Undergrad Exceptions to the postulate of equal a priori probabilities?

[AndreasC]

Not sure if this is the appropriate forum for this, hopefully if it isn't someone can move it to a more appropriate place.

The fundamental postulate of equal a priori probabilities in statistical physics asserts that all accessible microstates states in an ensemble happen with equal probability. It is an important assumption for proving a number of important results, like the form of the partition functions in microcanonical and canonical ensembles etc. My question is, are there any significant cases that statistical physics still deals with where equal a priori probabilities can not be assumed and other assumptions have to be made?

 

[hilbert2]

For instance, there's this system of coupled oscillators that can get "stuck" in some part of energy-allowed phase space and the energy doesn't partition to all degrees of freedom until some time has passed.

https://en.wikipedia.org/wiki/Fermi–Pasta–Ulam–Tsingou_problem

But this kind of things are something that would probably be called "metastable states" in statistical physics terminology.

 

[Nullstein]

In general, equal apriori probabilities are not a postulate but must be derived from the dynamics. However, in all but the most simple cases, this is mathematically very difficult and rarely of practical importance. Thus, this problem is usually sidestepped and the probabilities are just postulated. This is fine in practice, but you have to keep in mind that such a postulate is not independent of the dynamics and could in principle contradict it. Integrable systems are a typical example for this situation. E.g., in the two-body problem, no non-trivial set of initial conditions will lead to a thermal state if you just wait long enough.

 

[AndreasC]

This is fine in practice, but you have to keep in mind that such a postulate is not independent of the dynamics and could in principle contradict it.

Yeah, this has been my understanding so far, that's why I was interested in examples where this can't be assumed.

no non-trivial set of initial conditions will lead to a thermal state if you just wait long enough.

How is a thermal state defined? Furthermore, can equal probabilities be assumed in all such states?

 

[Nullstein]

How is a thermal state defined? Furthermore, can equal probabilities be assumed in all such states?

Typically, a closed, sufficiently chaotic system is expected to have a microcanonical ensemble as equilibrium distribution, which is just an equal probability distribution on a constant energy surface (or constant integrals of motion surface, more generally). An open subsystem that can exchange energy will then typically end up in a canonical ensemble. The canonical ensemble has this constant energy only on avergage, with configurations closer to this constant energy surface being more likely.