

The enigma of the Ford paradox - mef
https://scottlocklin.wordpress.com/2013/03/07/the-enigma-of-the-ford-paradox/

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Strilanc
Summary of the Ford paradox (which the post takes forever to actually get to):
Quantum mechanics has only periodic orbits, but chaotic systems aren't
periodic. How do we reconcile this with the existence of, say, double
pendulums?

My guess: because the abstract double pendulum is a model that only
approximates real double pendulums. Analogously, computers based on Turing
Machines exist despite Turing Machines requiring an infinite amount of space.

\---

Also, a minor nitpick: the post seems to equate 'not periodic' or 'chaotic'
with 'is random', which I think is misleading. Pseudo-random, I guess, but the
output of a simulation of a chaotic system is highly compressible and thus not
random in the information theoretic sense.

~~~
vannevar
_...the output of a simulation of a chaotic system is highly compressible..._

Is that really true? Losslessly? Do you mean the entire phase space, or the
sequence generated on any given run? Because I was under the impression that
what distinguishes a chaotic system is that the output cannot be predicted in
any way other than running the simulation (the article uses the word 'random',
but it might be better to say 'unpredictable'). Compressibility relies on
exploiting regularities, and regularity is predictability, isn't it?

~~~
Strilanc
See: Kolmogorov Complexity (
<http://en.wikipedia.org/wiki/Kolmogorov_complexity> )

The program that simulates the chaotic system is shorter than the unbounded
amount of output being generated, and counts as a compression of the output.

~~~
Gravityloss
Sure, but that is kind of circular.

I guess in other words, the more interesting question is this: can we generate
hard to compress pseudorandom sequences with simple algorithms - if the
algorithm description by itself can not be used as "compression"?

~~~
MostAwesomeDude
Absolutely. Consider any number which can be expressed as a patterned
continued fraction, like pi or e. pi's digits are such that it is _very_ hard
to compress, but it is absolutely trivial to generate.
(<http://en.wikipedia.org/wiki/Continued_fraction>)

~~~
AnthonyMouse
>Consider any number which can be expressed as a patterned continued fraction,
like pi or e. pi's digits are such that it is very hard to compress, but it is
absolutely trivial to generate.

You're confusing theoretical compressibility with the actual compression
ratios you get when you put a string through typical compression functions.

Here's a different example. Suppose you have ASCII text encrypted with AES
with the AES decryption key appended to it. Run that through gzip and you'll
probably get output which is _larger_ than the input. But here's a different
compression algorithm that will work much better: Decrypt the encrypted data
with the key and then compress the plaintext with gzip.

It isn't that the original string is of the sort that can't be compressed,
it's that you need a compression function which is suited to the input. Pi is
like that.

Compare this to other strings, like the output of an environmental hardware
random number generator, which will almost always be totally incompressible
whatsoever.

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lukev
My math and physics is pretty weak, but it seems to me that this assertion is
wrong:

> Classically chaotic systems generate information over time.

Really? Simply being chaotic does not preclude a system from being
deterministic. Every future state of the system is "present" given the initial
conditions, even if it isn't predictable.

The situation is analogous to that of the digits of irrational numbers. I
can't tell you a priori what the 2^1000th digit of pi is, but if I calculated
for a million years I could find it out. It's not being "generated", it's just
as much a part of pi as 3.14 is, just a little harder to access.

There may be true non-determinism in nature, but it isn't necessary for a
system to be considered chaotic.

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q_revert
anyone who found the discussion of dynamical systems here interesting would be
well served by having a look at <http://www.scholarpedia.org/>, which has some
very good articles on the area

[edit]
[http://www.scholarpedia.org/article/Encyclopedia_of_dynamica...](http://www.scholarpedia.org/article/Encyclopedia_of_dynamical_systems)

~~~
nnq
you just linked to the main page!

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simonster
I'm not a physicist and don't remember QM all that well, so feel somewhat
uncomfortable trying to comment on this, but I'm not sure I understand why
this is a paradox rather than an intriguing curiosity. There is a lot of work
out there on how chaos can be produced from order. Stephen Wolfram was
famously obsessed with how simple, deterministic cellular automata can produce
complex, chaotic behavior. There is an interesting problem here, but it seems
to violate our intuitions rather than actual physical principles.

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3pt14159
The answer might be stuffed in the "this system will _likely_ shut down prior
to generating more information than the sum of the information in the
individual subatomic particles."

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jheriko
Doesn't this miss something important about entropy and information creation?
From what I understand the classical system isn't producing information...

