entropy and chaos query

Fri, 28 Oct 1994 23:22:00 PDT

My apologies for sending this long comment to the list as a whole, but when I
tried to send it to the address in the original posting it got kicked back to
me as undeliverable.
D. Read

Maureen, you asK:

a. What is the relationship of entropy and chaos theory?
b. Does entropy, the second law of thermodynamics (Clausius,
1850), have an appropriate wider application--ie, decay/new life--
life cycle...??? Or is this just overgrown fuzzy metaphor?
c. When did chaos theory become part of public conversation? I know
I remember a PBS program ca.1989 on it (with fish).... How
far off the mark is popular understanding from the real thing,
whatever that is?

Here's is an Off-The-Cuff reply.

First, entropy and chaos theory (at least in terms of their mathematical
aspects, are very different). Entropy, as you note, has to do with the way
in which systems go from organized to unorganized states in the absence of
energy inputs to maintain ordered states. E.g., a forest maintains itself as
an organized system through constant input of energy from the sun; without
energy input the forest dies and decays, with complex molecules making up the
organic material breaking down into simpler molecules, etc.

Chaos theory (which has both a metaphorical usage as distinct from its
mathematical expression) has to do with the properties of a certain class of
systems defined by a set of differential equations. Differential equations
are mathematical constructs which define the rates of changes of variables.
For example, demographers use the equation (written in words): the rate of
population growth is proportional to the current population size, where "rate
of growth" means the number of individuals added over some unit of time
(e.g., the speed of a car is its rate of change of distance with time).
Systems are specified (mathematically) by several equations which define,
precisely, how all parts of the systems (i.e., the values of the variables)
change. For example, ecologists when talking about density limited
populations use an equation (again in words) that says: the current growth
rate is proportional to the current population size - a quantity which is
proportional to the current difference between the population size and the
carrying capacity population size. Such a system initially grows
exponentially (because of the growth rate proportional to the current
population size) but as it grows larger the growth rate slows down (because
of the term being subtracted from the current growth rate).

With this quick background to what constitutes a mathematical system,
mathematicians classify systems according to the type of equations used to
define the system. One group of systems is what are called linear systems.
These are systems where all of the equations are what are called linear
equations (roughly, no variable appears with an exponent greater than one and
there are no products of variables). The properties of linear systems (is
there an equilibrium value? if so, is it stable--i.e., if the system is
perturbed will it return to its equilibrium value?) have been completely
worked out.

THis leaves all other systems (those with non-linear equations) and this
class has NOT been well worked out in terms of its properties. While
specific examples of non-linear systems were investigated, as a class they
are very "messy" (hard to work with, mathematically) and exhibit not just an
increment of complexity over linear systems but can exhibit qualitatively
different properties.

Nonlinear systems started getting a lot of attention about 20 years ago (see
the book by Glick for a good, readbable account of how these systems started
to be investigated) because (a) it was realized that many real world systems
(such as the weather) cannot be adequately modeled with linear system models
and (b) the advent of PC's and easy graphic representations made it much
easier to examine the properties of these more complex systems; i.e., even if
you can't work out an exact mathematical answer, you can "run" the system on
a computer and have it draw a diagram that shows what is happening.

What was realized through this work was that there are certain non-linear
systems (and now called chaotic systems) that LOOK as if they exhibited
random behavior (the system never settles down to any particlar value, it
changes values in seemingly an unordered manner) yet are DETERMINISITIC; that
is, their behavior is precisely defined through their defining equations.
>From a mathematical perspective, chaos theory has to do with the mathematical
properties of non-linear systems which exhibit these chaotic-like properties
(which also includes the idea that a small change in the current state of
the system gets amplified and can eventually lead to a large difference in
the behavior of the system).

This links back to entropy indirectly through asking: If I look at a real
world system and it appears to be random in its behavior, is this an example
of entropy (orderliness decaying to randomness) or is the randomness ILLUSORY
and in fact is chaotic behavior, yet completely determinisitic (i.e., would
precisely repeat itself if started with precisely the same initial
values--note that this is only possible in the world of mathematics as all
real world systems can never have exactly the same values again and for
chaotic systems, even minute differences in values leads to very different
future states of the system, hence the argument about the butterfly affecting
the weather).

That's enough about the mathematical side. Chaos theory has also become a
metaphor for talking about systems even though those doing so have never
actaully demonstrated (and in many instances would completely reject) that
the system in question can be defined precisely via a set of differential
equations. As a metaphor, it becomes a bit slippery. Consider: The chaotic
deterministic system has the property that a small change in initial
conditions will lead to large differences in the state of the system. If we
look at social systems, many have argued that a small change in the social
system can lead to large changes in the future trajectory of that social
system. Is this the SAME phenomena as discussed by chaos theory
(taken in its mathematical sense)? Most likely those who discuss this kind
of social change would reject viewing the social system as a deterministics
system! So the term gets used metaphorically as if one is somehow invoking
some body of theory!

Because chaos theory has taken on this metaphorical role, there is MUCH
misunderstanding of what constitutes chaos theory as a mathematical theory.

Hope this hasn't been too vague.

D. Read