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Particle approximation of nonlocal interaction energies

Davide Carazzato, Aldo Pratelli, Ihsan Topaloglu

TL;DR

This work analyzes particle-approximation of nonlocal interaction energies with general kernels $g$ and establishes a rigorous discrete-to-continuum limit. It proves that the discrete energies $\mathcal{E}_n$ Γ-converge to the continuum energy $\mathcal{E}$ on $\mathcal{P}(\mathbb{R}^N)$ under broad assumptions, and that minimizers (up to translations) converge to minimizers of $\mathcal{E}$, with the ground-state energies converging as well. It further shows the existence of minimizers for the discrete energies for all sufficiently large $n$, and discusses conditions under which discrete minimizers exist, including Morse-type kernels. The results provide a solid justification for using discrete particle models to approximate continuum nonlocal energies and connect to concentration-compactness methods with applications in materials science.

Abstract

We consider Riesz-type nonlocal energies with general interaction kernels and their discretizations related to particle systems. We prove that the discretized energies $Γ$-converge in the weak-$*$ topology to the Riesz functional defined over the space of probability measures. We also address the minimization problem for the discretized energies, and prove the existence of minimal configurations of particles in a very general and natural setting.

Particle approximation of nonlocal interaction energies

TL;DR

This work analyzes particle-approximation of nonlocal interaction energies with general kernels and establishes a rigorous discrete-to-continuum limit. It proves that the discrete energies Γ-converge to the continuum energy on under broad assumptions, and that minimizers (up to translations) converge to minimizers of , with the ground-state energies converging as well. It further shows the existence of minimizers for the discrete energies for all sufficiently large , and discusses conditions under which discrete minimizers exist, including Morse-type kernels. The results provide a solid justification for using discrete particle models to approximate continuum nonlocal energies and connect to concentration-compactness methods with applications in materials science.

Abstract

We consider Riesz-type nonlocal energies with general interaction kernels and their discretizations related to particle systems. We prove that the discretized energies -converge in the weak- topology to the Riesz functional defined over the space of probability measures. We also address the minimization problem for the discretized energies, and prove the existence of minimal configurations of particles in a very general and natural setting.

Paper Structure

This paper contains 4 sections, 8 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convergence of Energies and Minimizers
  4. Existence of Discrete Minimizers

Key Result

Lemma 1

Suppose that $g$ satisfies ass:bddbelow_lsc. If $\mu\in\mathcal{P}(\mathbb R^N)$ is a minimizer of $\mathcal{E}$, then where $\psi_\mu(x) = \int g(y-x)\,\mathrm{d} \mu(y)$. Additionally, $\psi_\mu\leq \mathcal{E}(\mu)$ in ${\rm spt} (\mu)$ since $\psi_\mu$ is lower semicontinuous.

Theorems & Definitions (17)

  • Lemma 1: Euler-Lagrange Conditions
  • Lemma 2
  • proof
  • Lemma 3: Concentration compactness
  • Definition 4: $\Gamma$-convergence
  • Theorem 5: $\Gamma$-convergence
  • proof
  • Proposition 6
  • proof
  • Remark 7
  • ...and 7 more