Thanks for a very informative post! Of course you are right about
Liouville's theorem, and the fact that only dissipative systems can
evolve to a fixed point independent of initial conditions. Regarding
what you wrote about driven dissipative systems:
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The For *dissipative* systems then such behavior is the norm.
Any pendulum with friction will eventually end up hanging down still. This
state is the stable fixed point attractor for the system and all initial
states converge to it. *Driven* dissipative systems are more colorful.
Things can be attracted to sets in phase space more complicated than a single
fixed point like the damped pendulum. Various limit cycles, bounded surfaces,
etc., are possible for the motion to converge to. A *strange* attractor is
a an attracting set in phase space such that the motion converges to it for
a wide range of possible initial conditions, but this attracting limit set in
phase space is itself a rarified fractal and spread out all all through a
large region of phase space. Once the motion settles down onto the strange
attractor the motion still looks chaotic and is still very sensitive to slight
perturbations of the state as the system state seems to wander irregularly
throughout phase space (yet is confined to the attractor).
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I have a question: is it possible that someday a cosmology could be developed
which involves an analogous situation? In other words, that no matter what
the initial conditions (or for perhaps a broad range of possible conditions) a
universe capable of supporting life would be the result? I suppose this
would depend on whether the universe is a "driven, dissipative system". It
isn't, is it?
You also wrote, about superstring theories and boundary conditions:
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Superstring TOEs show more ultimate promise (not performance). They have far
fewer (hardly any) undetermined parameters and they have both GUTs and
quantized gravitation as appropriate limiting cases. It recently has been
shown that what formerly were thought to be different classes of superstring
theories are in fact just hidden versions of the same theory written in
different terms. It may be possible that there really is only one fully
self-consistent theory with the power to potentially explain "everything". In
principle, superstring theories predict the free parameters of the other more
well-known theories of particle physics. In practice they are so intractable
and complicated that they can't be solved sufficiently to hardly make a single
hard quantitative numerical prediction about such things.
Regarding Stan's comment about initial conditions, I think the hope of the
people in the field is that the ultimate theory will be so naturally
constrained by the requirements of maximal beauty, simplicity, and self-
consistency that only one boundary condition will be possible or compatible
with it. This is something like the motivation for Hawking's no-boundary
proposal for theories written in imaginary time. The idea is to have the
boundary condition as a natural part of the theory.
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Can you elaborate on this last point? I've heard this before but don't
understand how the boundary condition could emerge as a "natural part of
the theory". Mathematically they are logically independent, aren't they?
Thanks again,
Stan Zygmunt
Valparaiso University