Computing Equilibrium Points of Electrostatic Potentials

TL;DR

This work tackles the problem of computing equilibrium points of electrostatic potentials in d-dimensional space by introducing a grid-based Taylor-approximation framework tailored to well-behaved derivatives. In fixed dimensions, it achieves poly-time computation of ε-approximate stationary points and, under a strong non-degeneracy condition, poly-time strong approximations as well. The authors extend the approach to generalized potentials and establish complexity results, showing PPAD membership and CLS-hardness, with a set of open questions for high-dimensional settings. The results bridge numerical approximation with computational TFNP landscape, offering practical algorithms under precise structural conditions while mapping the inherent complexity for broader potential classes.

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.

Figures (4)

• Figure 1: The grid we use to ensure good approximation of the potential induced by a single charge. The charge is located at the black dot. The shaded region around the position of the charge is removed from the domain, since we can guarantee that no solution is located there. Note that the grid becomes finer in a given coordinate when we are close to the singularity in that coordinate.
• Figure 2: The final grid obtained by superimposing the grids induced by three different charges. The grid lines extend to infinity, but are faded here for improved visibility. Our algorithm solves a system of polynomial inequalities in each grid cell.
• Figure 3: An example with two positive charges (magnitudes $+3$ and $+1$) and one negative charge (magnitude $-1$). The force field is shown, but the length of the vectors is capped so as not to clutter the picture. The two solutions are shown as black dots. The set delimited by the black lines (inside the black outer boundary, and excluding the small round black regions) is an example of a set $S$ that we can use for our proof.
• Figure 4: Generalized potential function due to two charges (non-smoothed).

Theorems & Definitions (38)

• Definition 2.1: Weak Approximate Stationary Point
• Definition 2.2: Strong Approximate Stationary Point
• Theorem 3.1
• Definition 3.1
• Definition 3.2: Well-Behaved
• Theorem 3.2
• proof : Proof of \ref{['thm:well-behaved-easy']}
• Claim 1
• Theorem 3.3: e.g. GrigorevV88-polynomial-ineq
• proof : Proof of \ref{['clm:taylor-approx']}