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Waves and oscillations are commonly observed in the dynamics of self-drivenagents such as pedestrians or vehicles. Interestingly, many factors may perturbthe stability of space homogeneous streaming, leading to the spontaneousformation of collective oscillations of the agents related to stop-and-gowaves, jamiton, or phantom jam in the literature. In this article, wedemonstrate that even a minimal additive stochastic noise in stable first-orderdynamics can initiate stop-and-go phenomena. The noise is not a classic whiteone, but a colored noise described by a Gaussian Ornstein-Uhlenbeck process. Itturns out that the joint dynamics of particles and noises forms again a(Gaussian) Ornstein-Uhlenbeck process whose characteristics can be explicitlyexpressed in terms of parameters of the model. We analyze its stability andcharacterize the presence of waves through oscillation patterns in thecorrelation and autocorrelation of the distance spacing between the particles.We determine exact solutions for the correlation functions for the finitesystem with periodic boundaries and in the continuum limit when the system sizeis infinite. Finally, we compare experimental trajectories of single-filepedestrian motions to simulation results.

Spontaneous wave formation in stochastic self-driven particle systems