The collective dynamics of a stochastic Port-Hamiltonian self-driven agent model in one dimension

¹ Applied and Computational Mathematics, University of Wuppertal, Wuppertal, Germany
² Traffic Safety and Reliability, University of Wuppertal, Wuppertal, Germany

^* Corresponding author: ehrhardt@math.uni-wuppertal.de

Received: 15 June 2023
Accepted: 2 January 2024

Abstract

This paper studies the collective motion of self-driven agents in a one-dimensional space with periodic boundaries, using a stochastic Port-Hamiltonian system (PHS) with symmetric nearest-neighbor interactions and additive Brownian noise as an external input. In the case of a quadratic potential the PHS is an Ornstein-Uhlenbeck process for which we explicitly determine the distribution for any time t ≥ 0 and in the limit t → ∞. In particular, we characterize the collective motion by showing that the agents’ positions tend to build exactly one cluster. This is confirmed in simulations that show rapid and coordinated motion among agents, driven by noise, despite the absence of a preferred direction of motion in the model. Remarkably, the theoretical properties observed in the Ornstein-Uhlenbeck process also emerge in simulations of the nonlinear model incorporating a general interaction potential.