SIAM Journal on Applied Mathematics, Volume 81, Issue 3, Page 853-870, January 2021.
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.

中文翻译：

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随机自驱动粒子系统中的自发波形成

SIAM应用数学杂志，第81卷，第3期，第853-870页，2021年1月。
通常在诸如行人或车辆之类的自驱动代理的动力学中观察到波动和振荡。有趣的是，许多因素可能会扰乱空间均匀流的稳定性，从而导致自发形成与走走停停波，干扰或幻影相关的物质的集体振荡。在本文中，我们证明，即使是稳定的一阶动力学中的最小加性随机噪声也可以引发停走现象。噪声不是经典的白色噪声，而是由高斯Ornstein-Uhlenbeck过程描述的彩色噪声。事实证明，粒子和噪声的联合动力学再次形成了（高斯）Ornstein-Uhlenbeck过程，其特征可以根据模型的参数明确表达。我们分析了它的稳定性，并通过振动模式在粒子之间距离间隔的相关和自相关中表征了波的存在。当系统大小为无限时，我们为具有周期边界和连续极限的有限系统的相关函数确定精确解。最后，我们将单文件行人运动的实验轨迹与仿真结果进行了比较。