Re: Entropy, Time's Arrow, and Urns

From: Jesse Mazer <lasermazer.domain.name.hidden>
Date: Sun, 18 Aug 2002 19:56:22 -0400

Tim May wrote:

>Time for a digression. The classic urn experiment, with Price's objections.
>
>And let me throw in something several members of this list will likely
>appreciate: a bet on the outcomes (a la Bayesian reasoning, a la market
>processes, a la Robin Hanson's idea futures, a la probabalistic definitions
>of the truth).
>
>Imagine two urns. Imagine, say, 500 black stones and 500 white stones. A
>person is reaching inside one urn, removing a stone, transferring it to the
>other urn, picking up a stone "at random" (a regrettably loaded term, but
>one which will hopefully become clearer...imagine that the man is blind and
>cannot possibly see the color of the stone he is picking up).
>
>A group of people is show two filmed sequences:
>
>In Sequence One, the two urns are filled with stones of mixed color at the
>start of the film. As the main transfers stones, the number of black and
>white stones in each of the urns fluctuates, but there are never, in this
>particular film, any excursions outside the ratio 450 of one color to 550
>of the other.
>
>In Sequence Two, the the film begins rolling with one urn filled with white
>stones and the other urn filled with black stones. The man reaches in,
>takes a white stone, transfers it to the other urn. He reaches in, takes a
>black stone, transfers it to the first urn. As the film progresses, the two
>urns eventually reach a state where each has about 250 white stones and
>
>The group is told that one of the films is presented in correct
>chronological order while the other is presented in reverse chronological
>order.
>
>The group is told that bets will be taken on which is which. Oddsmakers are
>standing by. A terminal linked to the Idea Futures Market is available.
>
>(The Barbourian jumps up and yells "There is no time! All events happen at
>the same time!" The organizer says "Fine, but I'm still taking bets. The
>Barbourian sits down.)
>
>Which way would you bet? And what do think oddsmakers would make odds?
>
>Not surprisingly, nearly everyone will bet that Sequence Two was shown in
>correct chronological order and that Sequence One, if one of the two
>sequences was shown in reverse chronological order, must have been the one
>that was reversed.
>
>(The Quibbler points out that Sequence One could easily be shown in
>chronological order, just either a long time after the mixing started or
>starting with an initially mixed set. "Sure," the organizer points out,
>"but I told you one was in correct order, one was reversed, so place your
>bets.")
>
>Now the urn example is one that does not use the "molecular chaos" that Huw
>Price is so critical of in "gas mixing" examples, arguing that "molecular
>chaos" is assuming the conclusion.
>
>Here we have a simple, discrete, mixing problem.
>
>Is urn mixing "reversible"? Sure, though very, very unlikely. Unless the
>man doing the blind mixing (not looking at the color, or unable to see the
>color, etc.), the odds of a reversal back to the original 500 white/500
>black configuration is a simple matter of odds calcualations--on the order
>of 2^(-500).
>
>Now the Priceian would argue, from what I have read of his book, that while
>he agrees that nearly all people would correctly bet which of the films was
>time reversed and which was not, that they are missing the point, that the
>real issue is how the urns came to have 500 white balls in one urn and 500
>black balls in the other urn in the first place.
>
>Feh. This crotchet, this hangup, is one of the main reasons I can't get
>overly excited by Huw Price's analysis of time.

It's true that if one just wants to study the physics of how the world works
now one can just take for granted a low-entropy initial singularity and not
worry about why. But I think if one wants a complete cosmological theory
that's a pretty big issue you'll want to explain. It all depends on your
point of view.

I agree that Huw Price would say the real issue in your scenario is how you
got the balls segregated in the first place. If his point of view is
correct, the only reason you see it as "natural" is because you are used to
living in a universe with low-entropy initial conditions which allows for
the existence of things like intelligent beings who are able to set up such
ordered initial conditions. Consider a variation on this--say the video
features two types of molecules instead of two types of balls, and instead
of urns we're just dividing the screen into two halves and looking at how
many of each type appear on each side. Again, if the video was shot in our
universe then we'd expect the scenario with molecules starting out perfectly
segregated and then gradually mixing as they bounce around must be the one
shown normally while the one where they start about equally mixed and end
about equally mixed was shown in reverse. But suppose we then find out the
video was taken from a parallel universe which was in a state of maximum
entropy, with no low-entropy initial conditions. In that case, there'd be no
way to tell which video was shot in reverse; either way, the video showing
the molecules "starting" segregated would represent a statistically unlikely
fluctuation in entropy. Or what if the video was taken from a parallel
universe which we knew had a *future* low-entropy boundary condition, but no
such boundary condition in the past--then we'd bet that the video showing
the molecules going from segregated to mixed was actually the one shot in
reverse! A priori there is no reason why a universe with a future
low-entropy boundary condition should seem any more or less weird to us then
one with a past low-entropy boundary condition; for now we are equally
clueless about how to explain either one in terms of known principles of
physics.

