Talk:Mixing (mathematics)

Things to mention

Things that could be mentioned, beyond the basic definitions:

• K. E. Petersen (1970) "A topologically strongly mixing symbolic minimal set" Trans. Amer. Math. Soc. 148 (1970), 603-612 We give here a "machinal" construction of a bilateral sequence with entries from 0, 1 whose orbit closure is topologically strongly mixing and minimal. We prove in addition that the flow we obtain has entropy zero, is uniquely ergodic, and fails to be measure-theoretically strongly mixing.

The above talks about blocks, and as far as I know, there are no Wikipedia articles on blocks (which are used all over the place in ergodic theory...) 67.198.37.16 (talk) 06:40, 5 November 2020 (UTC)

Mixing stronger than ergodicity

The text says " Mixing asks for this ergodic property to hold between any two sets A and B, and not just between some set A and X." It seems that the difference is not B vs X, but that mixing requires the non-empty intersection for all n, whereas ergodicity only requires it for some n (\forall vs \exist). I'm not confident enough of this to edit the text. Could someone more familiar with this please check? LachlanA (talk) 11:09, 22 December 2022 (UTC)

This part is false as written : if you take A, B to be the singleton sets on two points in distinct orbits (which will exist as soon as X is uncountable) it will never occur that $A\cap T^{-n}B\not =\emptyset$. I think that to define topological mixing you want A, B to be open.

On the other hand you certainly want "almost all n" and not "all n" in the definition : for a measurable transformation T of a standard Borel spaces, for an arbitrarily large N there will always exist nontrivial open sets A, B for which $A\cap T^{-n}B$ is empty for all $0\leq n\leq N$.

This section "informal explanation" is a mess starting with the fourth paragraph. It should probably be pruned and the relevant information within incorporated in the rest of the article. jraimbau (talk) 13:26, 22 December 2022 (UTC)

Doesn't the total measure need to be = 1 ?

In the section Mixing in dynamical systems this passage appears:

"Let $(X,{\mathcal {A}},\mu ,T)$ be a measure-preserving dynamical system, with T being the time-evolution or shift operator. The system is said to be strong mixing if, for any $A,B\in {\mathcal {A}}$, one has

$\lim _{n\to \infty }\mu \left(A\cap T^{-n}B\right)=\mu (A)\mu (B).$"

Isn't it necessary to assume this is a probability measure space, that is, that µ(X) = 1 ?

Because then the measure of set A or B is the fraction of the total measure of X that A or B possesses, and then µ(A)µ(B) makes sense as the limit when the dynamical system is strongly mixing.

Merge proposal

I propose merging Mixing (physics) and Mixing (mathematics). These articles are about exactly the same concept in dynamical systems, which happens to be important in both physics and math. Further, Mixing (physics) has been a stub since its creation in 2004, even linking to Mixing (mathematics) for "a more detailed explanation", and its content could be easily incorporated into Mixing (mathematics). I am not sure what article title they should be moved to. Mathwriter2718 (talk) 14:43, 7 July 2024 (UTC)

A few relevant pings: @Cosmia Nebula @Linas @Michael Hardy Mathwriter2718 (talk) 14:58, 7 July 2024 (UTC)

Ideas:

1. Mixing (Dynamical Systems)

2. Mixing (Mathematics)

3. Mixing (ergodic)

4. Ergodic theory of mixing pony in a strange land (talk) 18:25, 7 July 2024 (UTC)

How about Mixing (ergodic theory)? –LaundryPizza03 (dc̄) 08:04, 8 July 2024 (UTC)

@User:Cosmia Nebula Mixing (Dynamical Systems) and Mixing (Mathematics) would violate WP:MOS because of the incorrect capitals. They should be Mixing (dynamical systems) and Mixing (mathematics). Michael Hardy (talk) 00:02, 9 July 2024 (UTC)

@User:Mathwriter2718 If you want this to get attention among the communities that can help, you should post to Wikipedia talk:WikiProject Mathematics and Wikipedia talk:WikiProject Physics. Michael Hardy (talk) 00:06, 9 July 2024 (UTC)

Without more content at mixing (physics), I'm inclined to think that the more developed article mixing (mathematics) is the primary one, although I could see scope for a separate article at some later time. Tito Omburo (talk) 01:35, 9 July 2024 (UTC)

Neither of these topics was the one that I first think of with respect to mixing, which can instead be found at Markov chain mixing time. Maybe whatever we do here, we at least need a hatnote? —David Eppstein (talk) 05:41, 9 July 2024 (UTC)

