标题： 矛与环：磁性胶体悬浮液中的新兴结构 显示英文标题

标题： The Spear and the Ring: Emergent Structures in Magnetic Colloidal Suspensions

摘要： 我们从数学角度研究了G. Klughertz提出的磁性胶体的纳米粒子模型。 我们的目标是从该模型出发，在长时间尺度下获得磁性纳米粒子动力学产生的稳定稳态结构的性质。 本文中，我们利用变分法的技术详细研究了两种特定结构。 第一种称为“矛”，由通过Lennard-Jones势相互作用的一系列对齐粒子组成。 我们证明了存在性和唯一性结果，推导出相邻粒子之间距离的界，并在粒子数趋于无穷时给出了精确的渐近描述。 第二种结构称为“环”，粒子均匀分布在圆周上。 我们证明了其存在性和唯一性，并推导出其半径的显式公式。 显示英文摘要

摘要： We study from a mathematical point of view the nanoparticle model of a magnetic colloid, presented by G. Klughertz. Our objective is to obtain properties of stable stationary structures that arise in the long-time limit for the magnetic nanoparticles dynamics following this model. In this article, we present a detailed study of two specific structures using techniques from the calculus of variations. The first, called the spear, consists of a chain of aligned particles interacting via a Lennard-Jones potential. We establish existence and uniqueness results, derive bounds on the distances between neighboring particles, and provide a sharp asymptotic description as the number of particles tends to infinity. The second structure, the ring, features particles uniformly distributed along a circle. We prove its existence and uniqueness and derive an explicit formula for its radius.