分布随数量增加而扩展的相互作用粒子能量的收敛速度
Convergence rates for energies of interacting particles whose distribution spreads out as their number increases
摘要：
我们考虑一类粒子系统，它出现在各种应用，如逼近论，塑性，位势论和空间填充设计。粒子在实直线上的位置被描述为相互作用能的全局极小值，它由非局域排斥相互作用部分和限制部分组成。出于应用的动机，我们涵盖了非标准的情况下，其中的约束势减弱的粒子数量的增加。这导致颗粒在其上散布的大面积。我们的目标是用一个相应的连续相互作用能来近似粒子的相互作用能。我们的主要结果是相应的能量差和相关的潜在价值之间的差异的界限。我们证明了这些界限是有用的问题，在逼近论和塑性。这些界限的证明依赖于相互作用和限制势的凸性假设。它结合了最新的进展，在文献中的一个新的上界的最小连续相互作用能。
英文摘要：
We consider a class of particle systems which appear in various applications such as approximation theory, plasticity, potential theory and space-filling designs. The positions of the particles on the real line are described as a global minimum of an interaction energy, which consists of a nonlocal, repulsive interaction part and a confining part. Motivated by the applications, we cover non-standard scenarios in which the confining potential weakens as the number of particles increases. This results in a large area over which the particles spread out. Our aim is to approximate the particle interaction energy by a corresponding continuum interacting energy. Our main results are bounds on the corresponding energy difference and on the difference between the related potential values. We demonstrate that these bounds are useful to problems in approximation theory and plasticity. The proof of these bounds relies on convexity assumptions on the interaction and confining potentials. It combines recent advances in the literature with a new upper bound on the minimizer of the continuum interaction energy.
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