Good piece [LWN.net]

Good piece

Posted Nov 29, 2012 18:33 UTC (Thu) by Cyberax (✭ supporter ✭, #52523)
In reply to: Good piece by davidescott
Parent article: LCE: Don't play dice with random numbers

I don't know what they're smoking but I definitely want some.

1) "Space invaders" scenario is perfectly deterministic, they just have a problem with incorrect abstraction. In particular, treating gravity as infinitely fast. If we instead treat gravity as a classical field with finite interaction propagation speed, then this paradox disappears. Ditto for special relativity (it has its own problems with singularities, though).

2) Even if we allow the "space invaders" it still does NOT mean the system is not deterministic. If we know all the details about the system then we can predict its behavior with arbitrary precision. That knowledge, of course, must include all the infinitely fast objects. Laplace demon is still able to predict everything with arbitrary precision.

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Good piece

Posted Nov 29, 2012 20:45 UTC (Thu) by man_ls (guest, #15091) [Link] (4 responses)

You are aware that Laplace's demon (as Maxwell's) is nothing short of God-like in nature. And that its very nature was refuted (as seen in Wikipedia) by the second law of Thermodynamics:

According to chemical engineer Robert Ulanowicz, in his 1986 book Growth and Development, Laplace's demon met its end with early 19th century developments of the concepts of irreversibility, entropy, and the second law of thermodynamics. In other words, Laplace's demon was based on the premise of reversibility and classical mechanics; however, under current theory, thermodynamics (i.e. real processes) are thought to be irreversible in practical terms (compared to the age of the universe, for instance).

Apparently this last part has not been stated convincingly enough since I brought it up several comments up, and it is indeed hard to grasp. The consequence is that no entity can predict what will happen in a system with absolute precision, not even a theoretical demon, not even in theory. In theory the amount of information that can be stored is always limited, so that (to state it in modern terms) below the noise level the signal is unpredictable. And the noise level cannot be made arbitrarily small.

Good piece

Posted Dec 4, 2012 22:31 UTC (Tue) by chithanh (guest, #52801) [Link] (3 responses)

> You are aware that Laplace's demon (as Maxwell's) is nothing short of God-like in nature. And that its very nature was refuted (as seen in Wikipedia) by the second law of Thermodynamics:

And the second law of thermodynamics has been refuted by Poincaré recurrence.

http://en.wikipedia.org/wiki/Poincaré_recurrence_theorem

Good piece

Posted Dec 5, 2012 8:35 UTC (Wed) by man_ls (guest, #15091) [Link] (2 responses)

Hardly a rebuttal since this theorem and the second law operate on different time scales. Boltzmann had a similar argument for the whole universe; it only made the second law stronger.

Whenever something contradicts the second law, that something loses. But if you feel better with a recurrence every 10^10^100... seconds or so, then so be it.

Good piece

Posted Dec 5, 2012 21:25 UTC (Wed) by mathstuf (subscriber, #69389) [Link] (1 responses)

> recurrence every 10^10^100

And don't fret, those 10's are rounded up from e, so it's not as long as it might seem (though I think you're missing few stacks and should be 10^10^10^10^10^1.1).

Good piece

Posted Dec 5, 2012 21:38 UTC (Wed) by man_ls (guest, #15091) [Link]

You are right, I added the ellipsis on a Googolplex because I could not find any good estimate at the moment. Now I have: in the Wikipedia no less, and it is as you say. Saying it is bigger than the age of the universe is a bit of an understatement. So don't hold your breath for a recurrence.

Besides, it would only happen if our universe is an energy-conserving system; probably just changing size would break the recurrence.

Finally, even if a recurrence was possible in an expanding universe, it would just leave us at the starting point; not just diminish the entropy.

Good piece

Posted Nov 29, 2012 21:13 UTC (Thu) by davidescott (guest, #58580) [Link] (4 responses)

> In particular, treating gravity as infinitely fast. If we instead treat gravity as a classical field with finite interaction propagation speed, then this paradox disappears.

You are moving the bar and no longer talking about classical mechanics.

We know classical mechanics is not the correct theory, the point is that one of the ways it is incorrect is that it allows for mathematical singularities in its solutions. If you start slapping restrictions on the theory: "finite force propagation, energy and speed limits, regularity of solutions, continuity of fields, etc...." to prevent those singularities from appearing then it is no longer classical mechanics. It is something else.

> If we know all the details about the system then we can predict its behavior with arbitrary precision. That knowledge, of course, must include all the infinitely fast objects.

Again you aren't talking about Classical Mechanics. Your Demon wants to predict what will happen to the ball at time t_2, and wants to make that prediction based on the state of the universe at time t_0. Suppose that the correct prediction is "the ball does not exist at time t_2, because an Invader appears at time t_1 \in (t_0, t_2) and vaporizes the ball before t_2." If your Demon is capable of making such a prediction then he/she must be able to:

Express in the notation of classical Newtonian mechanics the position and velocity of the Space Invader at time t_0, so that I can deterministically show that the invader MUST begin to slow at time t_1 and destroy the ball prior to time t_2.

