Re: Turbulent transition for fluids

From: Ken Muldrew (kmuldrezw_at_ucalgazry.ca)
Date: 12/03/04 

Date: Fri, 03 Dec 2004 18:48:39 GMT

glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) wrote:

>In article <41afa48d.14281924@news.ucalgary.ca>,
>Ken Muldrew <kmuldrezw@ucalgazry.ca> wrote:
>>spamspamspam3@netzero.com (Edward Green) wrote:
>>
>>>Sam Wormley <swormley1@mchsi.com> wrote in message
>>news:<eLwrd.429475$wV.279877@attbi_s54>...
>>>> Turbulent transition for fluids (Dec 1)
>>>> http://physicsweb.org/article/world/17/12/3
>>>> One of the first things most people do in the morning is turn on a tap
>>>> in the bathroom. Provided we are not too sleepy, we cannot fail to
>>>> notice what is often considered to be the greatest unsolved mystery in
>>>> classical physics: turbulence. If we turn the tap just a little, the
>>>> flow of water is smooth and laminar. However, as we open it more, the
>>>> flow becomes irregular and turbulent.
>>>
>>>
>>>In what sense is turbulence an "unsolved mystery"?
>>>
>>>I mean this seriously -- just what is it we would like to "solve" here?
>>
>>Well, for one, it would be nice to have a theory of irreversible
>>thermodynamics that could tell us the macroscopic evolution of
>>turbulent systems. Especially since the Navier-Stokes equation should
>>have everything there for a microscopic analysis. But somehow at high
>>Reynolds numbers the ability to predict dynamics rapidly diminishes
>>toward complete confusion and doesn't seem to give us many hints at
>>macro behavior (even though we might be able to see all kinds of
>>swirls and vortices that appear to be persistent patterns). So why
>>can't a complete knowledge of microscopic behavior be used to derive
>>some statistical knowledge of macroscopic behavior?
>
>Expanding on Edward's question, what would it mean for a problem in
>turbulence to be solved? Going for the whole enchilada-- a velocity field
>that correctly places each whorl and vortex? Or are they still working on
>more basic things like predicting a drag from first principles?

Let's distinguish between "a" problem in turbulence and "the"
problem in turbulence. For the former, the demands of the
solution are obviously dependent upon the particulars of the
problem. Many specific problems in turbulence are solvable; it's
just that these solutions can not be generalized to other
problems. "The" problem in turbulence refers to developing a
general statistical approach for performing inference on systems
that are far from equilibrium. Statistical mechanics gives us a
method for making inferences about dynamical systems that are
close to equilibrium, but so far nobody has been able to
generalize it to systems that are far from equilibrium. Questions
such as the minimum energy flow required to maintain a steady-
state of some macro variable, or even the path of approach to
equilibrium once that minimum energy flow is breached, remain out
of reach. This state of affairs is unsettling because we can, in
principle, already predict each whorl and vortex using the
Navier-Stokes equation. A statistical description of macro
behavior should be possible given a dynamical description of
micro behavior.

A few decades ago, everyone had their own pet theories of how the
dinosaurs became extinct. Now, a similar situation exists with respect
to chaotic dynamical systems, self organization, and turbulence.
Mindful of the egalitarian nature of unsolved problems, and in the
best tradition of vague and fuzzy reasoning that sci.physics is home
to, I will present my own take on this problem (suddenly readership
drops from 6 to 0!).

Let's just consider the phenomenon of self-organization in complex
systems. Here we find dynamical systems that appear to have no
organized structure within their macroscopic behavior, yet they evolve
to generate patterns (spatially, temporally, or both) without any
outside influence to create that order. Other than a flow of energy
through the system, there seems to be no information coming from
outside to create this order. It appears (though it is just an
illusion) that the system evolves from a state of high entropy to a
state of low entropy on its own. I suggest that this happens due to
the computational capacity of dynamical systems. A dynamical system
can perform computation to reduce the entropy of the system (ignoring
the energy that enters and exits an arbitrary boundary that we place
around the system), and likewise a dynamical system can evolve down an
entropy gradient (up actually, but it's easier to think of things
going downhill) to increase the computational capacity of the system.
With a sufficient energy flow, the cycle between high entropy/high
computational capacity and low entropy/low computational capacity can
continue indefinitely (steady state).

