Entropy, Time's Arrow, and Urns

From: Tim May <tcmay.domain.name.hidden>
Date: Sun, 18 Aug 2002 13:00:08 -0700

Hal has brought up Huw Price's book, "Time's Arrow and Arhimedes'
Point," and especially the thermodynamic/entropy arguments related to
recurrence a la Poincare, Boltzmann, and others.

A point Price makes several times is th

"..though it needs to be borne in mind that not everyone had a clear
grasp of the fact that the low-entropy past is itself in need of
explanation." (p 37)

"In sum, the puzzle is not about how the universe reaches a state of
high entropy, but about how it comes to be starting from a low one." (p
40)

"For all their intrinsic interest, then, the new methods of nonlinear
dynamics do not throw new light on the asymmetry of thermodynamics.
Writers who suggest otherwise have failed to grasp the real puzzle of
thermodynamics--Why is entropy low in the past?--and to see that no
symmetric theory could possibly yield the kind of conclusions they claim
to draw." (p 44)

"There is no separate problem as to why entropy in branch systems always
increases towards the future, in other words: only the big problem was
in the bottle in the first place." (p 45)

And so on. Price repeatedly bases many arguments on his dissatisfaction
with assuming a state of high order. (To be fair, his book is an
interesting romp through many theories of time asymmetry, touching on
Feynman-Wheeler absorber theories, delayed choice experiments in quantum
mechanics, psychology, etc. Not a bad place to get exposed to a lot of
the current issues. Just don't take his particular "crotchet" too
seriously, is my advice.)

Frankly, I don't worry how the beer got in the bottle (one of his
example, about gas expanding out of a beer bottle...Price worries that
the analysts of time are not asking proper questions about how the beer
came to be in the bottle in the first place...most of us, dullards that
we are, assume that a bottling company _put_ the beer in the bottle!.

I'm not being flip. It's an observed fact of our universe, and likely
derivable from anthropic arguments, that there's a lot of "free energy"
around: a lot of unfused hydrogen, a lot of gravitational potential
energy, a lot of stored chemical energy, etc. How this came to be from
"first principles" from an initial singularity is of course unknown at
this time.

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 about 250 black stones.

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.

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.

I don't _know_ where the free energy or high initial order of the
universe came from. But I sure don't believe that 10^80 or more
particles in the Universe, following 10^10^10^10^10^.....10 different
ballistic trajectories--not even counting Planck-scale effects, will
"every now and then fluctuate into this low entropy configuration that
is us." (my phrasing, not Price's)

But there are other interesting things in the book, more so than in many
other books on time and time asymmetry. As I said, a good read.


--Tim
Received on Sun Aug 18 2002 - 13:09:19 PDT