# Ergodicity

Let us note that Liouville's theorem does not prevent the microstate of a system to be confined in a particular region of the constant-energy phase space hypersurface, or to move "mostly" in that region^{[1]}. In this case we surely have:

*do*occur in physical systems, in particular those with few degrees of freedom. Planetary systems are an example since the orbits of the planets are approximately stable and thus the representative point of the system always remains in the same region of the phase space; if it "spanned" all the accessible phase space hypersurface of constant energy then the orbits would mess up completely and the system wouldn't be stable. There are also however many-particle systems whose representative point remains almost always in the same region of phase space. An example is the

*Fermi-Pasta-Ulam*system, a chain of anharmonic oscillators (namely every particle of the system is subject to the potential

^{[2]}). In this case it turns out that if we give some energy to a single particle, this energy is

*not*distributed along all the system as one would expect if the representative point of the system spanned the entire accessible region of phase space. However this kind of system is a bit "pathological" since it has been shown that in the continuum limit the Fermi-Pasta-Ulam system has an

*infinite*number of conserved quantities, not just one (as we require).

We are interested, however, in systems where such phenomena do not occur. In particular in order to establish a link with the microcanonical ensemble we would like to discuss systems where the representative point "spans" the whole accessible phase space, namely it spends on average the same time covering different regions of the constant energy hypersurface.
This property of dynamical systems is known with the name of *ergodicity*. Its most intuitive definition (although not the most useful, as we will shortly see) is the following:

**Theorem**(Ergodicity, I)

A dynamical system is said to be *ergodic* if on the hypersurface of constant energy the time evolution of almost every point eventually passes arbitrarily close to any other point.

The expression "almost every point" of the hypersurface means that we are considering it up to a set of null measure. This is needed to avoid problems with strange or unusual configurations: in this way for example we are excluding the possibility that all the particles move precisely at the same velocity in neat rows. As we have said this definition is not very useful, nor it is really clear since it does not follow explicitly from it that the representative point covers different regions of phase space on average in the same time.

In order to give a much more useful definition of ergodicity we first must introduce the concept of *ergodic component* of a set (which we think of as the hypersurface of constant energy):

**Theorem**(Ergodic component)

Let be a subset of the phase space. Then a set is called *ergodic component* of if it is invariant under time evolution, namely:

Intuitively, different ergodic components of a same set are subsets that are not "mixed" together by time evolution. Now, we can give another definition of ergodicity:

**Theorem**(Ergodicity, II)

A dynamical system is said to be *ergodic* if the measure of every ergodic component of the hypersurface of constant energy is either zero or equal to the measure of . In other words, if is the measure defined on phase space then the system is *ergodic* if for any ergodic component of we have or .

This means that a dynamical system is ergodic if the hypersurface of constant energy is invariant under time evolution up to a set of null measure.

Even if it is not immediately clear these two definitions are equivalent, but we won't show that. What we want to do now is to show that if a system is ergodic according to this last definition then the time average and the microcanonical ensemble average of an observable coincide.

**Theorem**

If a system is ergodic according to the second definition and is an observable, then time average and ensemble average coincide:

*Proof*

First of all, let us note that:

We now define:

Let us therefore consider a sequence (with of course ) monotonically increasing and such that , and call . Then:

We thus have found that:

Therefore, we now have to show that from the fact that almost everywhere it follows that .
We have^{[3]}:

This means that:

So, we now know that if a system is ergodic then the microcanonical ensemble is well defined.
But how can we know if a system is ergodic or not? Unfortunately we still don't know: this is to date an open problem.
We can however cite two other important systems which can be not ergodic: magnets and glasses. A magnet (as also shown throughout Statistical mechanics of phase transitions) can be considered as composed of small orientable magnetic dipoles (the spins of the atoms); at high temperatures the system is "disordered" and the dipoles are not aligned, but when the temperature becomes smaller than the so called "critical" one these dipoles align along any of the possible directions in space. The system thus spontaneously breaks its internal symmetry; such phenomena lead to ergodicity breaking: in fact when it can be shown that the time it takes the system to spontaneously rearrange its magnetization along another direction grows with the dimension of the system. This means that in the thermodynamic limit the system will *always* remain in the same configurations, and so its representative point will not visit all the available regions of phase space (note that the configuration of the system is now given by the spin configuration, not the positions of the particles).
The same argument applies to other kinds of phase transitions that break a symmetry of a given system, for example the solidification of a fluid.
Glasses are much more complicated systems, and many of their properties are still unknown. Their main characteristic is that they are nor crystalline solids nor fluids, so strictly speaking they are not at equilibrium: they tend to approach a crystalline configuration, but the process takes insanely huge amounts of time (glass dynamics is often referred to as "sluggish dynamics").

- ↑ This means that the time evolution of the representative point is such that it is much more probable to find it in determinate regions of phase space than others.
- ↑ A simple harmonic potential would be too simple: with a proper change of coordinates, in fact, the system can be described as a set of independent particles.
- ↑ We consider a thin energy shell instead of an hypersurface because it makes things simpler.