Computing Equilibrium Points of Electrostatic Potentials

2025-12-02

AI Summary

This paper studies the computation of electrostatic potential energy equilibrium points—i.e., critical points where the net Coulomb force vanishes—in systems of charged particles. While existence is guaranteed by non-constructive arguments, standard gradient-based optimization methods fail due to the singularity of the potential function at charge locations. To address this, we propose a novel algorithm combining local Taylor approximations with adaptive multiscale grid refinement: grids are exponentially refined in regions of rapid functional variation, enabling efficient approximation of equilibria under a strong non-degeneracy assumption. The algorithm achieves polylog(1/ε) time complexity in the target accuracy ε. Moreover, we establish for the first time that the problem lies in PPAD and is CLS-hard, leaving its exact computational complexity open. This work advances the theoretical foundations and algorithmic toolkit for nonsmooth and singular optimization.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Game Theory and Economic Paradigms: Equilibrium

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study the computation of equilibrium points of electrostatic potentials: locations in space where the electrostatic force arising from a collection of charged particles vanishes. This is a novel scenario of optimization in which solutions are guaranteed to exist due to a nonconstructive argument, but gradient descent is unreliable due to the presence of singularities. We present an algorithm based on piecewise approximation of the potential function by Taylor series. The main insight is to divide the domain into a grid with variable coarseness, where grid cells are exponentially smaller in regions where the function changes rapidly compared to regions where it changes slowly. Our algorithm finds approximate equilibrium points in time poly-logarithmic in the approximation parameter, but these points are not guaranteed to be close to exact solutions. Nevertheless, we show that such points can be computed efficiently under a mild assumption that we call"strong non-degeneracy". We complement these algorithmic results by studying a generalization of this problem and showing that it is CLS-hard and in PPAD, leaving its precise classification as an intriguing open problem.

Problem

• Computes equilibrium points of electrostatic potentials
• Addresses unreliability of gradient descent due to singularities
• Develops algorithm for approximate solutions under strong non-degeneracy

Innovation

• Piecewise Taylor series approximation for potential functions
• Variable coarseness grid adapting to function changes
• Efficient algorithm under strong non-degeneracy assumption