Thomson problem
The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of N electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law.
The physicist J. J. Thomson posed the problem in 1904[1] after proposing an atomic model, later called the plum pudding model, based on his knowledge of the existence of negatively charged electrons within neutrally-charged atoms.
The electrostatic interaction energy occurring between each pair of electrons of equal charges (
is the distance between each pair of electrons located at points on the sphere defined by vectors
over all possible configurations of N distinct points is typically found by numerical minimization algorithms.
[2] The main difference is that in Smale's problem the function to minimise is not the electrostatic potential
A second difference is that Smale's question is about the asymptotic behaviour of the total potential when the number N of points goes to infinity, not for concrete values of N. The solution of the Thomson problem for two electrons is obtained when both electrons are as far apart as possible on opposite sides of the origin,
, or Mathematically exact minimum energy configurations have been rigorously identified in only a handful of cases.
Geometric solutions of the Thomson problem for N = 4, 6, and 12 electrons are Platonic solids whose faces are all congruent equilateral triangles.
Numerical solutions for N = 8 and 20 are not the regular convex polyhedral configurations of the remaining two Platonic solids, the cube and dodecahedron respectively.
[8] For non-integrable Riesz kernels, the Poppy-seed bagel theorem holds, see the 2004 work of Hardin and Saff.
[9] Notable cases include:[10] One may also consider configurations of N points on a sphere of higher dimension.
The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance:[10] While the objective is to minimize the global electrostatic potential energy of each N-electron case, several algorithmic starting cases are of interest.
Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell.
represents the uniform distribution of a single electron's charge,
The expected global energy of a system of electrons distributed in a purely random manner across the surface of the sphere is given by and is, in general, greater than the energy of every Thomson problem solution.
Therefore, the charge-centered distribution represents a smaller "energy gap" to cross to arrive at a solution of each Thomson problem than algorithms that begin with the other two charge configurations.
[12] "No fact discovered about the atom can be trivial, nor fail to accelerate the progress of physical science, for the greater part of natural philosophy is the outcome of the structure and mechanism of the atom."
Though experimental evidence led to the abandonment of Thomson's plum pudding model as a complete atomic model, irregularities observed in numerical energy solutions of the Thomson problem have been found to correspond with electron shell-filling in naturally occurring atoms throughout the periodic table of elements.
[14] The Thomson problem also plays a role in the study of other physical models including multi-electron bubbles and the surface ordering of liquid metal drops confined in Paul traps.
The generalized Thomson problem arises, for example, in determining arrangements of protein subunits that comprise the shells of spherical viruses.
The "particles" in this application are clusters of protein subunits arranged on a shell.
Other realizations include regular arrangements of colloid particles in colloidosomes, proposed for encapsulation of active ingredients such as drugs, nutrients or living cells, fullerene patterns of carbon atoms, and VSEPR theory.
An example with long-range logarithmic interactions is provided by Abrikosov vortices that form at low temperatures in a superconducting metal shell with a large monopole at its center.
is the energy, the symmetry type is given in Schönflies notation (see Point groups in three dimensions), and
Most symmetry types require the vector sum of the positions (and thus the electric dipole moment) to be zero.
It is customary to also consider the polyhedron formed by the convex hull of the points.
is the smallest angle subtended by vectors associated with the nearest charge pair.
Thus, except in the cases N = 2, 3, 4, 6, 12, and the geodesic polyhedra, the convex hull is only topologically equivalent to the figure listed in the last column.
is the polyhedron formed by the convex hull of the solution configuation to the Thomson Problem for
