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...
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...
It seems that I must clarify my question. That formula (6.34) is <xi dH/dxj> = δi 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...
Every time the system is at one of the possible states. For this reason, the sum of all probabilities should be equal to unity. Or the integral over the whole range of energies, in your example. You should find out, what is the normalization of the distributions.
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...