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keiths



Posts: 2041
Joined: Jan. 2006

(Permalink) Posted: Dec. 25 2008,12:19   

Quote (Cubist @ Dec. 24 2008,16:35)
In sum: As a practical matter, this "grue/bleen" schtick is basically mental masturbation. If that's the sort of thing you enjoy doing, feel free; just don't expect anybody else to take it seriously.

Cubist,

You're missing the point.  Nobody believes the world is grue/bleen.  We all think it's green/blue.  The question is whether we can justify our belief rigorously, and the question for MSB specifically is (or was) how natural selection is able to choose green/blue over grue/bleen despite having no visibility into the future.

The grue/bleen example may be contrived, but the issues at stake are important because induction is central to science.

--------------
And the set of natural numbers is also the set that starts at 0 and goes to the largest number.  -- Joe G

Please stop putting words into my mouth that don't belong there and thoughts into my mind that don't belong there. -- KF

  
keiths



Posts: 2041
Joined: Jan. 2006

(Permalink) Posted: Dec. 25 2008,12:37   

Quote (olegt @ Dec. 25 2008,07:35)
Sounds like you're a mathematical physicist, Missing Shade of Blue.  What's wrong with ergodicity?  Phase-space averaging seems to describe time-averaged properties of systems in thermal equilibrium pretty well.  

Hi Oleg,

I can't speak for Shade and I'm definitely no mathematical physicist, but isn't it true that ergodicity must be assumed in many cases (albeit with some empirical justification)? Wouldn't it be better to have a rationale for phase-space averaging that didn't depend on such assumptions?

--------------
And the set of natural numbers is also the set that starts at 0 and goes to the largest number.  -- Joe G

Please stop putting words into my mouth that don't belong there and thoughts into my mind that don't belong there. -- KF

  
Quack



Posts: 1961
Joined: May 2007

(Permalink) Posted: Dec. 25 2008,13:30   

Quote
(I hope I am not derailing the thread. ;) )


I actually wrote an essay this morning in an attempt to pull the discussion over into the realm of reality, but when I finished off by writing those same words, I pulled back...

--------------
Rocks have no biology.
              Robert Byers.

  
Reciprocating Bill



Posts: 4265
Joined: Oct. 2006

(Permalink) Posted: Dec. 25 2008,14:12   

Quote (keiths @ Dec. 25 2008,13:19)
Quote (Cubist @ Dec. 24 2008,16:35)
In sum: As a practical matter, this "grue/bleen" schtick is basically mental masturbation. If that's the sort of thing you enjoy doing, feel free; just don't expect anybody else to take it seriously.

Cubist,

You're missing the point.  Nobody believes the world is grue/bleen.  We all think it's green/blue.  The question is whether we can justify our belief rigorously, and the question for MSB specifically is (or was) how natural selection is able to choose green/blue over grue/bleen despite having no visibility into the future.

The grue/bleen example may be contrived, but the issues at stake are important because induction is central to science.

An instance of "philospher's doubt."

--------------
Myth: Something that never was true, and always will be.

"The truth will set you free. But not until it is finished with you."
- David Foster Wallace

"Here’s a clue. Snarky banalities are not a substitute for saying something intelligent. Write that down."
- Barry Arrington

  
Missing Shade of Blue



Posts: 62
Joined: Dec. 2008

(Permalink) Posted: Dec. 25 2008,14:36   

Oleg,

I absolutely agree that phase averaging works very well. But I don't think the ergodic hypothesis is an adequate rationale for the procedure. We can only prove ergodicity for certain special systems (hard spheres in a box, geodesic motion on a manifold with negative curvature). We don't have proofs of ergodicity for virtually all the systems to which we usually apply classical stat. mech. In fact, there's good reason to think many of these systems are not ergodic. The KAM Theorem leads us to expect that a typical Hamiltonian system with finite degrees of freedom will contain islands of non-ergodic flow. Less abstractly, phase averaging works for a number of systems that we know contain KAM tori. The explanation for the success of the procedure in this case cannot be explained by the ergodic theory as traditionally conceived.

