chaosbook:diffusion
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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. |
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| ====== Chapter: Deterministic diffusion ====== | ====== Chapter: Deterministic diffusion ====== | ||
| - | (ChaosBook.org blog) | + | (ChaosBook.org blog, chapter [[http://chaosbook.org/paper.shtml#diffusion|Deterministic diffusion]]) |
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| + | ===== A description of diffusion locally? ===== | ||
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| + | **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. | ||
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| + | **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: | ||
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| + | //lim_{t \to \infty} (x(t) -x(0))^2/t// | ||
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| + | 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. | ||
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| + | 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. | ||
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| + | **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 ===== | ||
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| us about the diffusive properties of the map and hence the structure of the diffusion coefficient." | 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. |
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| + | 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.1279713760.txt.gz · Last modified: 2010/07/21 05:02 by predrag
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