I try to make this as short and concise as possible. For equilibrium systems in statistical mechanics, we have the Liouville's theorem which says that the volume in phase space is conserved when the system evolves in time. Thus formally, one could rephrase this as: time evolution of equilibrium/stationary systems is a measure invariant transformation (where the measure here is the volume in phase space).
Now for stationary systems, we have a density function $f(\mathbf{q},\mathbf{p})$ that fulfils Liouville's theorem, using which we can write an ensemble average of some phase space function $A$ as follows:
$$
\langle A \rangle = \int f(\mathbf{q},\mathbf{p})A(\mathbf{q},\mathbf{p}) d\mathbf{q} d\mathbf{p} \tag{1}
$$
Similarly the time average of the same function $A$ is defined by:
$$
\langle A \rangle_\rm{time} = \lim_{t\to \infty} \frac{1}{t}\int_0^t A(t)dt \tag{2}
$$
The most important, physically relevant, statement of the ergodic theorem is that (1) and (2) are equal, i.e. the ensemble average and time average of phase space functions are the same. This leads to an important interpretation regarding the time evolution of an ergodic system, which is that all the regions of the accessible part of phase space (i.e. consistent with the system's energy) are visited by the system regardless of the initial condition at $t=0$, and that the system spends an equal amount of time in all of them. This also intuitively explains why the averages (1) and (2) ought to be equal.
From a mathematical point of view, we know that the time average (2) converges for all trajectories taken by the system, since we started with the equilibrium assumption, which allowed us to treat the time evolution as a measure preserving transformation. The question is, how is the ergodic theorem generalized to also account for non-equilibrium systems, i.e. systems with dissipative dynamics and with sources/sinks of particles?
For one thing we no longer have the volume preserving transformation, thus Liouville's theorem is violated and the phase space volume is no longer incompressible under time transformations. Thus evaluating the convergence of (1) and (2) becomes non-trivial. Admittedly, the mathematical challenges aside, I am more interested in the physical aspect of the generalization of the ergodic theory. Does this concept even extend to non-equilibrium? If yes, how is it interpreted?
Best Answer
Non-equilibrium systems are most often considered in the approximation where local equilibrium is valid, yielding a hydrodynamic or elasticity description. Local equilibrium means that equilibrium is assumed to hold on a scale large compared to the microscopic scale but small compared with the scale where observations are made. In this case, one considers a partition of the macroscopic system into cells of this intermediate scale and assumes that each of these cells is in equilibrium, but with possibly different values of the thermodynamic variables.
From a macroscopic point of view, these cells are still infinitesimally small - in the sense that a continuum limit can be taken that disregards the discrete nature of the cells, without introducing too much error. Therefore the thermodynamic variables that vary form cell to cell become fields, tractable with the techniques of continuum mechanics.
On the other hand, from a microscopic point of view, these cells are already infinitely large - in the sense that the ideal thermodynamic limit, that strictly speaking requires an infinite volume, already hold to a sufficient approximation. (The errors in bulk scale with $N^{-1/2}$ for $N$ particles, which is small already for macroscopically very tiny cells.) Thus one can apply all arguments from statistical mechanics to the cells.
To the extent that one believes that an ergodic argument applies to the cell, it will justify (subjectively) the statistical mechanics approximation. However, the ergodic argument is theoretically supported only in few situations, and should be regarded more as a pedagogical aid for one's intuition rather than as a valid tool for deriving results.