The Universal Pattern Popping Up in Math, Physics and Biology (2013)(quantamagazine.org) |
The Universal Pattern Popping Up in Math, Physics and Biology (2013)(quantamagazine.org) |
N. Benjamin Murphy and Kenneth M. Golden* (golden@math.utah.edu), University of Utah, Department of Mathematics, 155 S 1400 E, Rm. 233, Salt Lake City, UT 84112-0090. Random Matrices, Spectral Measures, and Composite Media.
Quanta is not doing hypey PR research press releases, these are substantive articles about the ongoing work of researchers.
"We consider composite media with a broad range of scales, whose effective properties are important in materials science, biophysics, and climate modeling. Examples include random resistor networks, polycrystalline media, porous bone, the brine microstructure of sea ice, ocean eddies, melt ponds on the surface of Arctic sea ice, and the polar ice packs themselves. The analytic continuation method provides Stieltjes integral representations for the bulk transport coefficients of such systems, involving spectral measures of self-adjoint random operators which depend only on the composite geometry. On finite bond lattices or discretizations of continuum systems, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. In this lecture we will discuss various implications and applications of these integral representations. We will also discuss computations of the spectral measures of the operators, as well as statistical measures of their eigenvalues. For example, the effective behavior of composite materials often exhibits large changes associated with transitions in the connectedness or percolation properties of a particular phase. We demonstrate that an onset of connectedness gives rise to striking transitional behavior in the short and long range correlations in the eigenvalues of the associated random matrix. This, in turn, gives rise to transitional behavior in the spectral measures, leading to observed critical behavior in the effective transport properties of the media."
In this lecture we will discuss computations of the spectral measures of this operator which yield effective transport properties, as well as statistical measures of its eigenvalues.
So a lecture and not a paper, sadly.
Wow, that kind of reminds me of the process of evolution in that it seems so random and chaotic at the most microscopic scales but at the macroscopic, you have what seems some semblance of order. The related graph also sprung to mind just how very like organisms repel (less tolerance to inbreeding) but at the same time species breed with like species and only sometimes stray from that directive. What is the pattern that underlies how organisms determine production or conflict with other organisms and can we find universality in it?
I guess it's called "universality" for a reason. I suppose if we look hard enough, we'll see it in more things. I read the article and I'm hoping some brilliant minds out there can dissect musical tastes in the same way. I'd love to see if it could relate to what we find harmonious in music and what we find desynchronous via different phase, frequency and amplitude properties.
> I'm hoping some brilliant minds out there can dissect musical tastes
There has to be some reason there are "Top 10" listings for video games, music, art, tv, movies, anime, vacation destinations, toys, interior designs, historical buildings in NYC, et. al.
Certainly there is a great deal of variance in the order and membership of these lists, but you do find a lot in common. Without some underlying pattern or bias, I don't think we'd see this in so many places so consistently.
I am fairly convinced there is something to do with biological efficiency around information theory that drives our aesthetic preferences.
- https://www.quantamagazine.org/the-universal-pattern-popping...
- https://www.quantamagazine.org/tag/universality/
- https://en.wikipedia.org/wiki/Universality_class
- https://en.wikipedia.org/wiki/Renormalization_group
-https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...
But the big question is, does this "Universal" distribution match up to any well known probability distribution? Or could it be described by a relatively simple probability distribution function?
There have been a large variety of point processes explored in the literature, including some with repulsion properties that give this type of “universality” property. Perhaps unsurprisingly one way to do this is create your point process by taking the eigenvalues of a random matrix, which falls within the class of determinantal point processes [1]. Gibbs point processes are another important class.
[1] https://en.wikipedia.org/wiki/Determinantal_point_process
I wonder if the semi-random "universality" pattern they talk about in this article aligns more closely with what people want from song shuffling.
Why would it be? A random shuffling of a unique set remains a unique set.
It's only when "next song is picked at random each time from set" which you're bound to hear the same song twice, but that's not a random playlist shuffling (shuffling implies the new set is created at once).
In fact one could argue that this is a definition of an interesting system: It has to strike a balance between being completely ordered (which is boring) and being completely random (which is also boring).
DNA as a perfect quantum computer based on the quantum physics principles.
"How Physicists Approximate (Almost) Anything" (Physics Explained)
https://www.youtube.com/watch?v=SGUMC19IISY
If you are citing some crank with another theory of everything, than that dude had better prove it solves the thousands of problems traditional approaches already predict with 5 sigma precision. =3
Are they also cranks? Seems it at least warrants investigation.
That is a better question. =3
Eg https://arxiv.org/abs/0906.0510
See the authors- in terms of contemporary mathematics they are pretty much as far from a crank as it's possible to be. Universality seems to be some sort of intrinsic characteristic of the distribution of eigenvalues of certain types of random matrices which crop up all over the place. That seems interesting and the work is serious academic work (as you can see from the paper I linked) and absolutely doesn't deserve the sort of shallow dismissal you have applied.
I realize what I'm saying is very gestural. The analogous context I'm imagining is deriving blue noise distributed points from randomly distributed points: intuitively speaking it's necessary to inspect the actual distributions of the points in order to move the points toward the lower entropy distribution of blue noise, which means "consuming" information about where the points actually are.
The "random song" thing is similar: in order to make a shuffle algorithm that doesn't repeat, you need to consume information about the history of the songs that have been played. This requirement for memory allows the shuffle algorithm to produce a lower entropy output than a purely random process would ever be able to produce.
So hearing that a "purely random matrix" can have these nicely distributed eigenvalues threw me off for a bit, until I realized that observing the eigenvalues has some intrinsic computational complexity, and that it requires consuming the information in the matrix.
Again, this is all very hunchy, I hope you see what I'm getting at.
Thanks for the reflection though. I'm definitely gonna be thinking about the physical thermodynamics stuff differently after digging into this.