[Paleopsych] ruminations on entropy, order, life
Werbos, Dr. Paul J.
paul.werbos at verizon.net
Sun Oct 10 21:54:44 UTC 2004
Good morning!
Please forgive me if most of this email is more like rumination than
communication... but Howard has ruminated about entropy here before!
My excuse is that I am in something of a state of shock at this moment.
I was supposed to be in Quaker Meeting -- but the police waved me away from
it today.
And no, I did nothing wrong, no fines, no harsh conversations, just firmly
waved away.
Just a coincidence. Not like two weeks ago, when I made an accidental wrong
turn down an unmarked road
near the meeting - and got bombarded with sirens, fast police cars, and was
told I would be
arrested by the CIA if I ever made such a random mistake again.
Or the Inspector General doing strange things at a curious time... whatever...
So I will think of science for a moment.
There is lots of public noise and poetry about entropy.
Certainly I respect the folks who say we need more courses in "sciences for
poets."
And CERTAINLY we need more intuitive explanation before launching in,
say, to solutions to the Schrodinger equation. But sometimes the "science
for poets"
becomes so widespread and poorly marked (and devoid of clear abstract logic)
that it's hard to tell what is the science and what is the fuzz. Even
scientists get
fooled. And it end sup becoming dangerous and unwieldy in some ways.
So for example --
entropy is NOT disorder. Period.
"Entropy" basically has two closely related meanings, one in information
theory and
one in thermodynamics. As Eschel has said, the relations between the two
are interesting and important, but just for now let's think about system
dynamics.
In essence, "entropy" is basically just... logarithm of the probability of
a state.
That's still a simplification, but close enough for now. The entropy function
which applies to a system or universe depends on the specific dynamics of
that system or universe.
In some universes, the entropy function is strictly local --- just the sum
over space of some
local functions (local entropy functions) of local condition. In this case,
there is no correlation between the state at one point in space and the
state in another --
and in that case entropy does become something like a measure of disorder.
Crudely.
In systems like that, the probability distribution of states will converge
to a distribution
without correlations -- "disorder."
But not all systems are like that. Many of the classical "proofs" of
entropy being disorder
are based on elaborate simplifying assumptions or approximations. Take an
extreme
enough approximation, and you could say there is no life on earth
at all even today -- after all, MOST of the planet is nonliving. (OK -- sic
unless you are a
pantheist. But as a mathematical exercise, let us at least consider the
mathematical
possibility of an earth which is as it seems to the naive eye.) If we begin the
analysis by saying "let us assume, for simplicity, that life does not
exist. Now what
can we deduce about life?" -- well, of course we get nowhere.
So what, then, can we prove?
Years ago, I did scan through some of the work by Von Neumann on entropy
and such.
There are others who would know much better. But I noticed a key strategy.
First,
for any system, he would derive an "invariant measure," d mu, a kind of
null example
of one allowed equilibrium probability distribution for the state of the
system.
If ANY state in the system eventually gets to ANY OTHER state of the system,
the allowed distribution is unique, and that's it. BUT IF some regions in
state space are
disconnected from other regions, then... one can define variables which
represent
WHICH one of the disconnected regions one is in. For example, when energy
is conserved,
a state of energy E1 can never get to a state of energy E2, if E1 does not
equal E2;
energy E is the variable which distinguishes these states and energy does
not change with time,
in this example. In the general case, there is a set of conserved
quantities or "integrals,"
Q1 to Qn (possibly infinite!). All allowed equilibrium probability
distributions can be expressed
as d mu * f(Q1,...Qn), where ANY nonnegative function f is allowed. Thus --
the task
of understanding what can ACTUALLY happen, eventually, in any system or
universe,
is basically a matter of identifying (d mu) and Q1... Qn. To start.
The generalized Boltzmann probability distribution is still the foundation
for all practical thermodynamics,
under quantum theory as it was under classical theory... if by "generalized
Boltzmann distribution"
I mean the following. Usually, people look only at the function "f" and
consider
the particular case:
f(Q1,....,Qn) =
exp(-k1*Q1-k2*Q2... - kn*Qn)/Z,
where Z is just a "constant" we throw in to scale the probabilities so
that they add up to one. And usually they don't pay
a WHOLE lot of attention to "d mu" -- but sometimes in quantum
theory people are careful to note that "how many states are available
depends on whether it's a boson or a fermion," and that is part of the issue
of determining "d mu."
