Topic
Thomson problem
About: Thomson problem is a research topic. Over the lifetime, 91 publications have been published within this topic receiving 1621 citations.
Papers published on a yearly basis
Papers
TL;DR: In this article, an effective free energy describing the physics of disclination defects in particle arrays constrained to move on an arbitrary two-dimensional surface is proposed and analyzed, where the ground state is analyzed as a function of the ratio of the defect core energy to the Young's modulus.
Abstract: We propose and analyze an effective free energy describing the physics of disclination defects in particle arrays constrained to move on an arbitrary two-dimensional surface. At finite temperature the physics of interacting disclinations is mapped to a Laplacian sine-Gordon Hamiltonian suitable for numerical simulations. We discuss general features of the ground state and thereafter specialize to the spherical case. The ground state is analyzed as a function of the ratio of the defect core energy to the Young's modulus. We argue that the core energy contribution becomes less and less important in the limit $R\ensuremath{\gg}a,$ where R is the radius of the sphere and a is the particle spacing. For large core energies there are 12 disclinations forming an icosahedron. For intermediate core energies unusual finite-length grain boundaries are preferred. The complicated regime of small core energies, appropriate to the limit $R/\stackrel{\ensuremath{\rightarrow}}{a}\ensuremath{\infty},$ is also addressed. Finally we discuss the application of our results to the classic Thomson problem of finding the ground state of electrons distributed on a two sphere.
254 citations
TL;DR: Predictions from the continuum theory for the ground state energy agree with numerical simulations of long range power law interactions of the form 1/r(gamma) (0
Abstract: We attack the generalized Thomson problem, i.e., determining the ground state energy and configuration of many particles interacting via an arbitrary repulsive pairwise potential on a sphere via a continuum mapping onto a universal long range interaction between angular disclination defects parametrized by the elastic (Young) modulus Y of the underlying lattice and the core energy E(core) of an isolated disclination. Predictions from the continuum theory for the ground state energy agree with numerical simulations of long range power law interactions of the form 1/r(gamma) (0
219 citations
TL;DR: In this article, a model of purely repulsive soft colloidal particles interacting via a generalized elastic potential and constrained to a two-dimensional plane and to the surface of a spherical shell is presented.
Abstract: We perform numerical simulations of a model of purely repulsive soft colloidal particles interacting via a generalized elastic potential and constrained to a two-dimensional plane and to the surface of a spherical shell. For the planar case, we compute the phase diagram in terms of the system's rescaled density and temperature. We find that a large number of ordered phases becomes accessible at low temperatures as the density of the system increases, and we study systematically how structural variety depends on the functional shape of the pair potential. For the spherical case, we revisit the generalized Thomson problem for small numbers of particles N ≤ 12 and identify, enumerate and compare the minimal energy polyhedra established by the location of the particles to those of the corresponding electrostatic system.
74 citations
TL;DR: In this paper, a method for generating nearly uniform distributions of three-dimensional orientations in the presence of symmetry is presented, based on the Thomson problem, which consists in finding the configuration of minimal energy of N electrons located on a unit sphere, a configuration of high spatial uniformity.
Abstract: A method is presented for generating nearly uniform distributions of three-dimensional orientations in the presence of symmetry. The method is based on the Thomson problem, which consists in finding the configuration of minimal energy of N electrons located on a unit sphere – a configuration of high spatial uniformity. Orientations are represented as unit quaternions, which lie on a unit hypersphere in four-dimensional space. Expressions of the electrostatic potential energy and Coulomb's forces are derived by working in the tangent space of orientation space. Using the forces, orientations are evolved in a conventional gradient-descent optimization until equilibrium. The method is highly versatile as it can generate uniform distributions for any number of orientations and any symmetry, and even allows one to prescribe some orientations. For large numbers of orientations, the forces can be computed using only the close neighbourhoods of orientations. Even uniform distributions of as many as 106 orientations, such as those required for dictionary-based indexing of diffraction patterns, can be generated in reasonable computation times. The presented algorithms are implemented and distributed in the free (open-source) software package Neper.
64 citations
TL;DR: This work investigates codebooks based on other spherical arrangements, such as solutions maximizing the harmonic mean of the mutual distances among the codewords, which is known as the Thomson problem, and finds that in some special cases, Grassmannian code books based on these other spherical arrangement outperform codebooks from Grassmannians packing.
Abstract: Precoding codebook design for limited feedback MIMO systems is known to reduce to a discretization problem on a Grassmann manifold. The case of two-antenna beamforming is special in that it is equivalent to quantizing the real sphere. The isometry between the Grassmannian G2,1ℂ and the real sphere S2 shows that discretization problems in the Grassmannian G2,1ℂ are directly solved by corresponding spherical codes. Notably, the Grassmannian line packing problem in ℂ2, namely maximizing the minimum distance, is equivalent to the Tammes problem on the real sphere, so that optimum spherical packings give optimum Grassmannian packings. Moreover, a simple isomorphism between G2,1ℂ and S2 enables to analytically derive simple codebooks in closed-form having low implementation complexity. Using the simple geometry of some of these codebooks, we derive closed-form expressions of the probability density function of the relative SNR loss due to limited feedback. We also investigate codebooks based on other spherical arrangements, such as solutions maximizing the harmonic mean of the mutual distances among the codewords, which is known as the Thomson problem. We find that in some special cases, Grassmannian codebooks based on these other spherical arrangements outperform codebooks from Grassmannian packing.
41 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 3 |
| 2020 | 3 |
| 2019 | 10 |
| 2018 | 4 |
| 2017 | 3 |
| 2016 | 8 |