(ChaosBook.org blog, chapter Deterministic diffusion)

Roman 2010-11-20 I think the periodic orbit expression for the diffusion constant is wrong. Imagine you have a compact chaotic (ergodic, mixing) flow which has no relative periodic orbits. Then periodic orbit theory would seem to predict D=0, which is clearly wrong. Otherwise the flow would not be mixing & ergodic. Diffusion has to do with stretching by different periodic orbits in different directions, so I would expect D to depend on Floquet eigenvectors as well as Floquet exponents, not on whether the periodic orbits are relative or not.

Predrag 2010-11-21 I'm fairly sure the formula is correct for periodically tiled lattices. Diffusion is defined for infinite, unbounded space, so I am not sure what you mean by a compact chaotic (ergodic, mixing) flow having diffusion:

lim_{t \to \infty} (x(t) -x(0))^2/t

goes to zero for a compact domain. Using Floquet eigenfunctions would have been appealing, but they do not show up in our derivation at all.

My formulas are not quite right yet for streamwise and spanwise (very slow!) diffusion in plane Couette and (axial) diffusion in Couette-Taylor, and if you can make sense of ergodic relaxation toward natural measure in the compact case as some kind of local diffusion, that would be interesting.

Roman 2010-11-21 I was thinking of diffusion in a local sense (e.g., on finite times). For instance, if there are several different time scales with large enough separation, one might expect a range of times over which the effect of advection is diffusive-like. This certainly is the case for weakly non-integrable flows. If there is no separation of scales, maybe the advection is never diffusive-like. Couette-Taylor is a good example, we should definitely discuss that one in more detail.

NG notes

Chaos: what is it good for? TRANSPORT! Measurable predictions:

1. cold atom lattice diffusion constant:

1. washboard mean velocity:

That Smale’s “structural stability” conjecture turned out to be wrong is not a bane of chaotic dynamics - it is actually a virtue, perhaps the most dramatic experimentally measurable prediction of chaotic dynamics.

As long as microscopic periodicity is exact, the prediction is counter-intuitive for a physicist - transport coefficients are not smooth functions of system parameters, rather they are non-monotonic, nowhere differentiable functions.

At first glance this seems bizarre. Transport coefficients are averaged quantities, right? as thought about from kinetic theory or Green-Kubo formulae. So this Weierstrass-like behavior is astonishing. However, it has been seen, most cleanly in Klage's simulations.

It could perhaps be seen in experiments; the idea is explained along with Kasivajhula java AFM simulator and related student projects.

I have rewritten a section in the intro to ChaosBook emphasizing this signature of chaotic dynamics. Please have a look at 2nigeloupbook.pdf which, read in tandem with ChaosBook.org/chapters/diffusion.pdf hopefully explains this amazing thing about chaos to a physicist on the street.

If you have questions or editing suggestions, let me know.

— Predrag Cvitanovic 2009-01-12 04:30

Thanks very much to Predrag for giving me credit here and in Section 1.8 of the Chaos Book. The above section nicely presents in a nutshell what, also in my opinion, is the essence of irregular, or fractal, transport coefficients generated by topological instability of dynamical systems under parameter variation with respect to potential measurements. I may mention that such irregular parameter dependencies were certainly known since many decades for quantities like Lyapunov exponents, dynamical entropies and the like. My own contribution was just to show that this also holds for measurable physical quantities like diffusion coefficients.
There is certainly much more to say on this, however, the main problem to me at the moment is to carefully check these predictions in experiments… would be nice, and very important, if someone endeavoured to do this! I am trying to convince experimentalists to do so since many years but did not succeed up to now. I really think this is a nice, important phenomenon (also for potential applications), but for some reason this has still not been picked up by a wider audience (perhaps because the detailed calculations can be a bit tricky?).
Some further remarks:
1. My knowledge about fractal transport coefficients is essentially summarized in the first part of the book Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics (World Scientific, 2007). For anyone who wants to work on this topic, this might be a good reference to get a glimpse at the whole picture. In this book I am discussing this phenomenon for many different dynamical systems, and I also mention quite some real physical systems for which I would expect to see this in experiments (well, some exp. indication already exists, e.g., in antidot lattices, Josephson junctions, ratchets,… but close links between such experiments and first principles theories are still missing).
2. I may also mention that recently mathematicians started to get interested in such issues, see, e.g., G.Keller et al, Nonlinearity 21, 1719--1743 (2008) assessing mathematically rigorously the fractality of the parameter-dependent diffusion coefficient discussed in the Chaos Book. There is also related rigorous mathematical work by Ruelle, Baladi and others on the (non-)existence of linear response in simple maps, starting from the analysis of the differentiability of SRB measures.
See my book for a long list of references on all aspects mentioned above. — Rainer Klages 2009-02-11

Predrag 2010-07-21 I find Linear and fractal diffusion coefficients in a family of one dimensional chaotic maps use of Takagi functions by Knight and Klages quite interesting. Inter alia, they say: “The structure of the Markov partitions varies wildly under parameter variation. The method we employ to understand the Markov partitions involves iterating the critical point. The set of iterates of this point form a set of Markov partition points for the map. Hence we call the orbit of the critical point a ‘generating orbit’. If the generating orbit is finite for a particular value of parameters, we obtain a finite Markov partition. We can then use the finite Markov partition to tell us about the diffusive properties of the map and hence the structure of the diffusion coefficient.”

Predrag 2011-08-03 Remember to also read Capturing correlations in chaotic diffusion by approximation methods by Georgie Knight and Rainer Klages.

14 Feb 2012 01:14:40 GMT (124kb,D)

Predrag 2012-02-15 Read An analytic approximation to the Diffusion Coefficient for the periodic Lorentz Gas by Angstmann and Morriss. The say: “An approximate stochastic model for the topological dynamics of the periodic triangular Lorentz gas is constructed. The model, together with an extremum principle, is used to find a closed form approximation to the diffusion coefficient as a function of the lattice spacing. This approximation is superior to the popular Machta and Zwanzig result and agrees well with a range of numerical estimates.”