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Collective dynamics of trail-interacting particles

Paul Pineau*, Samuel Bell, and Raphaël Voituriez†

Ram M. Adar‡

  • *Contact author: paul.pineau@sorbonne-universite.fr
  • †Contact author: raphael.voituriez@sorbonne-universite.fr
  • ‡Contact author: radar@technion.ac.il
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Phys. Rev. E 114, L042102 – Published 8 October, 2026

DOI: https://doi.org/10.1103/1rfx-w3rw

Abstract

Trail interactions occur when past particle trajectories bias future motion, rendering the system out of thermodynamic equilibrium. While such systems are abundant in nature, their understanding is limited to the single-particle level or phenomenological mean-field theories. Here, we introduce a minimal model of many trail-interacting particles, which extends this paradigm to the fluctuating collective level in arbitrary dimension. Particles diffuse while depositing long-lasting repelling/attracting trails that act as a shared memory field, coupling their dynamics across time and space. Using stochastic density functional theory, we derive fluctuating hydrodynamic equations and analyze analytically and numerically the resulting behaviors. We show that memory, coupled with fluctuations, fundamentally reshapes collective dynamics. In the repulsive case, the particle density displays superdiffusive spreading for d=1 and diffusive spreading for d≥2, characterized by transient clustering and ballistic motion. In the attractive case, the system condensates in finite time into frozen, localized states. Our results establish general principles for trail-interacting systems and reveal how persistent fields generate instabilities and drive self-organization.

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Outline

Information

Phys. Rev. E 114, L042102– Published 8 October, 2026
  • Received 2 December 2025
  • Revised 6 August 2026
  • Accepted 18 September 2026
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