Erratum and original of Port-Hamiltonian structure of interacting particle systems and its mean-field limit

TL;DR

This paper develops a port-Hamiltonian framework for interacting particle systems, including mean-field limits, and corrects previous errors, providing new insights into stability and subsystem coupling.

Contribution

It introduces a minimal port-Hamiltonian formulation for a broad class of particle systems and extends the structure to mean-field limits, with corrections to earlier assumptions.

Findings

Port-Hamiltonian structure characterizes conserved quantities and long-term behavior.

The structure is preserved in the mean-field limit with an invariance principle.

Numerical studies support conjectures on trajectory compactness under certain interactions.

Abstract

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.