We're used to thinking of systems in terms of pure thermodynamics (a
system with no memory and limited energy) or of pure computation (a
system with infinite memory and energy). Although it's not too
difficult to consider a thermodynamic system out of equilibrium, we
don't often think of how memory could be used to affect the evolution
of the system. With computation, we're so used to working with
computing machines that are close to ideal Turing machines that
we rarely consider what computation would be like close to the
thermodynamic limits. A statistical theory of computation is
utterly foreign to most people who program computers; the
microscopic behavior is all there is.

I will suggest that in a statistical sense, all of computation
reduces to the compression of information. Further, the
computational complexity is defined as the degree to which
information contained within the system (integrated over the
history of the system) can be compressed. More specifically,
any system at a given time has an information content that can be
further compressed. If the information cannot be compressed then
the system has a computational complexity of zero (there is
nothing to be gained by further computation). This is similar to
Cruthfield's statistical complexity (cf. Computational
Mechanics) but simpler (and far less precise, it must be said).
In the spirit of Von Neumann, I will use the term "comptropy" to
describe the computational complexity of any system.

It's worth considering the graph of statistical complexity vs.
disorder at this point. It is an inverted U curve where we have
no complexity at the origin (with perfect order, the system is
trivial to describe) and again no complexity at complete disorder
(once again, the system is trivial to describe (statistically, or
macroscopically) when it is completely random). In the middle,
somewhere between complete order and complete randomness, we find
a maximum in the statistical complexity (where the information
required to specify the system is at a maximum). This curve is
useful for thinking about the two types of information
compression. First, information can be compressed by exploiting
statisticaly redundancies in the data. When this is done, the
system is pushed toward the highly disordered region of the
curve. When there are no more redundancies in the data (the
string is random) we have reached the limit of compression. The
second method consists of finding a pattern in the data. The data
are then discarded and the system is specified by the pattern
generator (and perhaps the first moment or two of statistical
spread about the pattern). This method of compressing information
moves the system toward the highly ordered region of the
complexity-disorder graph. When we have the simplest possible
pattern generator that completely specifies the data, then we
have reached the limit of compressibility (but two important
things about this limit: it cannot ever be reached if we are
restricted to gathering incomplete information about our system,
and as the limit is approached, the macroscopic description
approaches a perfect microscopic description). The statistical
complexity is also the computational complexity, because that is
where the system can be maximally compressed.

Computational work is done when the information in a system is
compressed. The important characteristic of pattern-based
compression schemes is that they can uncompress with no
computational work being done; in fact, work can be extracted
from the process. A system can perform thermodynamic work by
moving from a state of low entropy to one of high entropy, but
thermodynamic work can be used to perform computational work
(creating, maintaining, and erasing memory takes work).
Computational work can be used to reduce the entropy of a
thermodynamic system (by increasing the amount of information
known about the system). The lowered entropy then allows
thermodynamic work to be done. This is the essential method by
which low comptropy can be exchanged for low entropy. In systems
where infinite time is not available, energy dissipation is
necessary for both thermodynamic and computational work. Then
connection is confusing because energy is a property of the
microstate while both entropy and comptropy are properties of the
macrostate.

Conventional irreversible thermodynamics can only be used close
to equilibrium because the system is assumed memoryless (the
condition of ignoring the past is imposed on the analysis). Only
by including history can a fully non-equilibrium stat-mech be
developed, but here we need comptropy as well as entropy. When we
see persistent patterns in complex systems (so-called self-
organization), what we're really seeing are instances of
comptropy/entropy cycling where a steady flow of energy through
the system is being used to support the reversible conversion of
computational complexity for a well-specified system, and back
again.

Ken Muldrew
kmuldrezw@ucalgazry.ca
(remove all letters after y in the alphabet)

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