Meaning of equiprobability principle in statistical mechanics

[Count Iblis]

In general you don't have extensive variables. I think there as been some recent work on self gravitating systems and it was argued that you could treat these systems well within the microcanonical ensemble but not in the canonical ensemble.

 

[Zacku]

In general you don't have extensive variables. I think there as been some recent work on self gravitating systems and it was argued that you could treat these systems well within the microcanonical ensemble but not in the canonical ensemble.

This is just because in gravitational systems we are dealing with long range interactions and it is well known nowadays that in this case the ensemble equivalence break down, this is a "new field" (that started about 10 years ago i think) in statistical mechanics of system in or out of equilibrium.

The "natural intensive variable" are lagrange multipliers of the extensive one. How can you tell that are constants?

Yes but you can measure them, can't you ? Pressure and temperature are not "impossible to measure variables" as far as I know.
Although I think the Bayesian point of view is the correct procedure to use probability theory to every dynamical system (that's what I think at least) there are things in it that are quite obscure for me. As the meaning of statistical averages to express macrovariable values one measured on a system. In the case of E,V,N Balian would say "ok we have
<E>,V,<N> and then we can apply the maximum entropy principle and tadaa we find the grand canonical distribution". I actually don't understand very well the meaning of ensemble averages if we are interested in one system (as it is often the case in practice). I don't understand either if saying that <E> is constant means that we are at equilibrium or not.
What is more troublesome for me is that, traditionally, canonical ensemble and grand canonical ensemble are "justified" by the fact that there is an equivalence ensemble with the microcanonical ensemble (which is the only ensemble that can be found without statistical postulates about what is measured in both the ergodic and Bayesian point of view).

 

[Llewlyn]

Yes but you can measure them, can't you ? Pressure and temperature are not "impossible to measure variables" as far as I know.

For me it is not so obvious, ad example temperature may assume negative values in SM. In general i think as "lagrange's multipliers" and only when the system is thermodynamic i associate the meaning of temperature or pressure, due to the thermodynamic's laws. The definition of equilibrium i gave soddisfies myself, although I'm sure I've not completely understand the matter.

Ll.

 

[Zacku]

The definition of equilibrium i gave soddisfies myself, although I'm sure I've not completely understand the matter.
Ll.

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) ?
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 (where the word "tends" as to be clarified since it must have only a statistical meaning as in the Boltzmann H theorem).

 

[Zacku]

So I think the "information alone" approach in the first place is not a good description of itself. Also, it should be the microscopic physics that permits or does not permit us to "coarse grain" successfully. Pretty much all the same points Zacku made - except he likes the "information only" approach!

Thinking about that since the opening of this thread, I think that the indifference principle allows one to retrieve the microcanonical distribution with a correct use of probability theory.
However, its other use - that is when assuming that the values of measured macrovariables are ensemble averages (as stated for instance R. Balian)- is more difficult to understand for me. In quantum mechanics when we make a measure, we must find eigenvalues of the observable of interest, that is, we don't find a quantum average value unless we make more than one measure.
As a matter of fact, if one is able to know the expression of a macrovariable observable, then an apparatus will measure a time average of this quantity (even if we are talking about QM). 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 ).
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.

 

[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"!

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.