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)