Long-range spatial correlations on active self-propelled particles

¹ School of Materials Science and Physics, China University of Mining and Technology - Xuzhou 221116, China
² School of Mechanics and Civil Engineering, China University of Mining and Technology - Xuzhou 221116, China

Received: 21 June 2025
Accepted: 20 November 2025

Abstract

For nearly three decades, the Vicsek model has been extensively used to describe the flocking behavior of active matter. In this paradigmatic model, each individual imitates its neighbors while subject to uncorrelated random noise, which usually follows either a Gaussian (white) or uniform distribution. However, empirical findings and theoretical considerations indicate that the disturbances affecting individuals are not independent, but rather exhibit spatial and temporal correlations. In our work, we introduce the long-range spatial correlations in these disturbances, and perform extensive numerical simulations of the spatially correlated Vicsek model. Our results demonstrate that incorporating long-range spatial correlations enhances the sensitivity of Vicsek-type flocks to external parameter variations near the critical point. And the static spatial correlation function reveals that the correlation length is evidently influenced by spatially correlated noise, exhibiting an approximately linear dependence on the spatial correlation exponent θ. This behavior aligns more closely with empirical observations of natural flocks. Furthermore, we also estimate the critical exponents by performing data collapses of the spatial correlation functions in Fourier space. Our results show that the dependence of on θ exhibits a bilinear trend near the critical point θ_c, which implies that novel flock dynamics may emerge from strong spatial correlations. Our study not only extends the universal behavior of the Vicsek model but also offers novel perspectives for understanding collective motion in active matter systems.