Status of Ergodicity: FPU, KdV & Collisions

[selfAdjoint]

In the QM forum, Vanesch posted this:

A probability distribution has always been, in classical physics, an ignorance description, and the funny thing about ergodicity is that the very same mechanisms which make us ignore in practice how DIFFERENT SYSTEMS are prepared, also have as a consequence that different microparts are distributed in time and between them according to the same distributions. I think that this hypothesis has indeed not been proven in all generality, but is assumed and is at the basis of about all thermodynamical calculations.

I know a little about the modern history of the ergodic hypothesis and here it is.

Back in the 1950s Fermi, Pasta, and Ulam devised a model, intended to resemble a linear molecule, which consisted of anharmonic oscillators joined by weakly nonlinear couplings. The model was beyond their analytical capabilities but they applied their brand-new monte carlo computer simulation to it and found a surprise. They had expected the model (later named the FPU model after them) to show ergodic behavior at late times, with the different energy states smearing out to fill the phase space. But what the computer output showed them was a nondecreasing propensity to produce unsmeared energy spikes.

It was later shown, still by computer, that the FPU model was producing solitons. Still later the FPU equation was mapped into the discretized Kortweg-deVrees (KdV) equation; the KdV equation has a rich analytical tradition,and its general solution can be expressed as a sum of solitons.

Finally it was shown that if you allow the FPU oscillators to collide and rebound, then the model behaves ergodically. So perhaps elastic collisions are a prior requirement for ergodicity? Does anybody know?

 

[Juan R.]

In the QM forum, Vanesch posted this:


I know a little about the modern history of the ergodic hypothesis and here it is.

Back in the 1950s Fermi, Pasta, and Ulam devised a model, intended to resemble a linear molecule, which consisted of anharmonic oscillators joined by weakly nonlinear couplings. The model was beyond their analytical capabilities but they applied their brand-new monte carlo computer simulation to it and found a surprise. They had expected the model (later named the FPU model after them) to show ergodic behavior at late times, with the different energy states smearing out to fill the phase space. But what the computer output showed them was a nondecreasing propensity to produce unsmeared energy spikes.

It was later shown, still by computer, that the FPU model was producing solitons. Still later the FPU equation was mapped into the discretized Kortweg-deVrees (KdV) equation; the KdV equation has a rich analytical tradition,and its general solution can be expressed as a sum of solitons.

Finally it was shown that if you allow the FPU oscillators to collide and rebound, then the model behaves ergodically. So perhaps elastic collisions are a prior requirement for ergodicity? Does anybody know?

The ergodic hyphothesis newer was proven and people is searching new foundation for statistical mechanics.

- Kinchin axioms, which also failed.

- Lanford's theory of LT which also failed.

- Recently proposed theory of Malament, Zabell
and Vranas. Which i think that does not work.

Therefore, nobody has proven that statisical ensembles in classical physics are a coarse grained (ignorance) description of an underliyng description system.

There are several approaches to solve the dilema:

one advanced is from Brushles theory. They claim that point in phase space is not defined due to Poincaré resonances.

Other still more advanced is from canonical science, but i cannot put here because is a "personal theory". But is will discuss the others methods.

 

[vanesch]

Finally it was shown that if you allow the FPU oscillators to collide and rebound, then the model behaves ergodically. So perhaps elastic collisions are a prior requirement for ergodicity? Does anybody know?

I think we should not be surprised for some systems not to obey completely the ergodic hypothesis, because otherwise memories would not exist, no ?

 

[Juan R.]

I think we should not be surprised for some systems not to obey completely the ergodic hypothesis, because otherwise memories would not exist, no ?

Moreover "elastic" is an approximation. And for guariantize that equilbrium state is stable one needs anarmonic/inelastic.