open access

https://hdl.handle.net/2115/48802

Point vortex equilibria and optimal packings of circles on a sphere

英語

Shannon entropy

2010

Royal Society

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

467(2129)

1468-1490

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.

学術雑誌論文

Accepted Manuscript

1364-5021

1471-2946