Spontaneous wave formation in stochastic self-driven particle systems

TL;DR

This paper shows that minimal colored stochastic noise in stable self-driven particle systems can spontaneously generate wave phenomena like stop-and-go traffic, with explicit analytical solutions and real-world pedestrian data comparison.

Contribution

It introduces a minimal stochastic noise model that explains spontaneous wave formation in self-driven systems, providing explicit solutions and stability analysis.

Findings

Colored noise induces wave phenomena in stable systems

Explicit correlation functions derived for finite and infinite systems

Simulation results align with pedestrian trajectory data

Abstract

Waves and oscillations are commonly observed in the dynamics of self-driven agents such as pedestrians or vehicles. Interestingly, many factors may perturb the stability of space homogeneous streaming, leading to the spontaneous formation of collective oscillations of the agents related to stop-and-go waves, jamiton, or phantom jam in the literature. In this article, we demonstrate that even a minimal additive stochastic noise in stable first-order dynamics can initiate stop-and-go phenomena. The noise is not a classic white one, but a colored noise described by a Gaussian Ornstein-Uhlenbeck process. It turns out that the joint dynamics of particles and noises forms again a (Gaussian) Ornstein-Uhlenbeck process whose characteristics can be explicitly expressed in terms of parameters of the model. We analyze its stability and characterize the presence of waves through oscillation patterns in the correlation and autocorrelation of the distance spacing between the particles. We determine exact solutions for the correlation functions for the finite system with periodic boundaries and in the continuum limit when the system size is infinite. Finally, we compare experimental trajectories of single-file pedestrian motions to simulation results.