Poppy-seed bagel theorem explained
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.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
. 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
\omegaN=\{x1,\ldots,xN\}\subsetRp
, 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
where
is a
unit point mass at point
. Under these assumptions, in the sense of weak convergence of measures,
where
is the Lebesgue measure restricted to
; i.e.,
.Furthermore, it is true that
where the constant
does not depend on the set
and, therefore,
where
is the
unit cube in
.
Poppy-seed bagel theorem for manifolds
embedded in
and denote its
surface measure by
. We assume
. Assume
As before, for every
fix an
-point
-equilibrium configuration
and set
Then, in the sense of weak convergence of measures,
where
\mu(B)=\sigma(A\capB)/\sigma(A)
. If
is the
-dimensional
Hausdorff measure normalized so that
, then
where
is the
volume of a d-ball.
The constant Cs,p
For
, it is known 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
[1] associated with the
lattice and
Leech lattice, respectively.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:
The following connection between the constant
and the problem of
sphere packing is known:
where
is the
volume of a p-ball and
where the
supremum is taken over all families
of non-overlapping
unit balls such that the limit
exists.
See also
Notes and References
- Web site: Epstein zeta-function. Encyclopedia of Mathematics. EMS Press. June 17, 2023.