====== ChaosBook.org blog ======

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

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

;-) ** Predrag to Nigel**:
I've been barking up every tree for years, but this is the first time that a light went on upstairs, thanks! The questions are good, so sorry for the delay, but I used them to rewrite a section in the intro to ChaosBook: please have a look at

{{:2nigeloupbook.pdf}}

which, read in tandem with 

[[http://chaosbook.org/version12/paper.shtml#diffusion|Chapter: Diffusion]]

hopefully addresses your remark? Let me know what to edit next...

 --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-01-12 04:30//

----

:-| **Nigel to Predrag:**
you didn't tell me the most amazing thing that I just learned about from your chaos book.  I refer to
the passage in chapter on Deterministic diffusion (see [[http://chaosbook.org/version12/paper.shtml#diffusion|ChaosBook.org/chapters/diffusion.pdf]]):
     
     Chaos: what is it good for? TRANSPORT! Measurable predictions:

           - washboard mean velocity figure 24.7 a), [[http://chaosbook.org/version12/paper.shtml#diffusion|{{:pc_afm.png?400|}}]]
           - cold atom lattice figure 24.7 b),{{:pc_ladderSmall.png|}}
           - AFM tip drag force figure 24.7 c). 

    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 counterintuitive for a physicist - transport
    coefficients are not smooth functions of system
    parameters, rather they are nonmonotonic, nowhere
    differentiable functions.

This is totally bizarre.  Transport coefficients are averaged quantities,
at least as thought about from kinetic theory or Green-Kubo formulae.  So
this Weierstrass-like behavior is astonishing.  How could you resist
telling me about it??!!  I got the hint about this from Greg Eyink, who I
was talking about last week in Cambridge.

Now my question: do you know an explanation that a simple-minded
physicist like me could understand? And another question: how could I see
this in experiment?  Has it been seen in simulation?  etc....

--- //Nigel Goldenfeld, nigel [snail] uiuc.edu, 2008-12-20 12:25 //