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This shows a 2-dimensional 'electron crystal', formed when very cold electrons at low density are confined to a disk. The vertices are the electrons. The edges are just to help you see the pattern.
To minimize energy, the pattern wants to be a lattice of equilateral triangles. But since such a lattice doesn’t fit neatly into a disk, there are also some red and blue ‘defects'.
5 triangles meet at each red vertex. 7 meet at each blue vertex.
Now it's time to make some conjectures!
(1/n)
Take N points 𝑥ᵢ on the unit disk, arranged so as to minimize the energy
I conjecture that when N is large, when we form the Delauney triangulation of these points, most points will be connected to 6 others, while those on the boundary will be connected to 4. There will also be 'red and blue defects' - points connected to 5 or 7 others.
And I conjecture that there will be 6 more red defects than blue defects!
(2/n)
I don't have much evidence for this conjecture. But I have a hand-wavy argument for why there should be 6 more red defects than blue ones. It's here:
https://johncarlosbaez.wordpress.com/2017/12/07/wigner-crystals/
The easiest way forward would be to take N points on the unit disk, and use some algorithm to move them around until their energy is approximately minimized, and then work out the Delauney triangulation:
https://en.wikipedia.org/wiki/Delaunay_triangulation
and count the defects of various kinds. If you give it a try, let me know!
(3/n, n = 3)
@johncarlosbaez Maybe Euler's formula can be used here? A related well-known fact about e.g. 3D meshes (triangulations of closed surfaces) is that on average every vertex has degree 6 (the exact value is 6-\chi(X)/12V or something like that).
@lisyarus - yes, if you read my article about this you'll see I use Euler's formula to heuristically argue for 6 more red defects than blue ones:
https://johncarlosbaez.wordpress.com/2017/12/07/wigner-crystals/
But it's not a proof!
@johncarlosbaez Ah, I see!
Now I'm also incredibly tempted to write some simulation that would do the thing with any number of points...
@johncarlosbaez Interestingly, I consistently get results that look more like this, i.e. density gets higher closer to the disk edge
@johncarlosbaez Oh, I've read the wiki article and they say they have an extra parabolic potential added to the system. Gonna try that as well.
@lisyarus - Very nice result! Sorry, I forgot to mention the quadratic potential. If it's easy to run 100 simulations with different number of points and see if there are always 6 more red defects than blue ones, that would be cool.
@johncarlosbaez Do you know of the Thomson problem :) https://en.wikipedia.org/wiki/Thomson_problem
I have thought about it, made some progress, and realized I didn't even _know_ if the solution would be stable in other cases :)
BTW: Do you know if the electrons on the exterior/boundary of your disk are in general uniform for a solution?
@rrogers - I know a nice webpage listing solutions to the Thomson problem for up to 75 points. The webpage is defunct, but it's still available on the Wayback Machine:
https://web.archive.org/web/20160224233855/http://www.csun.edu/~hcmth007/electrons/points.html
Here is the solution for 25 points:
(1/2)
@rrogers wrote: "Do you know if the electrons on the exterior/boundary of your disk are in general uniform for a solution?"
You mean are the points on the boundary of the disk exactly equally spaced? That seems incredibly unlikely, except in some special cases.
@johncarlosbaez That's what I thought.
@johncarlosbaez Thanks! I have added it to my collection. Since I have fermented it in my mind (subconscious) for more than a couple of years, maybe I'll try some experiments to test my ideas; so far it's just been thoughts that seem reasonable. It is nice to have a reasonable number of figures (with precalculated points!) available.
@johncarlosbaez I just want to point to the reference https://arxiv.org/pdf/cond-mat/9707204.pdf (section V and Appendix A), where they discuss the same.
@The_Correlator - excellent, thanks! This topic is mainly a hobby for me so I haven't been digging into the literature. I'm glad to know that experts are thinking about this.