Richard Schwartz

13. June 2023

Divide and Conquer:  5 Point Energy Minimization

Thomson’s classic problem asks, for any N, which configurations of N points on the sphere minimize the electrostatic potential. A more general version asks which which points minimize the potential of the configuration with respect to a power law potential. I will describe the progress I have made to date on this problem. For one thing, I solved Thomson’s problem in the case N=5. More generally, I rigorously proved the existence of phase transition, first noticed by Melnyk-Knop-Smith in the 1970s.

Their original discovery is that the triangular bi-pyramid is the minimum potential configuration with respect to all power laws up to about 15.04 and then the answer changes to a pyramid with square base. My proof is a mixture of computer algebra, interval arithmetic, and symmetrization. I will explain the main ideas and also give some neat computer demos showing the proof in action.


About HITS

HITS, the Heidelberg Institute for Theoretical Studies, was established in 2010 by physicist and SAP co-founder Klaus Tschira (1940-2015) and the Klaus Tschira Foundation as a private, non-profit research institute. HITS conducts basic research in the natural, mathematical, and computer sciences. Major research directions include complex simulations across scales, making sense of data, and enabling science via computational research. Application areas range from molecular biology to astrophysics. An essential characteristic of the Institute is interdisciplinarity, implemented in numerous cross-group and cross-disciplinary projects. The base funding of HITS is provided by the Klaus Tschira Foundation.

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