A Beyond the energy equipartition, what is the uneven partitioning?

[napdmitry]

There is a celebrated energy equipartition theorem, it works fine for many systems. But it requires the dense filling of the surface of constant energy. What if there are other conserved quantities, like momentum or angular momentum? It seems, that the energy partitioning will be uneven, with approximately linear dependence of the mean energy on the mass share, associated with the degree of freedom. The heavier the particle, the lesser it's mean energy. There are numerical pieces of evidence of such partitioning in different systems, like isolated clusters of atoms. A good example is the usual ideal gas in a round vessel. The question is, how general is such a law of uneven energy partitioning? Can anyone provide other examples of uneven partitioning laws?

 

[Demystifier]

Can anyone provide other examples of uneven partitioning laws?

See K. Huang, Statistical Mechanics, Chapter 6.4. The equipartition theorem can be violated by quantum mechanics, as well as by Hamiltonians which are not quadratic in canonical positions.

 

[napdmitry]

I know it can be violated, but what is the uneven law then?

 

[Demystifier]

I know it can be violated, but what is the uneven law then?

See formula (6.34) in the mentioned book.

 

[napdmitry]

It seems that I must clarify my question. That formula (6.34) is <x_i dH/dx_j> = δ_{i j} kT, which is an equipartition theorem in a generalized form. It works fine for an ideal gas in a rectangular vessel, for example. But it does not work for ideal gas in a round vessel, where massive enough particles will have their mean energy below equipartition level. This can be easily found via numerical simulation. The theoretical explanation is that the derivation of energy partitioning includes integration over the whole energy surface. But the energy surface is not filled, except for the zero measure set, if there are other integrals of motion. Like the conservation of the angular momentum in a round vessel. So the question was, what will be if the formula (6.34), in particular, does not work? It seems that there is a more or less universal unevel law of energy partitioning. May be someone knows good related examples, except for the one I've provided.
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[hutchphd]

I believe the crux of the matter here is the ergodic theorem yes?

 

[Demystifier]

I believe the crux of the matter here is the ergodic theorem yes?

In my opinion, the ergodic theorems (including their weaker forms) are never relevant in realistic systems, because ergodicity is only relevant for very very long times, typically many orders of magnitude longer than the age of the Universe.

 

[napdmitry]

I believe the crux of the matter here is the ergodic theorem yes?

Yes, ergodicity is one of the requirements for the equipartition theorem to work. But there are some systems that are known to be nonergodic. Any additional law of conservation is enough for the energy surface not to be filled entirely. I believe such situations are not that uncommon. So what is known about energy partitioning when the equipartition theorem is not applicable?

 

[Lord Jestocost]

To my mind, non-ergodic systems might probaly be decomposed into a number of components, each of which might be treated separately ergodic.

 

[hutchphd]

It seems that there is a more or less universal unevel law of energy partitioning. May be someone knows good related examples, except for the one I've provided.

Sorry. Unevel?

Any additional law of conservation is enough for the energy surface not to be filled entirely.

How does this differ from other continuum limits routinely taken for finite systems? Can you elucidate (or provide guidance for the woefully misinformed) the concept of "measure" by which this is reckoned?

 

[napdmitry]

Uneven, of course. You are right, there are routinely taken limits for finite systems, while proofs of the equipartition theorem usually involve infinite limits of integration. But to my knowledge, the equipartition theorem still works fine in many finite systems. These limits by themselves do not lead to its violation. I don't know if there is any theoretical justification for it, it's only what my experience told me.

The presence of another integral of motion means that the state representing point moves along the intersection of two surfaces. These are the surfaces of constant energy and of another integral of motion being constant. The intersection of these two surfaces is a surface of lower dimensionality. It is obviously a subset of the energy surface. There is at least one such system with the equipartition violation being found, where the exact law of uneven energy partitioning is known theoretically. But that is only for one particular system. I would like to know if there are other such or different uneven laws found in other systems.

 

[Lord Jestocost]

There is at least one such system with the equipartition violation being found,

What system?

 

[napdmitry]

An ideal gas of N different particles in a round vessel.

 

[Lord Jestocost]

Simulation of an ideal gas which is confined by rigid vessel walls?

This would be a toy model simulation for a deterministic system.