of
grouping order. If entropy increase marks the arrow of time, and
symmetry increases with time, what does that indicate about the order of
the past? What is order without symmetry? Grouping order portrays the
transition toward ever decreasing symmetry to the maximum extreme of
opposite polarities.

It helps to imagine a contrast gradient which reveals how all patterns
(states) are necessarily bounded by extremes. Any image can be pushed to
high contrast where there is an opposition of two colors, usually white
and black, or low contrast where the image blends to one color, usually
gray. Grouping order is synonymous with high contrast, while symmetry
(entropy) is synonymous with low contrast. Only in between these
extremes do we find the infinite variety of possible states or patterns,
but the whole set is infinite only between extremes, between boundary
states. The space of all possible states forms the volume of a football,
with extremes of grouping and symmetry at each end, with smooth (low
contrast) and lumpy (high contrast) extremes at every point along the
gradient between, forming the shape in accordance with the measure of
possible states at any average cosmological density. The extreme of
smoothness at any average cosmological density is obvious and the
contrast gradient exemplifies the opposite extreme of lumpiness. Maybe I
am getting away from a simple example here but not really. In cosmology
there are physical extremes of the singularity of the big bang and
absolute zero, i.e., the stretched perfectly flat empty space, a
physical reality in the Big Rip scenario outlined recently by Caldwell
of Dartmouth.

In regards to entropy correlating either with symmetry or disorder,
there is a vague assumption surrounding the second law which implies
there is a never ending supply of disordered states in respect to the
deep time of cosmological evolution, first due to the idea of more
disordered states having been applied as a cause for the arrow of time
(Eddington), and second due to past predictions that time never ends in
an ever expanding universe. The issue of where absolute zero fits into
this model, or if a balance eventually ensues, has never really been
explored properly. In recognizing distinct boundary states, one
recognizes that possibilities toward the future are not unbounded, and
thus the set of all possible states is divisible, in which case
eventually if systems are probabilistic then all possible states must
balance out, with either order/disorder, grouping/symmetry, or
positive/negative groups on either side of the final state of the
evolved system. Which is it? Positive/negative of course.

The arrow of time is clearly moving toward symmetry, entropy, balance,
not disorder. Once we start looking for a balance point in state space
the unavoidable conclusion is that it is absolute zero, a universe
stretched flat, i.e., perfect symmetry. That conclusion is strongly
supported by the discovery of accelerating expansion. Zero is the great
attractor. Yet in order for absolute zero to be the balance of all
states, there has to be an identical but inverse set of states, a
positive and a negative side, as perhaps we might have expected
mathematically. So hypothetically the big bang of an anti-matter
space-time would begin from a dense point that is negative in reference
to our positive past, while both systems move toward the same
equilibrium state.

Loet, I apologize if not but I thought your paper was of similar focus
and invariably toward formulating this same model of two orders. The
algorithms for precise clustering is excellent work. There are I expect
algorithms capable of expressing the transition between grouping order
and symmetry order which would predict for example spiral galaxy
formations, which are perhaps the cleanest cosmological example of
grouping and symmetry in opposition yet structurally working together.

Devin Harris
Received on Thu May 27 01:40:15 2004

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