Rainer Backofen, Manuel Gräf, Daniel Potts, Simon Praetorius, Axel Voigt, Thomas Witkowski : A continuous approach to discrete ordering on S2

Author(s):

Rainer Backofen
Manuel Gräf
Daniel Potts
Simon Praetorius
Axel Voigt
Thomas Witkowski

Title:

A continuous approach to discrete ordering on S2

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 3, 2010

Mathematics Subject Classification:

35R01	[ ]
35K25	[ ]
42C10	[ ]

Abstract:

We consider the classical problem to nd optimal distributions of interacting particles on a sphere by solving an evolution problem for a particle density. Starting from a given pair potential we sketch the derivation of the resulting higher order surface partial differential equation, which is an approximation of a surface dynamic density functional theory. Different numerical approaches are discussed to solve the evolution problem: (a) an implicit approach to describe the surface using a phase-field description, (b) a parametric finite element approach, and (c) a spectral method based on nonequispaced fast Fourier transforms on the sphere. Results for computed minimal energy configurations are discussed for various particle numbers and extensions to other more complex and evolving surfaces are mentioned.

Keywords:

ordering on curved surfaces, Thomson problem, dynamic density functional theory, phase field crystal, PDE on surfaces

Language:

English

Publication time:

03/2010