Boltzmann approach to collective motion via nonlocal visual interaction

TL;DR

This paper introduces a Boltzmann-equation model for collective motion driven by non-local visual interactions, accounting for occlusion and stochastic neighbor selection, and analyzes the phase transition behavior.

Contribution

It develops a novel analytical framework incorporating non-local visual cues and stochastic interactions, revealing how non-locality influences the order-disorder transition in collective motion.

Findings

Visual screening raises the transition threshold.

The transition remains discontinuous but is weakened by non-locality.

The model clarifies the role of non-local visual interactions in collective behavior.

Abstract

Visual cues play crucial roles in the collective motion of animals, birds, fish, and insects. The interaction mediated by visual information is essentially non-local and has many-body nature due to occlusion, which poses a challenging problem in modeling the emergent collective behavior. In this paper, we introduce a Boltzmann-equation approach incorporating non-local visual interaction. Occlusion is treated in a self-consistent manner via a coarse-grained density field, which renders the interaction effectively pairwise. Our model also incorporates the recent finding that each organism stochastically selects a neighbor to interact at each instant. We analytically derive the order-disorder transition point, and show that the visual screening effect raises the transition threshold, which does not vanish when the density of the agents or the range of the intrinsic interaction is taken to infinity. Our analysis suggests that the model exhibits a discontinuous transition as in the local interaction models, and but the discontinuity is weakened by the non-locality. Our study clarifies the essential role of non-locality in the visual interactions among moving organisms.