by Tomas Milanovic
On the thread Trends, Change Points and Hypotheses, the issue of ergodicity was mentioned numerous times, and some clarification of this concept is needed.
So what is and what is not ergodicity and why does it matter in physics?
|It is annoying to see that many people use the word ergodicity without knowing what it means and how it should be interpreted. It is not annoying because it is mostly wrong but it is annoying because those readers who are not intimately familiar with these concepts, will get confused and ultimately get farther from understanding rather than nearer.So what is and what is not ergodicty and why does it matter in physics?The ergodic property is a very general mathematical property of measurable sets. A measurable set can be defined as (X,µ) where X is some set (f.ex the standard R^n cartesian space) and µ is a measure. For purists, I left out the sigma algebra, it is not necessary to understand the rest
Now let’ take some transformation T : X -) X , T is a map, a function
And request that T preserves the measure µ , e.g µ (T^-1 (A)) = µ(A)Okay, that’s everything we need in matter of definitions – you need a space X , a metricsµ , a transformation T and we want this T to preserve the metrics.
The ergodic property of this triplet is simply the statement :
If for any A that is a subset of X, we have T^-1(A) = A then µ(A)=1 or 0What does that mean in words ?If there are some subsets A (imagine a set of points) that are left invariant by the transformation T, then their measure (size) is either 0 or the measure of the whole set X (because µ(X)=1).
Even more crudely, an ergodic transformation doesn’t get “stuck” in some particular part of the space.
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 :
Ok and the ergodic theorem says simply that :
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.
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.
So when one studies a chaotic system, one can and must restrict the phase space to the allowable finite subspace which is the attractor.
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
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.