In
physics, the poppy-seed bagel theorem concerns interacting particles (e.g.,
electrons) confined to a bounded
surface (or body) when the particles repel each other pairwise with a magnitude that is proportional to the inverse distance between them raised to some positive power . 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.[1][2]
For such particles, a stable equilibrium state, which depends on the parameter , 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 which may or may not be nearly uniform with respect to the
surface area (or
volume) of . The poppy-seed bagel theorem asserts that for a large class of sets , the uniformity property holds when the parameter is larger than or equal to the dimension of the set .[3] 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 (). 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 and an -point set , the -energy of is defined as follows:
For a
compact set we define its minimal -point -energy as
where the
minimum is taken over all -point subsets of ; i.e., . Configurations that attain this infimum are called -point -equilibrium configurations.
Poppy-seed bagel theorem for bodies
We consider compact sets with the
Lebesgue measure and . For every fix an -point -equilibrium configuration . Set
Near minimal -energy 1000-point configurations on a torus ()
Consider a
smooth -dimensional manifold embedded in and denote its
surface measure by . We assume . Assume
As before, for every fix an -point -equilibrium configuration and set
For , it is known[6] that , where 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 and , the values of , , are given by the Epstein zeta function[7]
associated with the
lattice and
Leech lattice, respectively.[8]
It is conjectured that for , the value of is similarly determined as the value of the Epstein zeta function for the
hexagonal lattice. Finally, in every dimension it is known that when , the scaling of becomes rather than , and the value of can be computed explicitly as the volume of the unit
-dimensional ball:[4]
The following connection between the constant and the problem of
sphere packing is known:
[9]
^Fisher, M.E. (1964), "The free energy of a macroscopic system", Archive for Rational Mechanics and Analysis, 17 (5): 377–410,
doi:
10.1007/BF00250473
^Lewin, M. (2022), "Coulomb and Riesz gases: The known and the unknown", Journal of Mathematical Physics, 63 (6),
arXiv:2202.09240,
doi:
10.1063/5.0086835
^Borodachov, S. V.; Hardin, D. P.; Saff, E. B. (2008), "Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets", Transactions of the American Mathematical Society, 360 (3): 1559–1580,
arXiv:math-ph/0602025,
doi:10.1090/S0002-9947-07-04416-9