New Duality Relations for Classical Ground States

TL;DR

This paper introduces new duality relations connecting the energies of classical ground states for certain potentials, enabling insights into their structure and phase behavior, especially between short- and long-range interactions.

Contribution

The authors derive novel duality relations that relate the energies of classical ground states for a class of soft pair potentials and their Fourier transforms, providing new analytical tools.

Findings

Derived duality relations linking energies of dual potentials.

Applied relations to infer ground state properties across different potential ranges.

Identified bounds and phase transition behaviors in classical systems.

Abstract

We derive new duality relations that link the energy of configurations associated with a class of soft pair potentials to the corresponding energy of the dual (Fourier-transformed) potential. We apply them by showing how information about the classical ground states of short-ranged potentials can be used to draw new conclusions about the nature of the ground states of long-ranged potentials and vice versa. They also lead to bounds on the T=0 system energies in density intervals of phase coexistence, the identification of a one-dimensional system that exhibits an infinite number of ``phase transitions," and a conjecture regarding the ground states of purely repulsive monotonic potentials.