Convergence of the TFDW Energy to the Liquid Drop Model

TL;DR

This paper proves that, under certain conditions, the TFDW energy model converges to the liquid drop model, revealing a deep connection between atomic and nuclear physics models through Gamma-convergence analysis.

Contribution

It establishes the Gamma-convergence of the TFDW energy to the liquid drop model under a sharp interface scaling and constrained mass, for general external potentials.

Findings

Gamma-convergence of TFDW to liquid drop model

Conditions under which minimizers exist or do not exist

Implications for global minimization strategies

Abstract

We consider two nonlocal variational models arising in physical contexts. The first is the Thomas-Fermi-Dirac-von Weiz\"{a}cker (TFDW) model, introduced in the study of ionization of atoms and molecules, and the second is the liquid drop model with external potential, proposed by Gamow in the context of nuclear structure. It has been observed that the two models exhibit many of the same properties, especially in regard to the existence and nonexistence of minimizers. We show that, under a "sharp interface" scaling of the coefficients and constrained mass, the TFDW energy Gamma-converges to the Liquid Drop model, for a general class of external potentials. Finally, we present some consequences for global minimization of each model.