Meaning of equiprobability principle in statistical mechanics

[atyy]

So I agree with your comment atyy, there must be something else than only statistical inference (except for the microcanonical case i would say) to explain canonical ensembles. Noting that it doesn't solve the problem of the existence, or not, of a real ensemble distrubution probability for system at equilibrium.

Yes, I agree we do not have a good mathematical understanding from microscopic dynamics why the equilibrium ensembles work so well (even the microcanonical case). I guess quite a bit of computational work has been done showing that almost any microscopic dynamics reproduce the equilibrium ensembles. But that's different from having a theorem.

Actually, I was told by someone working on dynamical systems quite a few years back, that computational work suggested that the KAM theorem would "hold" beyond the limits of the mathematical proof - just as experience and computation suggest that we should have more than the ergodic theorem for statistical mechanics.

 

[Zacku]

Actually, I was told by someone working on dynamical systems quite a few years back, that computational work suggested that the KAM theorem would "hold" beyond the limits of the mathematical proof - just as experience and computation suggest that we should have more than the ergodic theorem for statistical mechanics.

Yes that's what I wanted to say in my last message. It seems that Khinshin's work is more helpful to understand an equivalence between time average and ensemble averages without the ergodic hypothesis.

Thanks! Interesting stuff, especially his other work on quantum measurement. We shall see if it works out, but I much prefer it to "many worlds"!

I don't know the work you are talking about. Have you got a link ?

 

[Llewlyn]

How can you apply the definition of equilibrium you gave to the simple case of a N particles gas in a box of volume V and constant energy E (thanks to the hamiltonian dynamics) ?

Yes, saying that is at equilibrium is wrong and fuorviant. What i would say is "if macrovariables don't evolve with time prevision of SM is a logically estimator" but it turns to "only to equilibrium".

The fact is that I never read (if I remember well) a paper about out of equilibrium statistical mechanics in which an ensemble distribution of probability tends to a canonical distribution at equilibrium

In ergodic theory there is a property stronger than ergodicity called "mixing". A mixing system has dynamic such that "relaxes" to uniform distributions.
http://en.wikipedia.org/wiki/Topological_mixing

Assuming that this time average equals an ensemble average is equivalent, according to me, to the ergodic problem.
It seems, as we said earlier, that the answer of the statistical mechanics "problem" is not the ergodic theroem since, despite the famous work of Sinai on this subject, is to restrictive to build the bridge between time averages and ensemble averages.
It seems that the answers is perhaps in the ideas rised by Khinshin (I'm reading his book on statistical mechanics ).

Following microscopic dynamic is one way for justifying ensembles and leads to its problems, as to demonstrate that a flux is ergodic. Of course we may be happy to prove a weak property, such as an asymptotic ergodicity due to great number of degrees of freedom as stated by Khinchine, or simply we simulate systems to computer watching what dynamic do, as told by atyy. Of course if we observe (or prove) that dynamic visit (quite) all phase space we have justified use of ensemble. It is an approach that perfectly satisfies me but i don't find an answer to our matter more that with information approach. If you follow the dynamic, why SM works only at equilibrium and what is it?

Ll.

 

[vanesch]

But there is ANOTHER way completely different and it starts OUT of physics. If i roll a dice and i ask you which number will spot you can only assign equal probability for all numbers, in other words you assign equal probability to every possible states when you cannot do better. If i tell you that my dice is cheated and only even numbers will spot you can do a better prevision. More information you have, more accurate your prevision will be. This how SM works: it makes prevision about the state of system on the partial knowledge you have. The quantity of your disinformation is called "entropy". Your find the state that maximize your disinformation compatible with your a priori information; if you have no information you'll obtain the uniform density (as for the dice), if your only information is the energy you'll obtain the canonical density.

The problem I guess the OP is addressing is: if the dice are loaded, but you don't know it. Then you still assume equal probabilities for the outcomes, and it is not correct, and there are a priori a lot of quantities which will give wrong "macroscopic" averages, like the fraction of even numbers (0.5 from the "equiprobable distribution" and 1.0 from the "correct" distribution).

The real question, I guess the OP is wondering about, is: how come that most if not all quantities we actually need come out correctly in statistical mechanics, if it is not the correct distribution (but just the one we only "know about"). Why is, say, pressure not something like the fraction of even outcomes ?

In other words, why does statistical mechanics with the equi-probability axiom work at all, if the equi-probability distribution is just "the best we can do with our limited knowledge (which might not be sufficient a priori)".

Question to the OP: is that what you mean ?

 

[atyy]

Thanks! Interesting stuff, especially his other work on quantum measurement. We shall see if it works out, but I much prefer it to "many worlds"!:rofl:

I don't know the work you are talking about. Have you got a link ?

http://arxiv.org/abs/cond-mat/0408316
http://arxiv.org/abs/quant-ph/0702135

Not sure how these are related, but some other models:
http://arxiv.org/abs/quant-ph/0105127
http://arxiv.org/abs/quant-ph/0304100

 

[Zacku]

In other words, why does statistical mechanics with the equi-probability axiom work at all, if the equi-probability distribution is just "the best we can do with our limited knowledge (which might not be sufficient a priori)".

Question to the OP: is that what you mean ?

Yes this is globally what I wanted know at the beginning. Though we answered partially to this question during this thread (why it does work practically). I also wondered why such a question is never mentioned in the current literature on the subject.