Physics theorem of interacting particles
In physics, the poppy-seed bagel theorem concerns interacting particles (e.g., electrons) confined to a bounded surface (or body) formula_1 when the particles repel each other pairwise with a magnitude that is proportional to the inverse distance between them raised to some positive power formula_2. In particular, this includes the Coulomb law observed in Electrostatics and Riesz potentials extensively studied in Potential theory. Other classes of potentials, which not necessarily involve the Riesz kernel, for example nearest neighbor interactions, are also described by this theorem in the macroscopic regime.
For formula_3 such particles, a stable equilibrium state, which depends on the parameter formula_2, is attained when the associated potential energy of the system is minimal (the so-called generalized Thomson problem). For large numbers of points, these equilibrium configurations provide a discretization of formula_1 which may or may not be nearly uniform with respect to the surface area (or volume) of formula_1. The poppy-seed bagel theorem asserts that for a large class of sets formula_1, the uniformity property holds when the parameter formula_2 is larger than or equal to the dimension of the set formula_1. For example, when the points ("poppy seeds") are confined to the 2-dimensional surface of a torus embedded in 3 dimensions (or "surface of a bagel"), one can create a large number of points that are nearly uniformly spread on the surface by imposing a repulsion proportional to the inverse square distance between the points, or any stronger repulsion (formula_10). From a culinary perspective, to create the nearly perfect poppy-seed bagel where bites of equal size anywhere on the bagel would contain essentially the same number of poppy seeds, impose at least an inverse square distance repelling force on the seeds.
Formal definitions.
For a parameter formula_11 and an formula_3-point set formula_13, the formula_2-energy of formula_15 is defined as follows:
formula_16
For a compact set formula_1 we define its "minimal formula_3-point formula_2-energy" as 
formula_20
where the minimum is taken over all formula_3-point subsets of formula_1; i.e., formula_23. Configurations formula_15 that attain this infimum are called "formula_3-point formula_2-equilibrium configurations".
Poppy-seed bagel theorem for bodies.
We consider compact sets formula_27 with the Lebesgue measure formula_28 and formula_29. For every formula_30 fix an formula_3-point formula_2-equilibrium configuration formula_33. Set
formula_34
where formula_35 is a unit point mass at point formula_36. Under these assumptions, in the sense of weak convergence of measures,
formula_37
where formula_38 is the Lebesgue measure restricted to formula_1; i.e., formula_40.
Furthermore, it is true that
formula_41
where the constant formula_42 does not depend on the set formula_1 and, therefore,
formula_44
where formula_45 is the unit cube in formula_46.
Poppy-seed bagel theorem for manifolds.
Consider a smooth formula_47-dimensional manifold formula_1 embedded in formula_46 and denote its surface measure by formula_50. We assume formula_51. Assume formula_52
As before, for every formula_30 fix an formula_3-point formula_2-equilibrium configuration formula_33 and set
formula_57
Then, in the sense of weak convergence of measures,
formula_37
where formula_59. If formula_60 is the formula_47-dimensional Hausdorff measure normalized so that formula_62, then
formula_63
where formula_64 is the volume of a d-ball.
The constant "Cs,p".
For formula_65, it is known that formula_66, where formula_67 is the Riemann zeta function. Using a modular form approach to linear programming, Viazovska together with coauthors established in a 2022 paper that in dimensions formula_68 and formula_69, the values of formula_42, formula_71, are given by the Epstein zeta function
associated with the formula_72 lattice and Leech lattice, respectively.
It is conjectured that for formula_73, the value of formula_42 is similarly determined as the value of the Epstein zeta function for the hexagonal lattice. Finally, in every dimension formula_75 it is known that when formula_76, the scaling of formula_77 becomes formula_78 rather than formula_79, and the value of formula_42 can be computed explicitly as the volume of the unit formula_81-dimensional ball:
formula_82
The following connection between the constant formula_42 and the problem of sphere packing is known:
formula_84
where formula_85 is the volume of a p-ball and 
formula_86 where the supremum is taken over all families formula_87 of non-overlapping unit balls such that the limit
formula_88
exists.
References.
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