I think Mixing (physics)#Physical mixing should be a redirect to Mixture which is much more detailed (but needs a little work and seems to have experienced an odd editing war of late). That page is general for chemistry/chemical engineering/materials science/physics/etc, we need to stop artificially creating field boxes. Hence Mixing (mathematics) needs to go beyond saying "in physics" and refer to Mixture. Some mention of Phase diagram#Binary mixtures and similar probably as well. There is also all the applied math/materials science on kinetics, for instance Cahn-Hilliard equation and Spinodal decomposition that at least is a "See also", perhaps more. Ldm1954 (talk) 07:12, 9 July 2024 (UTC)

• Don't merge. The "informal introduction" in mixing (mathematics) is already far beyond what the typical physics student (and even physicist) might typically encounter or even recognize as "mixing". (Disclosure: I wrote that section, and maybe some of the rest of that article.)

For example, David Eppstein's comment above: that when he thinks of "mixing in physics", he thinks of Markov chain mixing time without fully realizing that Markov chain mixing is a special case of the formal mathematical definition. This could be fixed by enlarging the Markov chain article to articulate the connection to the formal definition.

Ldm1954 provides another example: he suggests that mixing and mixtures are somehow "the same thing"; they are not. Mixtures are what you get after mixing, after taking the time-to-infinity limit. Before that limit, its mixing. I fear that merging (and redirecting) mixing (physics) into mixing (mathematics) will result in misguided efforts to "fix" mixing (mathematics) to more closely resemble physics and chem textbooks.

A short history of mixing that might help put things into context. The properties of mixtures have been of interest to chemists, physicists and engineers since about forever, but the focus on mixing didn't really take off until the Manhattan Project, and then later really took off in the 1980's with chaos theory. The atom bomb project required not only compressing plutonium, but making sure it mixed will with the initiator (neutron source) and so there were a lot of "RaLa tests" studying the hydrodynamic turbulence and mixing created by high explosives. Turbulence is one route to thermalization, but there are others: the Fermi-Pasta-Ulam-Tsingou problem was (one of?) the earliest attempts at studying the "mixing" of waves (via the overlap of dispersion equations). This plodded along until "chaos theory" erupted in the 1980's. A whole lot of inter-related questions got tangled into one. First, the russians in Novosibirsk made a lot of progress in the 1960's on the formal mathematical behavior of dynamical (physical) systems, proving a variety of ergodic theorems, while at the same time writing books on general topology that included some of these ergodic theorems. Being russians, they were ignored in the West until fractals turned out to be everywhere, and chaos everywhere. Suddenly, ergodicity in chaotic dynamical systems was generating thousands of papers a year. Quantum field theorists who knew the renormalization group were suddenly able to calculate how quickly sand and gravel mixed, plumping for big $$$ consulting fees from cement companies and petroleum drillers. (No joke. One of my friends, who stayed in academia, got money this way. His colleagues were envious. I was envious.) Mathematicians got in on the act, realizing there's some cool math here. Not just the KAM torus stuff, but symplectic manifolds, in general. I recall a book by Svetlana Katok (Anatole's wife) providing rigorous math definitions for classical mechanics, Hamilton-Lagrange, symplectic manifolds, ergodic theory and mixing, all under the same cover. (I recall poorly, I might be confabulating...) There are even overlaps to number theory, because the geodesics on Riemann surfaces are chaotic, ergo moduli space, blah blah. And then, just as quickly as it blew up, it cooled off. Not sure why. Were all the easy results mined out? Chaos and fractals went out of fashion just like bell-bottoms and puffy-sleeved shirts? I dunno. But it feels forgotten and obscure, these days. If you weren't there in the go-go years of the 1980's, you won't have even heard of any of it. Pre-Internet -> forgotten.

So we're back to an older conception, now. Mixtures, not mixing, are still in chemistry books. No one cares how quickly sand and gravel mix, or how quickly neutron sources and plutonium mix. The number theorists found something else to work on, and the physicists decided string theory had more to offer. A few lone wolves like Terrence Tao keep this stuff alive, but he's a super-star, and can do anything he wants. To recap: physics undergrads will mostly not hear about mixing, except in a simplistic sense, for which Mixing (physics) is the right entree, written at the right abstraction level. Then, mixture is for those people who don't care about mixing. Finally, mixing (mathematics) provides the abstract formal definitions palatable to mathematicians, but scary and opaque to the uninitiated physics & chem student. It provides a central theoretical framework, to which the physics and process chemistry people can appeal to, on an as-needed basis. 67.198.37.16 (talk) 21:53, 7 August 2024 (UTC)

Thanks for your comment, I found it very interesting. In my part of the world, it seems like the terms "chaos theory" and "fractal" have died out, but nowadays a lot of people describe themselves as doing "dynamics".

