Mesoscopic theory of flocking with alignment and anti-alignment copying

TL;DR

This paper develops an analytical mesoscopic framework for collective motion models with both alignment and anti-alignment interactions, revealing how these competing rules influence order and fluctuations.

Contribution

It introduces an exact derivation of mesoscopic equations from microscopic models with mixed alignment behaviors, including both annealed and quenched dynamics.

Findings

Competing alignment and anti-alignment suppress long-range order in large systems.

Finite systems exhibit fluctuation-driven structures influenced by interaction composition.

The framework is validated through Gillespie simulations.

Abstract

We study a stochastic model of collective motion in which individuals update their orientation through pairwise aligning or anti-aligning copying interactions. We analyze both annealed dynamics, where interaction types are chosen probabilistically at each update, and quenched dynamics, where individuals are permanently assigned to aligning or anti-aligning subpopulations. Starting from the microscopic master equation on the circle, we derive an exact mesoscopic description via a Fourier-mode expansion and a systematic large $N$ expansion, obtaining closed Fokker-Planck equations and effective stochastic differential equations for the polarization. We show that competing alignment and anti-alignment suppress long-range polar order in the thermodynamic limit in both cases, while finite systems display nontrivial fluctuation-induced structure controlled by the interaction composition. Our results, validated by Gillespie simulations, establish an analytically tractable framework for collective dynamics characterized by competing copying rules and intrinsic noise.