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lolcraft
Well, let's say you measured the times at which the bottom pendulum crossed
the median. You would get a list of times that were spaced here and there,
some near others, some far. This would be one possible set of information that
the pendulum produces.

The idea is that, presumably, this list of times "looks" very random and can't
be expressed by a much shorter, compressed, set of data, _not even by
providing the pendulum equation, and initial conditions with finite
precision_. (You could, if it weren't chaotic, just a single pendulum.) This
would mean it has high entropy content, in the information theory sense. (I
don't actually know, since I haven't done the experiment nor read about it
being done, but it's a reasonable guess.)

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pfortuny
There is something I do not get: the behaviour of the macroscopic pendulum
depends as well on random quantum events (f.e. the emmission of a photon or
whatever). These are not deterministic nor linear nor "periodizable".

Am I missing something?

I guess this is relevant to the problem as well.

~~~
Gravityloss
I think that's why he was proposing tiny pendulums where those effects can be
excluded.

~~~
pfortuny
Yep, but then it would be a thought experiment and... what about half-life in
that case?

But thanks for the pointer.

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graycat
When I was a senior in college, I went through the math of the double
pendulum. The subject is mostly just ordinary differential equations although
there is a cute role for a little matrix theory.

Generally it seems to me that the OP gets off into some not very well defined
and not very relevant topics and, instead, for the 'chaos' he is observing
there's a fairly easy explanation: The system is unstable. Or, in one step
more detail, the system really is an initial value problem for an ordinary
differential equation, but, going way back to Bellman's work on stability
theory, it's long been well known, just from the equations, that the solution
can be 'unstable', that is, small changes in the initial conditions (values)
can result in large changes in the solution. And that is just from ordinary
differential equations without considering quantum mechanics. And for the
'chaos' in the OP, that's about all the explanation that is needed.

Why? Because there is really no chance that the motion of the system could be
periodic or even simple because it nearly never gets back accurately enough to
an earlier state. So, the system often gets back to something 'close' to an
earlier state, but the system is so unstable that 'close' is not close enough
so that the earlier state and the present one close to that earlier state soon
result in very different solutions for the future.

Something similar happens with pseudo random number generators, e.g., the
usual linear congruential generators where we set

R(n + 1) = ( A * R(n) + B ) mod C

for n = 1, 2, ..., and R(1) some positive integer. Then roughly the R(n)/C are
independent, identically distributed, uniform on [0,1). One of Knuth's
recommendations (in one of the volumes of TACP) was A = 5^15, B = 1, C = 2^47.
So, a point here is, get such 'random' numbers without considering phase space
or quantum mechanics.

In a sense, this is a very old point: E.g., one dream before about 1900 was
that we could observe the present state of the world and, then, use
deterministic physics to predict the future. So, as I recall, it was E. Borel
who did a calculation and concluded that a change of moving 1 gram of matter 1
cm, or some such, on a distant star would invalidate predictions on earth
after just milliseconds (presumably starting after the travel time of light
from that star to the earth).

We suspect we see much the same in weather prediction: Small changes in
initial conditions too soon make changes large enough to switch between rain
and sunshine. The usual joke is that a butterfly could flap its wings and
convert a clear day to a hurricane.

We anticipate that probabilistically weather prediction is quite stable, that
is, what is stable is the conditional probability distribution of the
variables we use to measure weather conditioned on the present. In particular,
we still believe in the law of conservation of energy.

Also we should notice the classic work on ergodic theory, by Hopf, Poincare,
Birkhoff, etc.: The standard illustration is pouring cream into coffee and
stirring. Then the theorem says that, if stir long enough, then can make the
cream separate from the coffee back to as close as please to the original
state. Why? Because if take the 'volume' of the possible states at some point
in time and then let time pass, then the 'volume' of those states is still the
same. So, as the system evolves, it is 'measure preserving' in state space.
So, if want to apply this to a frictionless double pendulum, then can get it
to return as close as we please to its initial state, but between then and now
it is free to do a lot. This stuff goes back to the first half of the last
century.

Likely there are some interesting and important questions in chaos theory, but
what the OP is saying about the double pendulum seems to have a simple
explanation.

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DelvarWorld
Is it worth it to try to figure out what this article is about? It's very non-
accessible writing and I can't seem to find his point nor do I see a reason to
care about it from scanning.

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Nursie
Never tried the Ford Paradox, quite like Mustangs though...

~~~
brostorycool
FUNNY

~~~
Nursie
I know, lame joke, but I'm not sure it deserves a -4!

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skorgu
It's the fact that jokes like that end up at -4 that keeps me returning to hn.

