Competition in a system of Brownian particles: Encouraging achievers

P. L. Krapivsky, Ohad Vilk, and Baruch Meerson
Phys. Rev. E 106, 034125 – Published 15 September 2022

Abstract

We introduce and analytically and numerically study a simple model of interagent competition, where underachievement is strongly discouraged. We consider N1 particles performing independent Brownian motions on the line. Two particles are selected at random and at random times, and the particle closest to the origin is reset to it. We show that, in the limit of N, the dynamics of the coarse-grained particle density field can be described by a nonlocal hydrodynamic theory which was encountered in a study of the spatial extent of epidemics in a critical regime. The hydrodynamic theory predicts relaxation of the system toward a stationary density profile of the “swarm” of particles, which exhibits a power-law decay at large distances. An interesting feature of this relaxation is a nonstationary “halo” around the stationary solution, which continues to expand in a self-similar manner. The expansion is ultimately arrested by finite-N effects at a distance of order N from the origin, which gives an estimate of the average radius of the swarm. The hydrodynamic theory does not capture the behavior of the particle farthest from the origin—the current leader. We suggest a simple scenario for typical fluctuations of the leader's distance from the origin and show that the mean distance continues to grow indefinitely as t. Finally, we extend the inter-agent competition from n=2 to an arbitrary number n of competing Brownian particles (nN). Our analytical predictions are supported by Monte Carlo simulations.

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  • Received 27 July 2022
  • Accepted 1 September 2022

DOI:https://doi.org/10.1103/PhysRevE.106.034125

©2022 American Physical Society

Physics Subject Headings (PhySH)

Interdisciplinary PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

P. L. Krapivsky1,2,*, Ohad Vilk3,4,5,†, and Baruch Meerson3,‡

  • 1Department of Physics, Boston University, Boston, Massachusetts 02215, USA
  • 2Santa Fe Institute, Santa Fe, New Mexico 87501, USA
  • 3Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
  • 4Movement Ecology Lab, Department of Ecology, Evolution and Behavior, Alexander Silberman Institute of Life Sciences, Hebrew University of Jerusalem, Jerusalem 91904, Israel
  • 5Minerva Center for Movement Ecology, Hebrew University of Jerusalem, Jerusalem 91904, Israel

  • *pkrapivsky@gmail.com
  • ohad.vilk@mail.huji.ac.il
  • meerson@mail.huji.ac.il

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Vol. 106, Iss. 3 — September 2022

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