Fully Algebraic Description of the Static Level Sets for the System of Two Particles under a Van der Waals Potential

TL;DR

This paper provides an algebraic method to fully describe the static level sets of a two-particle system under Van der Waals potential in three dimensions, using polynomial solutions to determine equipotential zones.

Contribution

It introduces a complete algebraic framework for analyzing equipotential surfaces in two-particle Van der Waals systems, generalizable to higher-degree polynomials.

Findings

Level sets are determined by solutions of degree up to four polynomials.

Distribution of positive roots characterizes the equipotential zones.

Numerical simulations illustrate the theoretical results.

Abstract

We study the equipotential surfaces around of a two particle system in 3-d under a pairwise good potential as the one of Van der Waals. The level sets are completely determined by the solutions of polynomials of at most fourth degree that can be solved by standard algebraic methods. The distribution of real positive roots determines the level sets and provides a complete description of the map for the equipotential zones. Our methods can be generalized to a family of polynomials with degree multiple of 2, 3, or 4. Numerical simulations of 2-d and 3-d pictures depicting the true orbits and equipotential zones are provided.