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--- chaosbook:diffusion [2010/07/21 05:02]
predrag diffusions confusion reference
+++ chaosbook:diffusion [2012/02/15 09:31] (current)
predrag added a diffusion paper by Morriss.
@@ Line 3: / Line 3: @@
 ====== Chapter: Deterministic diffusion ======
-(ChaosBook.org blog)
+(ChaosBook.org blog, chapter [[http://chaosbook.org/paper.shtml#diffusion|Deterministic diffusion]])
+===== A description of diffusion locally? =====
+**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.
 ===== Lack of structural stability is good news for chaos =====
@@ Line 52: / Line 68: @@
 us about the diffusive properties of the map and hence the structure of the diffusion coefficient."
-~~DISCUSSION~~
+**Predrag 2011-08-03** Remember to also read  [[http://arxiv.org/abs/1107.5293|Capturing correlations in chaotic diffusion by approximation methods]] by Georgie Knight and Rainer Klages.
+Feb 2012 01:14:40 GMT   (124kb,D)
+**Predrag 2012-02-15** Read [[http://arxiv.org/abs/1202.2904|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."

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