On Thu, Jan 9, 2014 at 3:58 PM, John Clark <johnkcl...@gmail.com> wrote:

> On Thu, Jan 9, 2014 at 2:02 PM, Jesse Mazer <laserma...@gmail.com> wrote:
>
> > And as I've said, there is also the fact that if the laws of physics
>> don't conserve phase space volume, the 2nd law wouldn't hold either.
>>
>
> You've got it backwards, there is no fundamental law of physics concerning
> the conservation of phase space that forces matter to behave in certain
> ways, rather it's just a natural consequence of the FIRST law of
> thermodynamics and the statistical fact that if you make a change in a
> highly orders system you will probably make it more disordered because
> there are far fewer ordered states than disordered states.
>

I never claimed Liouville's theorem was a "fundamental law of physics" in
itself, rather it is derivable as a mathematical consequence of certain
features of the fundamental laws. What I've read indicates that Liouville's
theorem applies to any system that obeys Hamilton's equations (see the last
paragraph on p. 549 of Taylor's "Classical Mechanics" at
http://books.google.com/books?id=P1kCtNr-pJsC&pg=PA549 for example), but
I'm not sure if it's true that any logically possible laws that conserve
energy (obeying the first law of thermodynamics) also obey Hamilton's
equations...the Hamiltonian is not always equal to the total energy, see
http://physics.stackexchange.com/questions/11905/when-is-the-hamiltonian-of-a-system-not-equal-to-its-total-energy




> Liouville's equation is all about statistics, the variables in it
> determine the phase space distribution and that determines the PROBABILITY
> a system of things will be in a particular infinitesimal phase space volume.
>

Liouville's theorem is derived in deterministic classical mechanics. If you
take a volume of phase space, each point in that volume is a specific
microstate, and if you evolve each microstate forward for some time T using
the deterministic equations of physics, you get a later set of microstates
which occupy their own volume in phase space. Liouville's theorem just says
the two volumes must be equal. It only becomes statistical if you interpret
the original set of microstates as representing your own uncertainty--if
you just know the original macrostate, you may choose to consider the
statistical ensemble of microstates compatible with that macrostate, then
they will give the "volume of phase space" that you start with. But that's
just an extra layer of interpretation, Liouville's theorem itself is not
really statistical.


>
> >>> For example, in Life one could define macrostates in terms of the
>>>> ratio of white to black cells [...]
>>>>
>>>
>>> >> In the Game of Life the number of black cells is always infinite, so
>>> I don't see how you can do any ratios.
>>>
>>
>>
> > Maybe that would be true for some ideal Platonic version of the Game of
>> Life on an infinite board, but any real-world implementation of a cellular
>> automaton involves a finite number of squares
>>
>
> Maybe not. The universe is certainly a real world implementation and it
> might be infinite and it might be a cellular automation, that's what
> Stephen Wolfram thinks.
>


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. So the question of whether space in
*our* universe is infinite or finite is irrelevant to the discussion,
because it's certainly logically possible to have a universe with finite
space.

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



>
> > usually this is done with a periodic boundary condition, so squares on
>> the left edge of the finite grid are defined to be neighbors of squares on
>> the right edge, and squares on the top edge of the grid are defined to be
>> neighbors of squares on the bottom edge.
>>
>
> Then the rules governing the game have been changed.
>

I think most any book or website that defines the "rules" of the Game of
Life will just state the transition rules for how each cell's state depends
on the state of that cell and its nearest neighbors on the previous
time-step, they don't say anything about whether the topology of the board
is that of a torus (which is topologically equivalent to a square with the
edges identified in the way I described, as discussed at
http://plus.maths.org/content/space-do-all-roads-lead-home ) or an infinite
plane. But really this is just a matter of semantics, if you want to say
that your personal definition of the "Game of Life" includes the notion
that the board has the topology of an infinite plane that's fine with me,
we can just come up with some new term like "Game of Toroidal Life" for the
periodic version I described. If you were claiming that any logically
possible universe obeying mathematical rules would respect the 2nd law, the
"Game of Toroidal Life" would still be a counterexample.



>
> > 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 (areas beyond the region with black
squares are guaranteed to stay white until the black region reaches them,
so there's no need to explicitly simulate them), and then I said "the ratio
of black squares to white squares *on the simulated grid region* at any
given time is well-defined, so one can use this ratio to define the
macrostate". This is no different than defining the entropy of an isolated
finite system in an infinite universe by looking at all the possible
configurations within a finite volume of space large enough so that the
system lies wholly inside it, and ignoring everything outside that volume
(which should be fine if the system is truly isolated from outside
influences).

Jesse

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