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chaosbook:diffusion [2009/02/12 11:32]
wikiadmin
chaosbook:diffusion [2012/02/15 09:31] (current)
predrag added a diffusion paper by Morriss.
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 +<- [[:​chaosbook]]
  
 ====== Chapter: Deterministic diffusion ====== ====== 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 ===== ===== Lack of structural stability is good news for chaos =====
  
-8-) [[diffusion:​nigel|NG notes]]+8-) [[chaosbook:diffusion:​nigel|NG notes]]
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  --- //​[[r.klages@qmul.ac.uk|Rainer Klages]] 2009-02-11//​  --- //​[[r.klages@qmul.ac.uk|Rainer Klages]] 2009-02-11//​
  
-~~DISCUSSION~~+**Predrag 2010-07-21** I find  [[http://​arxiv.org/​abs/​1007.3393|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  [[http://​arxiv.org/​abs/​1107.5293|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 [[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."​
chaosbook/diffusion.1234467152.txt.gz · Last modified: 2009/02/12 11:32 by wikiadmin