论文标题
相互作用粒子系统连续近似值的定量估计值
Quantitative estimate of the continuum approximations of interacting particle systems in one dimension
论文作者
论文摘要
我们考虑了一类相互作用的粒子系统在一维相互作用的粒子系统中，其相互作用势为单数和非本地的能量描述。该类涵盖了Riesz气体（尤其是原木气体），并应用于可塑性和功能近似理论。虽然可以很好地确定，这种相互作用能的最小符会趋于某些粒子密度曲线，因为颗粒的数量倾向于无穷大，但只有通过定量估计，就这种收敛速率的任何结合。本文的主要结果将这些定量估计值扩展到了大量的相互作用能量。证明依赖于一维特征，例如相互作用势的凸度和粒子的顺序。证明的主要新颖性是通过精心选择的重态化处理相互作用潜力的奇异性。
We consider a large class of interacting particle systems in 1D described by an energy whose interaction potential is singular and non-local. 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 minimisers 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 renormalisation.
