There are quite a few conceptual confusions in this question.

A system is either closed or open. A system is not "equilibrium" or "non-equilibrium". Also, a system is either conservative or dissipative. The ergodic theorem does not apply to open systems, neither to dissipative systems, since they tend to tend to a fixed point or something like that.

A state of a system can be an equilibrium state, or not, depending on whether it is invariant with the passing of time. A system has two concepts of "state": the one relevant for statistical mechanics is a macrostate, which means not a point in the phase space, but a probability distribution on the phase space. Usually the energy is fixed. If Liouville measure were a probability distribution, which it is not, it would be an equlibrium state since it is invariant under the passing of time. If a fixed-energy surface has finite volume, which it usually does, then if one restricts Liouville measure to that surface, one gets an equilibrium state. This can be done for any closed Hamiltionian conservative system. This has nothing to do with ergodicity.

The ergodic theorem does not apply to every dynamical system, yet the above remarks do. A dynamical system could have equilibrium states whether or not the system is ergodic.

Very few of the dynamical systems are known to be ergodic. Even if a system is ergodic, it is a fallacy that that means the paths nearly always enter into every region. That is the concept of "mixing", which is even harder to prove, and rarer. Ergodic just means the time averages are almost always equal to the phase averages, nothing more nor less. And this has to be true for non-equilibrium states too, so it has nothing to do with equilibrium.

The question of open systems cannot use Liouville's theorem since it is false for open or dissipative systems. And so is the ergodic theorem.

Interestingly, if a system is composed of a very large number of similar components which interact somewhat weakly, one can prove that some of the important time averages are approximately equal to their phase averages even though the ergodic theorem is not applicable to that system. (Khinchin, The Mathematical Foundations of Statistical Mechanics.)

A good reference, a little old but easy to understand, for non-equilibrium statistical mechanics, is the book by de Groot and Mazur, recently reprinted by Dover. It studies fluctuations near equilibrium, which can be related to the amount of dissipation present in the system.

This post imported from StackExchange Physics at 2016-01-02 13:14 (UTC), posted by SE-user joseph f. johnson