=============
At the end of the day... what we really have is a kind of taxonomy of
systems or universes, analogous in a way to the taxonomies of
types of control system and types of intelligent system fundamental to
the clear understanding of those fields. And the taxonomy may be viewed
as a chain of possible simplifications or approximations.
One VERY broad class of possible universes is a class... which might be
called "statistically flat." (I coin that phrase here and now.
Maybe others have another term for this; maybe not.)
A statistically flat universe, crudely, is one whose "d mu" is... well,
you could call it disorderly, like heat death. A product of local
"d mu" components -- such that the logarithm of d mu, the entropy,
really is a sum of local components.
Playing with the math... I see that all normal classical field theories
(based on a Lagrangian) are statistically flat. Likewise, all wave function
evolution theories based on the usual kind of dynamics
(loosely, "Schrodinger" or Louiville equations) are statistically flat.
An interesting corollary is that the statistical equilibrium distributions
predicted by (any bosonic) quantum theory are IDENTICAL to those predicted
by the corresponding
classical field theory -- as Einstein predicted but few believed -- if one
represents
the statistics of the classical field theory by used of "classical density
matrices"
as defined in quant-ph 0309031 and 0309087. (arXiv.org). And, according
to Vachaspati, there does exist a bosonic field theory in the SU(5) family
equivalent to the standard model of physics -- i.e. a theory which fits
every successful
prediction ever made by quantum field theory.
OK ... so ... so far as the standard model of physics goes,
we seem to be living in a statistically flat universe. The "natural" (d mu)
equilibrium probability distribution is indeed flat or disorderly.
(One may ask... what about local gradients... but you can use a series
of lattice approximations to prove that they change nothing, at least for
bounded fields like SU(5)). This seems to imply a much
stronger demonstration of entropy and heat death and the rest than
the usual thermodynamics provides!
-------
But...
There are some caveats.
First, Boltzmann universes are a subset of statistically flat universe.
They represent
a particular POSSIBILITY for the function f. Yet in actuality... our
universe presumably has
a certain level of Q1,...Qn in total (or per unit of volume of space). So
often enough
(i.e. in many possible universe states)... we may expect a Boltzmann
distribution should
fit in any case. Often -- but not always. If Qk can be negative OR
positive, things can become tricky.
But as a practical matter -- the Boltzmann distribution is really used as a
LOCAL probability distribution,
when we make the simplifying assumption that some PIECE of the universe is
essentially decoupled
from the rest of it. That's an important simplification. But if we think of
VERY BIG PIECES (like planets)...
it can yield better results than we get if we make believe that individual
atoms are disconnected from
other atoms.
--------
Anyway... we then get into some interesting questions.
Under what conditions can a Boltzmann universe support life? And how can we get
a rigorous representation for what happens in "open systems," like chunks
of the universe whose influx and outflux of energy and matter is too large
to neglect?
What does life "look like" at this level? To what extent are non-Boltzmann
(e.g. non-SU(5)) effects fundamental to the existence of even the most mundane
forms of life in this universe?
First -- the broader the context, the more we will find that the simple
binary distinction of
"alive" versus "not alive" becomes less and less meaningful. In the mathematics
of consciousness (which is far more developed than the mathematics of life,
and far more developed than most folks know)... the need to get beyond a binary
way of thinking was apparent long ago. (See q-bio 0311006.) With life,
it has not been apparent yet, because of the way our world tends to cluster.
But clones and semiartificial life forms and ideas about
machine replication have already begun to blur things.
In more general mathematics... the diversity is immediate.
In a Boltzmann universe, there are basically two loopholes which could
allow life.
(Roughly, as I ruminate this afternoon... ). One is the effect of the
Boltzmann term proper, the exponent with energy and other conserved quantities.
