Spatially Structured Cohesion from Extremal Alignment in Topological Active Matter

TL;DR

This paper introduces a novel extremal alignment rule in active matter models that couples local density to orientation decisions, enabling stable, self-confined, spatially structured collective motion without explicit cohesive forces.

Contribution

It demonstrates that candidate-dependent extremal alignment rules can generate cohesion and spatial structure in active matter, overcoming limitations of traditional relaxational models.

Findings

Extremal alignment rules couple orientation to local density, creating effective cohesion.

Topological interactions stabilize finite-sized flocks, preventing collapse.

The model exhibits diverse dynamical phases, including flocks, swarms, and swirling states.

Abstract

Alignment interactions in active matter are typically modeled as relaxational dynamics toward local consensus. In unbounded systems, this makes alignment effectively decoupled from local density and therefore unable to sustain self-confined collective motion without additional attractive forces. Here we show that this limitation can be overcome by extremal alignment rules in which the interaction neighborhood depends on the candidate orientation. For a broad class of candidate- dependent rules with pairwise additive utilities, the decision utility factorizes into the product of an average interaction score and the number of selected neighbors. This multiplicative structure couples orientational decisions to local density and thereby generates an effective cohesive bias without explicit cohesive forces. In metric models, however, the same mechanism drives collapse toward globally connected, effectively mean-field states that suppress spatial structure. We show that topological interactions regularize this tendency, stabilizing self-confined flocks of finite extent in open space. The resulting dynamics exhibits a rich dynamical phase diagram as a function of noise intensity and turning rate, including polarized flocks, swarms, and persistent swirling states. Our results identify candidate-dependent extremal alignment as a simple mechanism for generating cohesive, spatially structured active matter beyond the standard relaxational paradigm.