# Accessible States in the Ergodic Hypothesis

According to Wikipedia, the ergodic hypothesis is the assumption that

all accessible microstates are equiprobable over a long period of time.

My question is about the precise meaning of "accessible" here.

Consider the microcanonical ensemble for a thermally insulated system (say an ideal gas of $$N$$ particles and $$E$$ units of energy, confined to a box of volume $$V$$). As far as I can tell, the only thing that "accessible" means here is that the energy is constant, so when computing spatial averages we only average over the states of a given energy.

But why do we only worry about the energy here? Why not worry about only averaging over the states of constant momentum, or angular momentum, or any other quantity that could potentially be conserved?

(I realize, by the way, that momentum is not conserved in an ideal gas confined to a box, but surely there are thermodynamic systems where there is some conserved quantity other than energy, right?)

What exactly is so special about energy here? And are there situations where "accessible states" refers to more than just all the states of a certain energy?

Generically, you could absolutely fix any number of conserved quantities you like. The energy need not even be among them. The energy is just the most common thing to fix because the second most common conserved quantity, momentum, is often not conserved due to boundary conditions in the systems that are common in statistical mechanics. There's no reason why this need always be the case though.

• But how do we know that there are not other conserved quantities that need to be accounted for when taking averages, even if we never explicitly "fix" them? It just seems very strange to me that such considerations are never mentioned in any treatment of statistical mechanics. Nov 30, 2020 at 0:18
• @Uyttendaele In principle, we don't know there aren't others. Though I will point out this is one reason why the microcanonical ensemble is not typically used and almost all applications only make use of the canonical ensemble. There we are only fixing certain expectation valued, which is a much weaker assumption about the system. Nov 30, 2020 at 1:39