There are also conceptual problems with the way ergodicity is used to justify phase averaging. We need an explanation for why our macrosopic measurements of thermodynamic quantities match the phase average. The traditional justification for this is that our macroscopic observation times are so long relative to characteristic microscopic time scales that we might as well consider our macroscopic observations as infinite time averages which, for ergodic systems, are just phase averages. To my eyes, at least, this sort of justification is almost laughably ad hoc. One obvious problem: if macroscopic measurements really approximate infinite time averages, how could we ever observe a system going into equilibrium?

There is a more plausible (although, mysteriously, less widely quoted) justification. One can prove, using Birkhoff's theorem, that if a system is ergodic then the only invariant probability measure over the phase space (meeting some further trivial criteria) is the microcanonical measure. And, if the system is large enough, then in equilibrium there is an overwhelmingly large probability (under this measure) that the system is at a phase point where thermodynamic quantities are very very close to phase averages. No appeal to time averages at all. Unfortunately, this argument doesn't work at all if there are any non-ergodic regions in phase space that are greater than measure zero. Even if the phase space contains a really really tiny non-ergodic island, one can no longer prove that the microcanonical measure is the unique invariant measure. So again, this justification, which is quite elegant, unfortunately only works for a very very restricted set of systems.

So yeah those are some of the reasons I think ergodicity is a red herring.

  
Missing Shade of Blue



Posts: 62
Joined: Dec. 2008

(Permalink) Posted: Dec. 25 2008,14:49   

Cubist,

Perhaps this sort of speculation is just mental masturbation. But since there was more than one person involved in the discussion, doesn't it qualify as mental sex? Or at least a mental circle-jerk.

  
olegt



Posts: 1405
Joined: Dec. 2006

(Permalink) Posted: Dec. 25 2008,22:49   

Missing Shade of Blue,

Physicists are content with the microcanonical distribution being stationary, so it describes an equilibrium state.  Whether it's unique is of course another question, but that's the difference between physics and mathematics.  I'm sure you've heard this joke: all mathematicians think and all physicists know that a face-centered cubic lattice is the most efficient stacking of hard spheres.  ;)

And while I agree that not all systems are ergodic, non-ergodicity may have important physical implications.  Spontaneous symmetry breaking makes the system non-ergodic: a magnetized piece of iron will not change its magnetization direction for a very long time (much longer than any reasonable experimental time scale).  Glass becomes non-ergodic at a sufficiently low temperature, too.  So ergodicity is more than just a technical step in getting to the microcanonical distribution.

--------------
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Missing Shade of Blue



Posts: 62
Joined: Dec. 2008

(Permalink) Posted: Dec. 26 2008,03:39   

Oleg,

I certainly don't want to argue that ergodic theory is useless in physics. I am merely criticizing its use in one particular area: the justification of the phase averaging procedure.

I do have a slight quibble with the notion that glass "becomes non-ergodic" at low temperatures. The phrase suggests that the system in its high-temperature equilibrium state is ergodic, and that it loses this ergodicity when it freezes into a glass state. But there is no reason to believe that the equilibrium state is ergodic if you reject the standard stat. mech. assumption that thermodynamic equilibrium implies ergodicity.

No doubt the low-temperature state is in some sense "more non-ergodic" than the high-temperature state. Regions of phase space that were mutually accessible become separated as the glass cools, and the final frozen state is conspicuously path-dependent. So "ergodicity breaking" is an important phenomenon. But calling it that might give the false impression that the original state was ergodic in the full technical sense, and this need not be true (and probably isn't true for most real glass transitions). The relevant transition is not from ergodicity to non-ergodicity, but from a stable global equilibrium to a metastable local equilibrium.

  
Missing Shade of Blue



Posts: 62
Joined: Dec. 2008

(Permalink) Posted: Dec. 26 2008,04:23   

Oleg,

I'm assuming you're a physicist. What area do you work on, if you don't me asking?

  
Reciprocating Bill



Posts: 4265
Joined: Oct. 2006

(Permalink) Posted: Dec. 26 2008,08:27   

Quote (Missing Shade of Blue @ Dec. 25 2008,15:49)
Cubist,

Perhaps this sort of speculation is just mental masturbation. But since there was more than one person involved in the discussion, doesn't it qualify as mental sex? Or at least a mental circle-jerk.

A circle jerk has a victim. Wonder who that is here?

No, on second thought, don't turn on that light.

--------------
Myth: Something that never was true, and always will be.

