Remarks on the Minimal α-Energy Problems

초록

We study the minimal α-energy problem where positions of all particles on the unit sphere minimize the total potential, called the α-energy interaction potential and defined by weighted distances ∥xi −xj∥2α+2 for α ?= −1 and log ∥xi −xj∥2 for α = −1. Our contribution consists of three parts. First, for a three-particle system, we find a critical exponent αcri,3 ≈ 0.4094 under which the system undergoes a bifurcation. Precisely, if α> αcri,3, then the bipolar state is a global minimizer; when α< αcri,3, then the regular triangle becomes a global minimizer. Second, in order to consider general N particles, we restricted ourselves with the unit circle instead of the unit sphere. In this case, we show that the equally distributed bipolar state can be a global minimizer for large α depending on N. Our results holds for both even and odd N. Lastly for the cases of N = 5 and N = 6, which are extremely hard to study, we briefly mention that possible local minimizers can be stable or not in a specific sense. © (2025), American Psychological Association

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