论文标题
相互作用粒子能量的收敛速率，其分布随着数量的增加而扩散
Convergence rates for energies of interacting particles whose distribution spreads out as their number increases
论文作者
论文摘要
我们考虑一类出现在各种应用中的粒子系统，例如近似理论，可塑性，潜在理论和空间填充设计。粒子在实际线路上的位置被描述为相互作用能量的全局最小值，由非本地，排斥的相互作用部分和狭窄部分组成。在应用程序的激励下，我们涵盖了非标准的情况，在这种情况下，随着粒子数量的增加，限制的潜在弱势会减弱。这导致颗粒散布的大面积。我们的目的是通过相应的连续性相互作用能近似粒子相互作用的能量。我们的主要结果是相应的能量差和相关电位值之间的差异的边界。我们证明这些界限对近似理论和可塑性中的问题很有用。这些界限的证明依赖于对相互作用和限制电势的凸度假设。它将文献的最新进展与连续互动能量的最小化器上的新上层结合在一起。
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.