Anyway, I might be able to get behind your vision that Mixing (physics) should be written from the physics student POV and thus shouldn't be merged into Mixing (mathematics), even though the subject of the article is the same in the abstract. However, the current article Mixing (physics) definitely doesn't accomplish that goal at the right abstraction level (as you have put it): do physics undergrads really know what a "measure-preserving transformation" is? I don't know anything about mixing from the physics student POV, so I find it difficult to visualize what your vision would actually look like concretely and how it would be significantly different from the math POV. For these reasons, I feel I must be neutral about this merge at present. Mathwriter2718 (talk) 22:41, 7 August 2024 (UTC)

I'll spend a few minutes thinking on how to make the physics article "more physics-like", but I'm kind of too far away from whatever actual physics people are actually talking about to do a good job of it. It is possible that I'm the one who added the "measure preserving dynamical system" stuff in the lede, and yes, maybe it should be moved down or moved away. I'll try to do that now. 67.198.37.16 (talk) 22:55, 7 August 2024 (UTC)

I wrote a new lede for Mixing (physics) and moved the dynamical systems definition to the bottom. The other sections remain stubs, as before. 67.198.37.16 (talk) 00:41, 8 August 2024 (UTC)

Also: @User:Mathwriter2718, if you want to write about mixing in dynamical systems, I strongly suggest creating a brand new article Mixing (dynamical systems) instead of trying to extend this article. This article is already over-long; adding one more section just makes it even more eye-wateringly overwhelming. There's a lot that can be said about dynamical systems. I assume you are thinking about low-dimensional symplectic manifolds; the high-dimensional stuff is "completely different", as they say, having uneasy overlaps into deep-learning neural nets and large language models and all that jazz. That stuff should really get it's own article, independent of any review of the more traditional 1990's dynamical systems work. Perhaps even ergodicity (dynamical systems) would be an even better target (after explaining that ergodicity is just 1-mixing?) 67.198.37.16 (talk) 22:55, 7 August 2024 (UTC)

Sorry, but I deeply disagree with many of the recent statements, and I think the point of my comments has been missed. IMO we should not fit science into boxes that are isolated, it is Venn diagrams with multiple areas of overlap. What I am unhappy about is the omission of all the applied math/materials science of problems such as the Cahn-Hilliard equation as well as the many, many related areas of phase separation, Avrami equation and the vast literature on this.

I agree that most physics undergraduates, PhD student and even faculty will not know much about this. The reason is that the offshoot of the Manhattan project and later Sputnik was the creation of Materials science to deliberately cross the barriers. Physicists are not involved in this (beyond more recently finding it a potential funding source), but there are many people with joint appointments in Applied Math & Materials Science. (I am not one of them, but I have worked with a few.) I definitely disagree with writing an article from physics students POV, instead write a WP:NPOV one from a John Cahn view, or perhaps Stephen H. Davis (just two). Ldm1954 (talk) 22:49, 7 August 2024 (UTC)

The Cahn-Hilliard equation has nothing to do with mixing, and everything to do with mixtures. First of all, its a differential eqn, fer cryin out loud, and it is not derived from (or derivable from) the standard definition of mixing. It is only defined for differentiable systems, for which an actual derivative can be obtained in some way. It's in three dimensions, without any mention whatsoever of topology. How are you going to apply Cahn-Hilliard to something infinite dimensional, that doesn't have any derivatives? It's a kind-of unrelated topic. Now, if you want to put it into mixing (physics), that's fine, its physics/chemistry. But its got almost nothing to do with conventional concepts of mixing. 67.198.37.16 (talk) 23:03, 7 August 2024 (UTC)

• Delete - Given the nature of the above conversation, perhaps the right thing to do is to delete Mixing (physics). I've written maybe 3/4ths or 7/8th of the article, by adding the mathematical description 19 years ago, and writing a one-size-fits-all lede today. Remove these two, and its a stub. The math description is a two-paragraph synopsis of mixing (mathematics) and is what prompted the merge proposal. The brand new lede onl;y tries to unify this with the two stubby parts, about fluids and aggregates. But the fluids already has a "main article", as does the aggregates, so why do we need an article consisting of three stubby parts that summarize other existing articles? Just delete the thing. Or maybe redirect to mixture. 67.198.37.16 (talk) 01:11, 8 August 2024 (UTC)

  How about redirect Mixing (physics) to Mixing (mathematics)? Mathwriter2718 (talk) 01:13, 8 August 2024 (UTC)