You CANNOT do so because v=infinity and x=everywhere (or x=emptyset) is not a valid expression of position and velocity of a body in Newtonian Mechanics. In Newton's formulas, the infinitely fast object DOES NOT EXIST within the classical universe, but his formulas allow it to be the finite-time limit of a classical process.

Good piece

Posted Nov 29, 2012 21:36 UTC (Thu) by Cyberax (✭ supporter ✭, #52523) [Link] (3 responses)

>Again you aren't talking about Classical Mechanics. Your Demon wants to predict what will happen to the ball at time t_2, and wants to make that prediction based on the state of the universe at time t_0.
Yup.

>Suppose that the correct prediction is "the ball does not exist at time t_2, because an Invader appears at time t_1 \in (t_0, t_2) and vaporizes the ball before t_2." If your Demon is capable of making such a prediction then he/she must be able to
It must be able to integrate equations of motion. That's all.

>Express in the notation of classical Newtonian mechanics the position and velocity of the Space Invader at time t_0, so that I can deterministically show that the invader MUST begin to slow at time t_1 and destroy the ball prior to time t_2.
Wrong. "Space invader" exists only at ONE point - it has infinite speed and can't slow down.

>You CANNOT do so because v=infinity and x=everywhere (or x=emptyset) is not a valid expression of position and velocity of a body in Newtonian Mechanics. In Newton's formulas, the infinitely fast object DOES NOT EXIST within the classical universe, but his formulas allow it to be the finite-time limit of a classical process.
Newtonian mechanics has no problems with infinitely fast objects, as long as you don't collide them with something else.

That makes them a little bit like black holes - they are singularities, but they are fairly well-behaved

Good piece

Posted Nov 29, 2012 23:21 UTC (Thu) by davidescott (guest, #58580) [Link] (2 responses)

> Newtonian mechanics has no problems with infinitely fast objects, as long as you don't collide them with something else.

WHAT?

If you think that is the case solve the following single particle 1-dimensional, force-less system:
t=0: the particle is "at" x=0 and has dx/dt=\infty and d^2x/dt^2=0.
Solve for t=1 to get x_1,v_1,a_1

Now solve the following systems for t=1 and t=-1:
t=0: x=x_1, dx/dt=-v_1, d^2x/dt^2=a_1
t=0: x=2*x_1, dx/dt=-v_1, d^2x/dt^2=a_1
t=0: x=x_1, dx/dt=-2*v_1, d^2x/dt^2=a_1
t=0: x=2*x_1, dx/dt=-2*v_1, d^2x/dt^2=a_1

Either you cannot do this, or something will be contradictory.

Good piece

Posted Nov 30, 2012 0:33 UTC (Fri) by Cyberax (✭ supporter ✭, #52523) [Link] (1 responses)

That's just an artifact of a chosen coordinate system. If you really want to solve it - write down Lagrangian of a system and see what happens.

Good piece

Posted Nov 30, 2012 2:18 UTC (Fri) by davidescott (guest, #58580) [Link]

I'm telling you I can't solve that. I don't know how. If you think it is so easily solved I would love to see your solution.

Good piece

Posted Nov 30, 2012 7:07 UTC (Fri) by apoelstra (subscriber, #75205) [Link] (1 responses)

> 1) "Space invaders" scenario is perfectly deterministic, they just have a problem with incorrect abstraction. In particular, treating gravity as infinitely fast. If we instead treat gravity as a classical field with finite interaction propagation speed, then this paradox disappears. Ditto for special relativity (it has its own problems with singularities, though).

This isn't the problem with space invaders, though. The problem is that you have an object that disappears to infinity, where it remains for all time. But since classical mechanics is time-reversible, it could just-as-legitimately said that the object "has been at infinity since eternity, then moves to a finite position at time t".

But nothing in classical mechanics predict when "time t" is, hence the indeterminacy.

As you have pointed out, special relativity wrecks up this pathology (though general relativity introduces many more, much worse, ones). But that's irrelevant to whether the claim "classical mechanics is deterministic" is true.

Good piece

Posted Nov 30, 2012 14:07 UTC (Fri) by davidescott (guest, #58580) [Link]

As an aside. SR eliminates infinite velocity singularities in classical mechanics, but there are other non-regular solutions in Classical Mechanics which are finite velocity. To eliminate those you would have to enforce C^2 on all constraint surfaces (and hope that covers all possible singularities). In many ways these surface is C^1 but not C^2 problems are easier to understand than space invaders because you don't have any messy infinities to deal with. So go read Norton's paper.