Physical distinction between mixing and ergodicity

How can one in a very contrasting manner distinguish between the physical meaning of mixing dynamics and that of ergodic dynamics? More precisely, is one a stronger condition than the other? (which begs to ask questions such as: are ergodic systems also mixing? or vice-versa).

Usually by "mixing" one means the quick decorrelation of system properties (more correctly averages of system properties) from the initial conditions. On the other hand, rather similarly by ergodicity one means equal time and ensemble averages independently of initial conditions. Please feel free to use examples that you see fit, in order to shed a clear light on the difference behind these two concepts.

Mixing is a very physically intuitive concept: a set of particles saitsfying a small spread of uncertainty in their initial conditions, follow paths that enter (nearly) into (nearly) any region, and in a relatively "uniform" way: After sufficiently long a period of time, the percentage of them found within in that region, even within a short period, is proportional to the volume of the region.

Consider the following intuitive example. We consider a (small) amount of uncertainty in the initial conditions, which means we consider not a point but a small volume to begin with. Picture it as being a bit of black colour in the phase space, where the rest of the phase space is white. Now suppose the dynamics really is mixing: somebody is stirring the phase space as if it were a can of paint (into which we put this bit of colour). If the dynamics is truly "mixing", then after a while, the colour will be, as far as the eye can tell, more or less evenly spread throughout the phase space. This is the intuitive example of "mixing" dynamics. Another way of putting this is, even a small deviation in the initial condition eventually results in a large deviation in the state.

Ergodicity is more of a mathematical notion than a physical notion. It means that the time averages are (nearly always) equal to the phase averages. This is the "dual" to the above notion. The ergodic theorem says that this holds as long as the phase space cannot be decomposed into two disjoint invariant subsets of positive measure ("metric transitivity") . Well, you could be generous and say that this is physical too.

A trivial example of ergodic dynamics is irrational rotation on the torus. consider the two-dimensional surface of a three-dimenional doughnut. Or, what is the same, consider the unit square with its edges identified, so a particle that reaches the boundary immediately re-appears at the opposite edge. The dynamics is simple unaccelerated free motion. The initial condition consists of the initial position plus a velocity vector. If the slope of the velocity vector is rational number, the particle will eventually return to its original position and the motion is not ergodic since there are regions it will never reach. But such initial conditions constitute a set of measure zero. "Nearly all" initial conditions have a velocity vector with irrational slope. On such trajectories, the time average of an observable equals the phase average. So this dynamical system is ergodic.

• I am being very sloppy about the mathematical details, without actually introducing any conceptual confusions. I hopp..... – joseph f. johnson Dec 6 '15 at 0:34
• I agree with your starting points, but i think you re wrong about your last point: ergodicity does imply the average time spent in a region is proportional to its corresponding measure. To see this, take the two average expressions (time and phase space ave) for the indicator function as the phase function we are interested to average, then it follows immediately that the time average reduces to the fraction of time spent in the region, and the phase average simply reduces to the measure of that region. – user929304 Dec 6 '15 at 1:12
• True. I was too sloppy. I will think how to express in words this "uniform"...that is partly the difference, I think. – joseph f. johnson Dec 6 '15 at 2:36
• +1 for the edit. Definitely eager to read about any further additions you may have, if hopefully time allows. – user929304 Dec 6 '15 at 19:42
• Yes, I am working on it: mixing paint – joseph f. johnson Dec 6 '15 at 19:46

Yes, these conditions are one stronger than the others. Actually, there are various types of "mixing" (in descending order of strength): multiple mixing (omitted here), mixing, weak mixing.

I will not write all the definitions down, and I suggest to see this book (section 1.6, page 22).

In addition, we have this chain of implications (proved in the reference): $$\text{mixing}\Rightarrow\text{weak mixing}\Rightarrow\text{ergodicity}$$

In addition, it can be shown that the notions are not equivalent: there are ergodic systems that are not weakly mixing, and weakly mixing systems that are not mixing.

As far as I know, the concept of mixing is introduced for convenience: it is often easier to prove the (weak) mixing property than ergodicity directly, and once mixing is proved, ergodicity follows.

• Thanks a lot, very important that you clarified the hierarchy of implications! Unfortunately I do not have access to the referred book yet (I will try to but that specific chapter you mentioned if I can), so would you be so kind to briefly elaborate on how from the definition of mixing (or weakly) ergodicity follows? – user929304 Dec 4 '15 at 11:31
• Well, it is a little bit technical, if I have time I will elaborate later... – yuggib Dec 4 '15 at 12:02
• Dear yuggib, look very much forward to the addition you were going to consider few days ago, hopefully if time allows. – user929304 Dec 11 '15 at 11:52
• @user929304 Yes, I have been a little bit busy these days sorry...Hopefully I will write something in the weekend ;-) – yuggib Dec 11 '15 at 12:01

This is not a full answer, but just an excuse to show a closely related animation I made (more information here): What you see is phase space (top panel) and an energy-vs-position graph (bottom panel). The dynamics are simply those of a massive particle in a potential well, where that potential is shown by the red curve in the lower panel. An ensemble of systems is being considered, each represented by a blue pixel. A small distribution of initial states is considered.

As I understand mixing, it means that the ensemble seems to approach a smoothened, steady state distribution. Indeed this seems to be the case, as the ensemble swirls up. In detail, though, it always contains fine stripes and if the motion were reversed, the ensemble would return back to the initial compact region. See the H-theorem wikipedia article for discussion about what this means for entropy.

As I understand it, ergodicity is the idea that mixing will cover "all accessible phase space". Usually "accessible" is taken to mean any state with the same energy. As you can see however, the ensemble never mixes into the same-energy states in the left half of the potential well. So I think this is a good illustration of why either "accessibility" is a subtle concept, or ergodicity is not generally true.

• Thanks for your answer and animations, neat! Would be great if you could elaborate a tad more on your last paragraph. When you say "if the motion were reversed, the ensemble would return back to the initial compact region" so surely this is not a mixing case, since if i m not mistaken in case of mixing the system soon decorrelates from initial conditions (i.e. the initial region is not recurrent then) – user929304 Dec 6 '15 at 19:49
• I think systems can be mixing but also reversible. All classical dynamics has the reversibility paradox. If however you don't reverse but just move forward in time, the ensemble only becomes more mixed. – Nanite Dec 6 '15 at 20:14
• Note that for classical ensembles, there is reversibility but NOT recurrence. This is because ensembles have an infinte number of degrees of freedom, which are the probabilities for every possible point in phase space. – Nanite Dec 6 '15 at 20:17