Ergodicity

Now an ergodic transformation has a property which is expressed by the ergodic theorem which is why we are actually interested in ergodicity at all.

Here we must first define an iteration average and a space average on X for some function F. The iteration average of F is Fia = lim (n-oo) 1/n Sum over k [F(Tk(x))].  This rather heavy formula is just saying that you take some point x in X and make it move by applying T on x, then T on T(x) etc.

If you take F of each of these points , and make an arithmetical average, you obtain the iteration average.

The space average is easier :
Tsa = [1/µ(X)] Integral over the whole space X [F.dµ]

Ok and the ergodic theorem says simply that :
If T is ergodic then for any F , Tia=Tsa almost everywhere.
Why “almost” ? Well remember that the measure of subsets left invariant by an ergodic transformation T is 0. This means that there may be isolated invariant points but they don’t matter because they are only here and there.
So the ergodic theorem is saying that if you make an F average on a trajectory constructed by applying T again and again starting in some point X0 then this average is equal to the simple average of F over all points of X in the infinite limit of iterations.

And that’s all we need to know as far as the maths go. There are many interesting additional results and consequences but the real maths are not easy and most readers would probably stop following. The point of this short but rigorous introduction was to demonstrate that ergodicity is not some fog that could be interpreted by anybody as it suits his particular view.

Now why does that matter in physics in general and in the atmospheric system in particular? Well as we have a rigorous and very general mathematical theory, we can now take some particular cases with physical meaning.  So here we go:  X is a cartesian finite dimensional space and its points are states of a dynamical system (each state is defined by N coordinates). X is also called then a phase space.
µ is a probability (yes a probability is (also) a measure)
T is the dynamical law that our system obeys
T(x) are then trajectories (orbits) of the dynamical states in X.         Of course “iterating T” means following the system’s states in time.

Suppose T is ergodic (please note and I stress that I wrote suppose).  It is not a surprise that all this looks like Hamiltonian mechanics because it IS Hamiltonian mechanics. It follows that all those who are not familiar with statistical mechanics and for whom KAM is some chinese abbreviation should abstain talking about ergodicity.  Well and now we can apply everything we already know from above on this particular case.

First there are no subsets of the phase space where the system stays “trapped”. It roams almost everywhere. Besides one can easily demonstrate the non intersection theorem which shows that the system never pases twice through the same state.

We also know with the ergodic theorem that if we follow the states of the system for an infinite time (remember: infinite iterations of T) and take the time average of some parameter of the system, then this average will be equal to the (probability weighted) average of this same parameter over the whole phase space.  This is very interesting because we are mostly interested in the latter while we can experimentally measure only the former.

What is even more interesting is that it also follows that there exists an invariant PDF of µ that doesn’t depend on initial conditions. Of course it depends on T (e.g on the form of the dynamics) but not on the initial conditions.

And THAT is very rich! This means that by taking some (any) initial condition and following the trajectory of the system for a (very) long time, you will obtain empirically ONE PDF, but thanks to ergodicity, you know that you don’t need to redo it for the rest of the infinity of initial conditions because your PDF is the one unique PDF for the whole system.  Caveat: when I say infinite time, I mean it seriously. You have to observe really for a long time and it would take a whole new post to discuss how long one has to observe to apply the ergodic theorem in practice.

So now I hope everybody understands how confused a discussion can become when people mix up stochasticity, randomness and ergodicity.

Last let us make a HUGE qualitative leap and cross from finite dimensional X’s (aka Hamiltonian mechanics) to infinite dimensional X’s (aka field theories). From the mathematical side not much changes – the formulation of the measurable set theory doesn’t prescribe any particular X’s, µ’s and T’s. But physically everything changes – our points become functions, the measures base on square integrable fields and trajectories can no longer be geometrically visualised. Navier Stokes and by extension weather and climate belong to this category.

Of course one can also talk about ergodicity but it takes considerably more skill and training, and no simple analogies to statistical mechanics or thermodynamics work anymore.

And what has chaos to do with all this? Well chaos theory is interested by a particular category of T’s (recall:  T defines the dynamical laws). Namely, T’s with sensitivity to initial conditions, which represent a very large sub set of possible T’s in physics.
These T’s have an important property – they don’t visit the whole phase space (exercise: show why this is not in contradiction with the ergodic property).  The reason is intuitively easy – as the orbits diverge exponentially, they would finish in infinity, which is physically impossible.  That’s why there is (and must be) a physical mechanism which restricts the system to a finite part of the phase space.  This mechanism is energy dissipation and the restricted volume where the system must live forever (or at least untill its dynamical laws don’t change) is called attractor.

So when one studies a chaotic system, one can and must restrict the phase space to the allowable finite subspace which is the attractor.
Once this done, and provided the dynamics has the ergodic property (a chaotic system may be but is not necessarily ergodic), one can apply the regodic theorem and study the probabilities of the different states. Don’t forget that “states” here happen in the infinite dimensional framework of field theories.

Of course there are no mysterious “stochastic” perturbations which make  the fundamental features of behaviour of chaotic systems disappear – Navier Stokes is deterministic and chaotic all the way down to the quantum scales.

I would like to describe now how ergodicity can be used for these particular cases but I am afraid that the post is already too long anyway.

JC comment:  This is a topic that I want to understand better, and I am hugely appreciative of Tomas’ contributions here, see also his previous posts

• Chaos, ergodicity and attractors
• Spatio-temporal chaos

I can’t say that I now understand this, but at least I now know enough not to use the term ‘ergodic’ since I don’t really understand it.  I’m hoping that some further discussion of this will illuminate us all.