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Point vortex equilibria and optimal packings of circles on a sphere
| Title: | Point vortex equilibria and optimal packings of circles on a sphere |
| Authors: | Newton, P. K. Browse this author | | Sakajo, T. Browse this author →KAKEN DB |
| Keywords: | Point vortex equilibria | | Tammes problem | | Icosahedral symmetry | | Singular value decomposition | | Shannon entropy |
| Issue Date: | 2010 |
| Publisher: | Royal Society |
| Journal Title: | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |
| Volume: | 467 |
| Issue: | 2129 |
| Start Page: | 1468 |
| End Page: | 1490 |
| Publisher DOI: | 10.1098/rspa.2010.0368 |
| Abstract: | We answer the question of whether optimal packings of circles on a sphere are equilibrium solutions to the logarithmic particle interaction problem for values of N = 3−12 and N = 24, the only values of N for which the optimal packing problem (also known as the Tammes problem) has rigorously known solutions. We also address the cases N = 13−23 where optimal packing solutions have been computed, but not proven analytically. As in Jamaloodeen & Newton (2006), a logarithmic, or point vortex equilibrium is determined by formulating the problem as one in linear algebra, A~ = 0, where A is a N(N − 1)/2 × N non-normal configuration matrix obtained by requiring that all interparticle distances remain constant. If A has a kernel, the strength vector ~ ∈ RN is then determined as a right-singular vector associated with the zero singular value, or a vector that lies in the nullspace of A where the kernel is multi-dimensional. First we determine if the known optimal packing solution for a given value of N has a configuration matrix A with a nonempty nullspace. The answer is yes for N = 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16 and no for N = 10, 15, 17 − 24. We then determine a basis set for the nullspace of A associated with the optimally packed state, answer the question of whether N-equal strength particles, ~ = (1, 1, 1, ..., 1)T , form an equilibrium for this configuration, and describe what is special about the icosahedral configuration from this point of view. We also find new equilibria by implementing two versions of a random walk algorithm. First, we cluster sub-groups of particles into patterns during the packing process, and ‘grow’ a packed state using a version of the ‘yin-yang’ algorithm of Longuet-Higgins (2008). We also implement a version of our ‘Brownian ratchet’ algorithm (Newton & Sakajo (2009)) to find new equilibria near the optimally packed state for N = 10, 15, 17− 24. |
| Type: | article (author version) |
| URI: | http://hdl.handle.net/2115/48802 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)
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Submitter: 坂上 貴之
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