Extending the ergodic theorem to non-equilibrium systems

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.

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.

While I know this question has already been answered, I felt obligated to get around to writing a formal answer. I will not go into detail about irreversible/dissipative systems, as Arnold Neumaier's answer already addressed that issue. Rather, my answer will focus on the mathematics behind ergodicity and mixing.

Note: Most of my comments were taken from Penrose [1979] in the following.

Background

First let us define $\boldsymbol{\Gamma}$ to be the whole of phase space, described by the position and momentum coordinates, $\mathbf{q}$ and $\mathbf{p}$, respectively. Then if we define the phase space density as that which satisfies: $$ \int_{\boldsymbol{\Gamma}} \ d^{n}q \ d^{n}p \ \rho\left( \mathbf{q}, \mathbf{p} \right) = 1 \tag{1} $$ where $n$ is the number of degrees of freedom and $\rho\left( \mathbf{q}, \mathbf{p} \right)$ is the phase space probability density.

Now if we use a generic variable, $G\left( \mathbf{q}, \mathbf{p} \right)$, to describe any dynamical variable (e.g., energy), then the ensemble average of $G$ is denoted by: $$ \langle G \rangle = \int_{\boldsymbol{\Gamma}} \ d^{n}q \ d^{n}p \ G\left( \mathbf{q}, \mathbf{p} \right) \ \rho\left( \mathbf{q}, \mathbf{p} \right) \tag{2} $$

Three Principles

However, there is an issue to be aware of at this point [i.e., page 1940 in Penrose, 1979]:

The fundamental problem of statistical mechanics is what ensemble – that is, what phase-space probability density $\rho$ – corresponds to a given physical situation... It is, however, possible to state three principles that the phase-space density should satisfy; and it turns out, rather surprisingly, that these principles when combined with a study of the dynamics of our mechanical models give enough information to answer the fundamental problem satisfactorily in some important cases.

1st Principle
The first of the three principles is just Liouville's theorem in the limit where $d \rho/dt = 0$. It's another way of saying that the Hamiltonian of the system does not explicitly depend upon time, which is how we define the system to be isolated.

2nd Principle
The second principle is stated as [i.e., page 1941 in Penrose, 1979]:

The second of the three principles is more general, since it does not require the system to be isolated... The principle, which I shall call the principle of causality, is simply that the phase-space density at any time is completely determined by what happened to the system before that time, and is unaffected by what will happen to the system in the future.

3rd Principle
Finally, the third principle is stated as [i.e., page 1941 in Penrose, 1979]:

The last of the three principles is that the probabilities in the ensemble really can be described by a phase-space density $\rho$ with $\rho$ a well-behaved (say, piecewise continuous) function, rather than some more general measure.

Now the last principle, it is important to note, is actually very important but often overlooked. It is important because if we require it, we cannot include systems like a gas of hard spheres in a cubical box where all spheres bounce between the same two faces for eternity (i.e., the spheres move only along one-dimension). That is to say, a time-average of this imaginary system will not be the same as an ensemble average (see explanation below). Let us define any system like this as an exceptional system, for brevity.

As an aside, the problems with time-averages in classical electricity and magnetism are well known and it is now known that spatial ensemble averages are the correct operations for converting between the micro- and macroscopic forms of Maxwell's equations [e.g., see pages 248-258 in Jackson, 1999 for a detailed discussion].

Ergodicity and Mixing

Ergodicity

If $G$ is a dynamical variable, then we can define the ensemble average over time as: $$ \langle G \rangle_{t} = \int_{\boldsymbol{\Gamma}} \ d^{n}q \ d^{n}p \ G\left( \mathbf{q}, \mathbf{p} \right) \ \rho_{t}\left( \mathbf{q}, \mathbf{p} \right) \tag{3} $$ where we can obtain $\rho_{t}$ using the assumption that Liouville's theorem holds (i.e., $d \rho/dt = 0$).

Note that when $\langle G \rangle_{t}$ exists, we can define it as an equilibrium value of $G$. However, it is worth noting that $\langle G \rangle_{t}$ does not necessarily exist, as in the case of any oscillating system without damping (e.g., simple harmonic oscillator). In other words, $\lim_{t \rightarrow \infty} \langle G \rangle_{t}$ will not approach a single value, it will oscillate indefinitely.

The time-average, however, always exists and one can avoid calculating a nonexistent value by redefining the equilibrium value of $G$ as: $$ \langle G \rangle_{eq} \equiv \lim_{t \rightarrow \infty} \ \frac{1}{T} \int_{0}^{T} \ dt \ \langle G \rangle_{t} \tag{4} $$ which is equal to $\lim_{t \rightarrow \infty} \langle G \rangle_{t}$ if $\langle G \rangle_{t}$ exists.

If we define the time-average of $\rho$ as $\bar{\rho}$, we can write this as: $$ \bar{\rho}\left( \mathbf{q}, \mathbf{p} \right) = \lim_{t \rightarrow \infty} \ \frac{1}{T} \int_{0}^{T} \ dt \ \rho_{t}\left( \mathbf{q}, \mathbf{p} \right) \tag{5} $$ which allows us to redefine $\langle G \rangle_{eq}$ as: $$ \langle G \rangle_{eq} = \int_{\boldsymbol{\Gamma}} \ d^{n}q \ d^{n}p \ G\left( \mathbf{q}, \mathbf{p} \right) \ \bar{\rho}\left( \mathbf{q}, \mathbf{p} \right) \tag{6} $$

It is important to note some properties of ergodic theory here [e.g., page 1949 in Penrose, 1979]:

It follows from the ergodic theorem of Birkhoff (1931) that $\bar{\rho}$ is well-defined at almost all phase points... consequently the integral in (1.16) is well-defined... Birkhoff's theorem also shows that $\bar{\rho}$ is constant on the trajectories in phase space...

where the integral (1.16) in the quote refers to the version of $\langle G \rangle_{eq}$ in Equation 6. The last statement, namely that $\bar{\rho}$ is an invariant, is crucial here. Were it not an invariant, it "...would require us to solve the equations of motion for $10^{23}$-odd particles..." [e.g., page 1945 of Penrose, 1979].

