Ground state at high density

TL;DR

This paper investigates the behavior of classical ground states at high densities, revealing conditions under which particle distributions become uniform or patterned, and establishing the absence of certain lattice structures in specific interaction regimes.

Contribution

It provides a rigorous analysis of high-density ground states, linking Fourier transform properties of interactions to particle distribution patterns and lattice structures.

Findings

Particles tend to distribute uniformly at high density with positive Fourier transform interactions.

High-density ground states exhibit patterns if the Fourier transform has negative parts.

No Bravais lattice structures form among high-density ground states with certain diverging or cusp potentials.

Abstract

Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize the mean-field energy functional. By studying the latter we derive global properties of high-density ground state configurations in bounded domains and in infinite space. Our main result is a theorem stating that for interactions having a strictly positive Fourier transform the distribution of particles tends to be uniform as the density increases, while high-density ground states show some pattern if the Fourier transform is partially negative. The latter confirms the conclusion of earlier studies by Vlasov (1945), Kirzhnits and Nepomnyashchii (1971), and Likos et al. (2007). Other results include the proof that there is no Bravais lattice among high-density ground states of interactions whose Fourier transform has a negative part and the potential diverges or has a cusp at zero. We also show that in the ground state configurations of the penetrable sphere model particles are superposed on the sites of a close-packed lattice.