关于相互作用粒子系统的端-哈密顿结构及其均场极限的更正与原始论文
Erratum and original of Port-Hamiltonian structure of interacting particle systems and its mean-field limit
AI总结：
本文通过端-哈密顿结构分析相互作用粒子系统的均场极限，提出新的稳定性视角并修正原论文中的错误。
摘要：
我们推导了一类一般相互作用粒子系统的基本端-哈密顿公式，这些系统由对齐和基于势能的力动力学驱动，包括具有势能相互作用的Cucker-Smale模型和二阶Kuramoto模型。端-哈密顿结构允许通过粒子层面的LaSalle型论证来表征守恒量，如Casimir函数以及长期行为。随后证明端-哈密顿结构在均场极限下得以保持，并在概率测度空间上配备2-Wasserstein度量的情况下研究其LaSalle不变性原理的类比。关于粒子和均场极限的结果为一般相互作用粒子系统的均匀稳定性提供了新的视角。此外，由于最小端-哈密顿公式是封闭的，我们识别出子系统的端口，这些子系统允许具有广义质量-弹簧-阻尼结构，模拟两个粒子之间的二元相互作用。利用端口信息，我们讨论了在保持端-哈密顿结构的方式下不同物种的耦合。该更正修正了我们关于相互作用粒子系统端-哈密顿结构的论文中的一个错误。虽然在原始假设下梯度的收敛性仍然有效，但没有额外的吸引力假设，系统轨迹在2-Wasserstein空间中的相对紧致性并不成立。我们基于Barbalat引理证明了梯度收敛性的证明。对于排斥二元相互作用，我们给出了系统轨迹相对紧致性的反例。在短程排斥和长程吸引的情况下，我们展示了几项数值研究，以支持我们关于相对紧致轨迹的猜想。
英文摘要：
We derive a minimal port-Hamiltonian formulation of a general class of interacting particle systems driven by alignment and potential-based force dynamics which include the Cucker-Smale model with potential interaction and the second order Kuramoto model. The port-Hamiltonian structure allows to characterize conserved quantities such as Casimir functions as well as the long-time behaviour using a LaSalle-type argument on the particle level. It is then shown that the port-Hamiltonian structure is preserved in the mean-field limit and an analogue of the LaSalle invariance principle is studied in the space of probability measures equipped with the 2-Wasserstein-metric. The results on the particle and mean-field limit yield a new perspective on uniform stability of general interacting particle systems. Moreover, as the minimal port-Hamiltonian formulation is closed we identify the ports of the subsystems which admit generalized mass-spring-damper structure modelling the binary interaction of two particles. Using the information of ports we discuss the coupling of difference species in a port-Hamiltonian preserving manner. The erratum corrects an error in our paper on the port-Hamiltonian structure of interacting particle systems. While convergence of the gradient of the Hamiltonian remains valid under the original assumptions, relative compactness of the system trajectories in the 2-Wasserstein space does not hold without an additional attractivity assumption on the binary interaction force. We provide a proof for the convergence of the gradient of the Hamiltonian based on Barbalat Lemma. A counterexample is given for the relative compactness of the system trajectories for repulsive binary interactions. In the case of short-range repulsion and long-range attraction we show several numerical studies that underpin our conjecture of relatively compact trajectories.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
