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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

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.

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

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.
Paper Structure (13 sections, 20 theorems, 169 equations, 4 figures)

This paper contains 13 sections, 20 theorems, 169 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Description of the model and notation
  3. New results
  4. Numerical results
  5. Introduction
  6. Background on interacting particle systems
  7. Port-Hamiltonian formulation
  8. Mean-field limit
  9. PHS preserving coupling of subsystems
  10. Identification of ports
  11. Coupling of identical subsystems
  12. Coupling of different species
  13. Conclusion and outlook

Key Result

Theorem 1.1

(particle level, original formulation JaTo2024) Let $\mathcal{V} \in C^{1}(\mathbb R^d; \mathbb R)$ with $\nabla \mathcal{V}$ antisymmetric, locally Lipschitz continuous and bounded. Let $\psi \in C(\mathbb R_{\geq 0}; \mathbb R_{\geq 0})$ be bounded and locally Lipschitz continuous and assume that Then for every initial condition $z_0=(r_0,v_0)\in \mathbb R^{Nd}\times \mathbb R^{Nd}$ such that

Figures (4)

  • Figure 1: Trajectories of the positions for the Morse potential \ref{['eq:regMorse']} with dominant repulsion. The initial positions and velocities are sampled from the uniform distribution on $[-1,1]^2$. The parameters are $R=10.0$, $a=5.0$, $d_0=2.2$, $T=5000$.
  • Figure 2: Trajectories of the positions for the Morse potential \ref{['eq:regMorse']} with balanced attraction and repulsion. The parameters are $N=20$, $R=0.4$, $a=5.0$, $d_0=0.93$, $T=600$.
  • Figure 3: Trajectories of the positions for the Morse potential \ref{['eq:regMorse']}. The parameters are $N=100$, $R=0.4$, $a=5.0$, $d_0=0.93$, $T=50000$. Blue (resp. red) markers show positions at time $5000$ (resp. at time $25000$). The initial velocities are sampled from the uniform distribution on $[-1,1]^2$. For $s \in \{-4,4\}^2$, the initial positions of $25$ particles are sampled uniformly from $s+[-1.5,1.5]^2$.
  • Figure 4: Trajectories of the positions for the Morse potential \ref{['eq:regMorse']}. The initial positions and velocities are sampled uniformly from $[-1,1]^2$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Conjecture 1.6
  • Proposition 2.1
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 33 more