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Estimating the Number of Stable Configurations for the Generalized Thomson Problem

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Abstract

Given a natural number \(N\), one may ask what configuration of \(N\) points on the two-sphere minimizes the discrete generalized Coulomb energy. If one applies a gradient-based numerical optimization to this problem, one encounters many configurations that are stable but not globally minimal. This led the authors of this manuscript to the question, how many stable configurations are there? In this manuscript we report methods for identifying and counting observed stable configurations, and estimating the actual number of stable configurations. These estimates indicate that for \(N\) approaching two hundred, there are at least tens of thousands of stable configurations.

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Acknowledgments

The authors are grateful to Mark Ellingham for his clear explanation of Algorithm 2. The authors are also grateful to the referees for their suggested changes to the manuscript. The work of Matthew Calef was performed under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. LA-UR-14-27638 The work of Alexia Schulz is sponsored by the Assistant Secretary of Defense for Research & Engineering under Air Force Contract #FA8721-05-C-0002. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed by the United States Government.

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Authors and Affiliations

  1. Computational Physics and Methods, Los Alamos National Laboratory, Los Alamos, NM, 87545-0001, USA

    Matthew Calef

  2. Financial Institutions Group, Bank of America, New York, NY, 10036, USA

    Whitney Griffiths

  3. Cyber Security and Information Sciences, MIT Lincoln Laboratory, Lexington, MA, 02420-9108, USA

    Alexia Schulz

Authors
  1. Matthew Calef
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  2. Whitney Griffiths
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  3. Alexia Schulz
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Correspondence to Matthew Calef.

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Calef, M., Griffiths, W. & Schulz, A. Estimating the Number of Stable Configurations for the Generalized Thomson Problem. J Stat Phys 160, 239–253 (2015). https://doi.org/10.1007/s10955-015-1245-6

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