Timeline for Why do many people link entropy to chaos?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Jul 1, 2016 at 11:16 | comment | added | gatsu | I am sorry but I have put -1 for now. Not because the answer is completely wrong but because it discusses only mainly Hamiltonian chaos without discussing issues like the KAM theorem and its relevance for the actual important stuff like ergodicity and most of all entropy. One could have easily mentioned the topological ($\epsilon-$) entropy for instance very much used in dynamical systems and discuss how it is bound from above by the equilibrium Gibbs/Boltzmann entropy for instance. | |
| Jun 24, 2016 at 15:26 | comment | added | TLDR | I agree. After checking Goldstein again, I should correct my earlier comment: strange attractors are defined in the text as being fractal. | |
| Jun 24, 2016 at 7:38 | comment | added | Wrzlprmft | @fs137: that a strange attractor is an attractor with dimension > 1 – I do not now about Goldstein, but that definition includes tori, which I am pretty certain most people would consider not strange. Usually, you would require the dimension to be non-integer for the attractor to be strange. | |
| Jun 24, 2016 at 3:54 | comment | added | TLDR | Well, I certainly wasn't trying to say they are the same, just that chaos is one route to the type of ergodicity that is needed in order for statistical mechanics to be valid. Both ergodic and chaotic systems have distinct general definitions. As I understand it, a chaotic system can always be made ergodic with an appropriate choice of state space, but an ergodic system is not necessarily chaotic. | |
| Jun 24, 2016 at 0:17 | comment | added | knzhou | @fs137 I was under the impression that chaos was much, much more special than just ergodicity. Are you saying they're the same thing? | |
| Jun 23, 2016 at 23:48 | comment | added | TLDR | I guess there might be a few definitions of attractor, and strange attractor. From wikipedia, I had the impression that an attractor needs a non-trivial basin (i.e. the basin of attraction is an open set, of which the attractor is a proper subset), while strange attractors are fractal. However, Goldstein might say that an attractor is any set closed under time evolution, and that a strange attractor is an attractor with dimension > 1. In physics, I would only expect fractal structure to be apparent in a scaling limit (e.g. overdamped with strong driving force & interactions, or dynamical RG). | |
| Jun 23, 2016 at 23:10 | comment | added | QuantumBrick | As far as I know only dissipative systems can have attractors. | |
| Jun 23, 2016 at 21:52 | vote | accept | Malkoun | ||
| Jun 23, 2016 at 21:45 | comment | added | CuriousOne | I am not aware that chaotic attractors require driven-dissipative systems under the KAM-theorem nor do I understand how a dissipative system (from a physical rather than mathematical perspective) can even have a strange attractor, as it is under the thumb of the fluctuation-dissipation theorem, which will automatically smear it out into a "well behaved" average. | |
| Jun 23, 2016 at 21:41 | comment | added | TLDR | @Malkoun: You're welcome. For discussing with laypeople, I think it depends a little on their background. Since a lot of people are familiar with statistics (and digital security), a maximal uncertainty justification can be effective I think. To CuriousOne: I've only heard of strange attractors in the context of driven-dissipative systems (but I would be very interested in hearing if you know of conservative examples). This would indicate to me that strange attractors could only appear in nonequilibrium stat mech. | |
| Jun 23, 2016 at 21:35 | comment | added | Malkoun | Thank you so much for the link with ergodic theory, and the ergodic hypothesis. I understand much better where it comes from, now, the link between chaos and entropy. Still, if I were to discuss the concept with a "layman", so to speak, or young people, I would prefer to avoid using the word "chaos" at least at first. | |
| Jun 23, 2016 at 21:33 | comment | added | CuriousOne | I don't know if naive statistical mechanics allows for a whole lot of leeway, there, does it? Can one modify the ergodic hypothesis for strange attractors and still recover standard thermodynamics? | |
| Jun 23, 2016 at 21:28 | comment | added | TLDR | Depends on what space you are considering ergodic behavior in. Conserved quantities certainly prevent a classical system from filling all of phase space. For dissipative systems, you need to consider the trajectory of the attractor itself, and (possibly) the trajectory of the system within the attractor. In my answer, however, I only addressed non-dissipative systems. | |
| Jun 23, 2016 at 21:24 | comment | added | CuriousOne | You mean a chaotic system is not automatically ergodic because the attractor is not guaranteed to fill the entire phase space, right? | |
| Jun 23, 2016 at 21:21 | history | answered | TLDR | CC BY-SA 3.0 |