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arXiv · 2301.06121

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

Jannik Daun·Daniel Jannik Happ·Birgit Jacob·Claudia Totzeck
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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.

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Jannik Daun, Daniel Jannik Happ, Birgit Jacob, Claudia Totzeck. 2023-01-15. Erratum and original of Port-Hamiltonian structure of interacting particle systems and its mean-field limit. https://arxiv.org/abs/2301.06121

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