>And I certainly don't draw the same conclusions he draws about how the
>initial low entropy state of such systems must come from Poincare/Boltzmann
>recurrences.

If I remember correctly Price did *not* suggest low-entropy initial
conditions could be explained by Poincare recurrence--in fact I think he
raised the exact same objection to this possibility that you did, pointing
out that the anthropic principle would only guarantee the minimum entropy
fluctuation needed to generate an intelligent observer, which is tiny
compared to the fluctuation you'd need to explain a universe like ours with
low entropy as far as the eye can see (and even lower entropy in the past).
I think Price's suggestions were:

1. new time-asymmetric laws of physics

2. a "Gold universe" with time's arrow reversing during the contraction
phase so the big crunch is a mirror to the big bang (Greg Egan wrote a cool
short story about how living in such a universe could allow you to send
messages into the past using light from stars in the contraction phase)

3. laws which guarantee that one end of the universe will be low in entropy
while the other will be high, but are equally likely to lead to a
low-entropy "future" condition as a low-entropy "past" condition (though
observers in either type of universe will call the low-entropy end the
'past').

Incidentally, J. Richard Gott (who I think might have come up with the
Doomsday argument, I'm not sure) in his book "Time Travel in Einstein's
Universe" makes another interesting suggestion about how to explain
low-entropy initial conditions. Combining the idea of wormholes spawning
"baby universes" with the idea of wormholes as time machines, he suggests
that the Big Bang might be such a baby universe spawned my a wormhole which
originates in its own future history! As it turns out this "self-creating
universe" model seems to naturally imply no advanced electromagnetic waves,
only retarded ones, which is equivalent to saying that entropy must decrease
in the past direction, not the future one. Quoting from p. 195-196:

"The geometry of our time-travel model provides a natural explanation for
the asymmetry between the future and the past that we observe in our
universe. Suppose we live in the universe represented by the farthest right
horn in Figure 27. [he refers to a diagram of a self-creating universe where
the 'farthest right horn' is another baby universe that spawned from the
original self-created universe] If we allowed light waves to go to the past,
they would work their way down this branch to the branch to its left and
eventually to the trunk, where they would enter a time loop at the bottom
and circle the time loop counterclockwise an infinite number of times,
leading to an infinite buildup of energy and causing the whole structure to
blow up, creating a singularity. That is not the geometry we started
with--the solution is inconsistent. The only way for a self-consistent model
to work is if light waves always travel toward the future, just as we
observe. (If photons created in the branches travel only toward the future,
then these photons travel out the branches away from the time loop, creating
no problem.)

"Now consider a photon emitted within the time loop at the bottom. It could,
in principle, circle the loop clockwise an infinite number of times. But
each time it went around, it would lose energy because it would be traveling
toward the future, in the same direction that the branch is expanding. Each
time it circled, it would add only one 535th as much energy as on the
previos circuit because the expansion stretches its wavelength by a factor
of 535, robbing it of energy. The sum rapidly converges to a finite value.
So, even though it circles an infinite number of times, it would not cause
an infinite buildup of energy. However, a photon going backward in time
(counterclockwise) around the loop would pick up energy on each circuit
because in the counterclockwise direction the branch is always getting
smaller, compressing its wavelength. A photon circling an infinite number of
times toward the past would cause an infinite buildup of energy, causing the
model to blow up. In fact, the only way to obtain a self-consistent solution
is to have light waves, and gravitational waves, travel only to the future
throughout the entire model. Thus, in our model, the asymmetry between the
future and the past that we observe (in which causes precede effects) comes
from the time asymmetry in the geometry of the Universe--it has a time loop
in the beginning.

"This arrow of time was not something we built into the model; it was
implicit in the model, but its emergence quite surprised us. It's an
important prediction by the model, which turns out to agree with our
observations.

"In the standard big bang model, by contrast, there is nothing to produce an
arrow of time. In that model, the early universe is filled with radiation,
and whether it is going forward or backward in time from its source does not
matter. Waves going toward the past would increase in energy as they
approached the big bang singularity, where they would blow up. But the
density in the big bang model blows up there anyway, so it causes no
problem. Waves going to the past are not forbidden, in principle, in the
standard big bang model. But with a time loop in the beginning,
self-consistency forbids waves going to the past--just as is observed
today."

Gott then goes on to show how the entropy arrow of time can be derived from
the electromagnetic arrow of time, based on the fact that it implies higher
temperatures as you go back in time. Anyway, this probably is not the most
likely answer to the arrow of time question (there are a few other
contenders), but it'd be pretty cool if it turned out to be true.

--Jesse

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Received on Sun Aug 18 2002 - 17:00:57 PDT

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