Important Side Note: Recall again that Equation 6 given above for $\langle G \rangle_{eq}$ does not always hold, as in the trivial case of an undamped simple harmonic oscillator because the integral on the right-hand side oscillates forever.

Assume we can write $\bar{\rho}\left( \mathbf{q}, \mathbf{p} \right) = \phi\left( x \right)$, where $\phi$ is an arbitrary function of only one variable. If $\phi\left( x \right) \rightarrow \phi\left( H \right)$, where $H$ is the Hamiltonian, for all $\bar{\rho}$ in a system, then the system is said to be ergodic. Another way of stating this is that if the system were ergodic, the trajectories would cover all parts of an energy manifold if given enough time.

Mixing

Let us define the microcanonical average over energy of $G$ as: $$ \langle G \rangle_{E} = \frac{ \int_{\boldsymbol{\Gamma}} \ d^{n}q \ d^{n}p \ G\left( \mathbf{q}, \mathbf{p} \right) \ \delta\left( H\left( \mathbf{q}, \mathbf{p} \right) - E \right) }{ \int_{\boldsymbol{\Gamma}} \ d^{n}q \ d^{n}p \ \delta\left( H\left( \mathbf{q}, \mathbf{p} \right) - E \right) } \tag{7} $$ where $\delta()$ is the Dirac delta function, $H\left( \mathbf{q}, \mathbf{p} \right)$ is the Hamiltonian, and $E$ are energy manifolds (i.e., systems that have energy $E$).

Thus, we can redefine $\langle G \rangle_{eq}$ as: $$ \begin{align} \langle G \rangle_{eq} & = \int_{\boldsymbol{\Gamma}} \ d^{n}q \ d^{n}p \ G\left( \mathbf{q}, \mathbf{p} \right) \ \bar{\rho}\left( \mathbf{q}, \mathbf{p} \right) \tag{8a} \\ & = \int_{\boldsymbol{\Gamma}} \ d^{n}q \ d^{n}p \ G\left( \mathbf{q}, \mathbf{p} \right) \ \phi\left( H \right) \tag{8b} \\ & = \int_{\boldsymbol{\Gamma}} \ d^{n}q \ d^{n}p \ G\left( \mathbf{q}, \mathbf{p} \right) \ \left[ \int_{-\infty}^{\infty} \ dE \ \phi\left( E \right) \ \delta\left( E - H\left( \mathbf{q}, \mathbf{p} \right) \right) \right] \tag{8c} \\ & = \int_{-\infty}^{\infty} \ dE \ P\left( E \right) \ \langle G \rangle_{E} \tag{8d} \end{align} $$ where $P\left( E \right)$ is given by: $$ P\left( E \right) = \int_{\boldsymbol{\Gamma}} \ d^{n}q \ d^{n}p \ \bar{\rho}\left( \mathbf{q}, \mathbf{p} \right) \ \delta\left( E - H\left( \mathbf{q}, \mathbf{p} \right) \right) \tag{9} $$ Note that $P\left( E \right)$ is just the probability density of $H$ in the time-averaged ensemble.

Now to define mixing we consider whether the following holds: $$ \lim_{t \rightarrow \infty} \ \langle \rho_{0}\left( \mathbf{q}, \mathbf{p} \right) \ G\left( \mathbf{q}, \mathbf{p} \right) \rangle_{E} = \langle \rho_{0}\left( \mathbf{q}, \mathbf{p} \right) \rangle_{E} \ \langle G\left( \mathbf{q}, \mathbf{p} \right) \rangle_{E} \tag{10} $$ where $\rho_{0}$ is just the initial value of $\rho_{t}$.

If the system, for every $E$ and functions $\rho_{0}$ and $G$, satisfies the above relationship, the system is said to be mixing [i.e., pages 1948-1949 in Penrose, 1979]:

Mixing can easily be shown to imply ergodicity (e.g. Arnold and Avez (1968, p20); the equivalence of our definition of mixing and theirs follows from their theorem 9.8), but is not implied by it; for example, as mentioned earlier, the harmonic oscillator is ergodic but not mixing... The precise definition of mixing is... 'whether an ensemble of isolated systems has any tendency in the course of time toward a state of statistical equilibrium'...

Note that mixing is not sufficient to imply a system will approach equilibrium [i.e., page 1949 in Penrose, 1979]:

Mixing tells us that the average $\langle G \rangle_{t}$ of a dynamical variable $G$, taken over the appropriate ensemble, approaches an equilibrium value $\langle G \rangle_{eq}$; it does not tell us anything about the time variation of $G$ in any of the individual systems comprised in that ensemble. To make useful predictions about the behaviour of G in any individual system we must show that the individual values of G are likely to be close to $\langle G \rangle$, i.e. that the fluctuations of $G$ are small, and to do this we have to use the large size of the system as well as its mixing property...

Additional and/or Related Answers

• Discussion of collisionless Boltzmann equation (i.e., the Vlasov equation) at https://physics.stackexchange.com/a/177972/59023
• Discussion of affect of instabilities on Liouville's theorem at https://physics.stackexchange.com/a/300053/59023

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