"The truth will set you free. But not until it is finished with you."
- David Foster Wallace

"Here’s a clue. Snarky banalities are not a substitute for saying something intelligent. Write that down."
- Barry Arrington

  
olegt



Posts: 1405
Joined: Dec. 2006

(Permalink) Posted: Dec. 26 2008,17:25   

Missing Shade of Blue,

I work in physics of condensed matter, studying all kinds of magnetic phenomena, as a theorist.  My interest in the foundations of statistical mechanics is not exactly idle: I teach a graduate course in statistical physics every other year.  So I am interested in what mathematicians think about this.  

   
Quote
I do have a slight quibble with the notion that glass "becomes non-ergodic" at low temperatures. The phrase suggests that the system in its high-temperature equilibrium state is ergodic, and that it loses this ergodicity when it freezes into a glass state. But there is no reason to believe that the equilibrium state is ergodic if you reject the standard stat. mech. assumption that thermodynamic equilibrium implies ergodicity.


I am not sure that there are sufficient grounds for rejecting the link between ergodicity and thermodynamic equilibrium.  A key point to remember is this: physicists like to work in the so-called thermodynamic limit (infinite number of particles).  If your system is too small, you can't even define a temperature, so there is no thermodynamics to speak of.  For example, even though N>1 hard spheres in a box are an ergodic system, the velocity distribution becomes Maxwellian only in the limit of an infinite N.  

   
Quote
We don't have proofs of ergodicity for virtually all the systems to which we usually apply classical stat. mech. In fact, there's good reason to think many of these systems are not ergodic. The KAM Theorem leads us to expect that a typical Hamiltonian system with finite degrees of freedom will contain islands of non-ergodic flow.

Yes, KAM theory tells us that some invariant tori of an integrable system remain such even after the addition of a nonlinear perturbation.  That is quite significant since it follows that realistic dynamical systems can be non-ergodic.  However, there is an important caveat: the perturbation must be sufficiently weak.  And furthermore, the bounds on the strength of the perturbation become more stringent as the number of degrees of freedom increases, so in the thermodynamic limit invariant tori shrink and a typical system becomes ergodic.  

Let me quote Henk Broer's brief overview KAM theory: the legacy of Kolmogorov’s 1954 paper (PDF file):
   
Quote
On the other hand, and from a more global point of view, the measure-theoretical part of KAM theory implies that for typical Hamiltonian systems in finitely many degrees of freedom, no ergodicity holds, since the energy hypersurfaces can be decomposed in several disjoint invariant sets of positive measure. This is of particular interest for statistical physics, where the ergodic hypothesis roughly claims that the system, when confined to bounded energy hypersurfaces, is ergodic. This paradox probably is resolved as the number of particles is increasing since the obstruction to ergodicity provided by the KAM tori then seems to decrease rapidly in importance.

Towards the end of the paper he discusses a particular case of N coupled nonlinear oscillators, where the allowed perturbation strength goes down exponentially with N.  

Of course, ergodicity in systems with an infinite number of degrees of freedom is not yet well understood.  But it looks to me like physicists need not be alarmed just yet: while we now know that ergodicity is broken in some realistic finite systems, there are as yet no reasons to question ergodicity of realistic systems in the thermodynamic limit.

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dvunkannon



Posts: 1377
Joined: June 2008

(Permalink) Posted: Dec. 26 2008,19:08   

Quote (keiths @ Dec. 25 2008,13:19)
Quote (Cubist @ Dec. 24 2008,16:35)
In sum: As a practical matter, this "grue/bleen" schtick is basically mental masturbation. If that's the sort of thing you enjoy doing, feel free; just don't expect anybody else to take it seriously.

Cubist,

You're missing the point.  Nobody believes the world is grue/bleen.  We all think it's green/blue.  The question is whether we can justify our belief rigorously, and the question for MSB specifically is (or was) how natural selection is able to choose green/blue over grue/bleen despite having no visibility into the future.

The grue/bleen example may be contrived, but the issues at stake are important because induction is central to science.

If I understand what Wikipedia is saying about Goodman's argument, we could be discussing dogs that smell explosives in America and limburger cheese in France. Time and color are not important to the problem.

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I’m referring to evolution, not changes in allele frequencies. - Cornelius Hunter
I’m not an evolutionist, I’m a change in allele frequentist! - Nakashima

  
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