SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 681-709, January 2021.
We consider a large class of interacting particle systems in one dimension described by an energy whose interaction potential is singular and nonlocal. This class covers Riesz gases (in particular, log gases) and applications to plasticity and approximation theory of functions. While it is well established that the minimizers of such interaction energies converge to a certain particle density profile as the number of particles tends to infinity, any bound on the rate of this convergence is only known in special cases by means of quantitative estimates. The main result of this paper extends these quantitative estimates to a large class of interaction energies by a different proof. The proof relies on one-dimensional features such as the convexity of the interaction potential and the ordering of the particles. The main novelty of the proof is the treatment of the singularity of the interaction potential by means of a carefully chosen renormalization.

中文翻译：

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一维相互作用粒子系统的连续近似的定量估计

SIAM数学分析杂志，第53卷，第1期，第681-709页，2021年1月。
我们考虑一维相互作用中的一大类相互作用的粒子系统，其能量由其相互作用势为奇异且非局部的能量来描述。此类内容涵盖Riesz气体（尤其是对数气体）及其在可塑性和功能近似理论中的应用。众所周知，随着粒子数趋于无穷大，这种相互作用能的极小值会收敛到一定的粒子密度分布，但这种收敛速度的任何限制只有在特殊情况下才能通过定量估计得知。本文的主要结果通过不同的证明将这些定量估计扩展到一类较大的相互作用能。证明依赖于一维特征，例如相互作用势的凸度和粒子的有序性。