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====== Chapter: Deterministic diffusion ======

(ChaosBook.org blog)

===== Lack of structural stability is good news for chaos =====

8-) [[chaosbook:diffusion:nigel|NG notes]]
----


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

  - cold atom lattice diffusion constant:
[[http://chaosbook.org/version12/paper.shtml#diffusion|{{:pc_afm.png?400|}}]]
  - washboard mean velocity:
{{:pc_ladderSmall.png|}}

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 [[http://www.maths.qmw.ac.uk/~klages/|Klage]]'s simulations.

It could perhaps be seen in experiments; the idea is explained along
with [[http://chaosbook.org/projects/Kasivajhula/index.html|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 {{chaosbook:2nigeloupbook.pdf}} which, read in tandem with [[http://chaosbook.org/version12/paper.shtml#diffusion|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@physics.gatech.edu|Predrag Cvitanovic]] 2009-01-12 04:30//


===== Rainer Klages =====

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
[[http://www.maths.qmw.ac.uk/~klages/cftbook|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., [[http://www.mpipks-dresden.mpg.de/%7Erklages/papers/non8_8_003.pdf|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.
 --- //[[r.klages@qmul.ac.uk|Rainer Klages]] 2009-02-11//

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