For example -- the energy term can favor condensed states like black holes
instead of
the uniform state that (d mu) by itself yields; maybe it could do more. The
other
is the "open system" kind of situation... which may exist on earth only
because of nonBoltzmann
effects ELSEWHERE, but still is important to understanding life on earth.
Regarding open system effects....
I was recently somewhat startled to hear of the theoretical and
experimental results
of Prof. Zoya Popovic of the University of Colorado. (The key paper was
http://nemes.colorado.edu/microwave/papers/2004/MTT_JHfhWMrzZP_Mar04.pdf)
At first, it seems like a total refutation of the second law of
thermodynamics as
we usually understand it. She got 20 percent efficiency, they say, in
extracting energy from
totally disordered electromagnetic radiation, in the frequency 3-18 GHz.
The temperature of her system was MUCH higher than the temperature of the
radiation.
How can something like THAT happen in a Boltzmann universe?
Furthermore -- if we accept the principle that such things CAN WORK, then
how do we know that they cannot work at 1000 times the frequency --
i.e. 3-18 THz? In principle, all it takes is to shrink the size of the
"spiral rectenna" to have feature size 1000 times smaller --
i.e. nanotechnology. And that is an option becoming very real very fast.
(Nano spiral rectennas would be very similar in flavor to metamaterials,
which have become very real and very solid and very useful and very strange.)
I have NOT CHECKED certain critical numbers AT ALL -- but I think I read
somewhere that the flux of electromagnetic energy at night in the 3-18THz
zone on warmish parts of earth is about 100 watts per square centimeter.
"Heat radiation" it is sometimes called. If one could extract 5-20 percent
of THAT, the implications would be truly monumental -- but is it possible?
Is even 1 percent possible, really?
Please forgive a bit of crazy science fantasy here. If it were 20 percent
(which I doubt) at 3-18THz... and if those crazy numbers I think I heard
were true...
then a 10 centimeter by 10 centimeter board
placed INSIDE a hybrid car somewhere could output 2 kilowatts... enough to
recharge the batteries over a 24-hour cycle without a need to plug in or
buy gasoline.
That would be really handy to have, as a safety measure in case of a 1979-style
gasoline shortage! Or to survive a prolonged war. I really doubt it... but on a
scale between 0 to 20 percent, I have no idea what we really could get here
on earth.
The implications are not entirely academic. It would be worth knowing.
It comes down to: what DOES the ambient 3-18 THz radiation on earth look like?
And how would the output of a nanospiralrectenna look as a function of
those characteristics?
Does it have variations we need to be aware of, over space and time?
I could **IMAGINE** two possible sources of properties that **COULD
CONCEIVABLY** allow nonzero
efficiency of a nanospiralrectenna. Again, nonlocal correlations
(coherence) CONSISTENT WITH a kind of partial Boltzmann equilibrium... and
open system effects.
The open system effects would be partly obvious. If we look around us in
the infrared
frequency, we don't see a homogeneous fuzz. We see bright spots and dark spots.
Angular variation should allow energy extraction, in principle. Yet I
wonder... how much
could this buy us, in reality, for a scaled-down version of the Popovic
antenna?
But then... is that the only effect? What of coherence length
(or even polarization?) effects? When we look at far-away stars, we do
see "blackbody light" -- but I think it has enough coherence ANYWAY that
we can do interferometry. Could the same possibly be true even of heat
radiation on earth?
In fact... such a scheme suggests that a partially coherent ... bath... of
ambient heat radiation...
would really be a legitimate "closed system" equilibrium -- APPROXIMATELY.
A kind of metastable state. Like the state of a forest full of dry wood and
oxygen.
And in this context... what life is... is a spark. It is the specific
improbable
local pattern/object/device... which is... an "unstable mode" of the large
system...
and such can exist to be sure in some universes. One species of life
then accomplishes something like a phase change... which create
a new metastable state, which creates a niche for yet another species, and
so on, in
a kind of endless progression ... from which emerges the kind of picture
that George Gaylord Simpson and E.O.Wilson have so well articulated.
(Though Wilson does tend to understate the role of learning at times.)
Who knows?
Best,
Paul
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