On Thu, Jan 9, 2014  Jesse Mazer <laserma...@gmail.com> wrote:

>I never claimed Liouville's theorem was a "fundamental law of physics" in
> itself,

Good, I agree.

> rather it is derivable as a mathematical consequence of certain features
> of the fundamental laws.

And of the initial conditions!

> Liouville's theorem is derived in deterministic classical mechanics.

Then Liouville's theorem can only be approximately true.

> > It [Liouville's theorem] only becomes statistical if you interpret the
> original set of microstates as representing your own uncertainty

But that's the only way you can interpret it because the laws of physics
insist that you will *always* be uncertain about the microstates, all you
know are purely statistical things about the system, like its temperature
and pressure.

> This line of discussion got started because I was disputing your
> statement that we can derive the 2nd law in a *purely* logical way like
> 2+2=5, with no need to invoke knowledge about the laws of physics that was
> based on observation. This would imply that *any* logically possible
> mathematical laws of nature would obey the 2nd law.

Yes, *any* logically possible mathematical law of nature must actually do
something, or it shouldn't be called a law. If the initial state of a
system is in a state of lowest possible entropy, and if one of those laws
goes to work on that state then the entropy of the system in that state
will NOT go down. And that is the second law of thermodynamics.

> If you did not mean to suggest that we can know a priori the 2nd law is
> true because it would be true in any logically possible universe whose
> behavior follows mathematical laws, please clarify.

That is exactly what I meant to suggest, provided that the initial
conditions were of very low entropy.

> But I thought you were talking about logically possible universes as
> well, not just our universe

If the initial conditions were of high entropy then applying a law of
physics to that mess would be just as likely to decrease its entropy as
increase it, therefore the second law would not be true and time would have
no arrow; in fact the very concept of time would have no meaning in that

> the very fact that you were willing to discuss the Game of Life suggested
> this, since even though it's possible our universe could be a cellular
> automaton, I think we can be pretty confident it's not a 2-dimensional
> cellular automaton like the Game of Life!

Well... you can make a Turing Machine from the Game of Life. And according
to the Bekenstein Bound the maximum amount of information that the laws of
physics allow you to store inside a sphere is NOT proportional to its 3D
volume as you might expect but is instead proportional to the sphere's 2D
surface area. So you could know all there is to know about what's going on
inside a sphere just by looking at its surface, this has led some to
propose what they call  "The Holographic principle", the idea is that the
entire volume of our 3D universe is a projection from a 2D surface.

 Maybe they're right. As I've said I don't know what reality will turn out
to be but whatever it is it's going to be weird.

>>> Another alternative would be to imagine you do have an infinite grid,
>> but with a starting state where there are only a finite pattern of black
>> squares surrounded by an infinite number of white squares,
> >> So the ratio of white squares to black is a finite number divided by
> infinity.

> No, because I said that in this case the region of the grid being
> *simulated* could still be finite

So the rules of the Game of Life apply to some of the cells in the grid but
do not apply to others. What rules govern which cells must obey the rules
and which cells can ignore the rules, that is to say who is allowed to
ignore the laws of physics in that universe?

  John K Clark

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