Fluctuations of a swarm of Brownian bees
Abstract
The "Brownian bees" model describes an ensemble of N independent branching Brownian particles. When a particle branches into two particles, the particle farthest from the origin is eliminated so as to keep the number of particles constant. In the limit of N →∞ , the spatial density of the particles is governed by the solution of a free boundary problem for a reactiondiffusion equation. At long times the particle density approaches a spherically symmetric steadystate solution with a compact support. Here, we study fluctuations of the "swarm of bees" due to the random character of the branching Brownian motion in the limit of large but finite N . We consider a onedimensional setting and focus on two fluctuating quantities: the swarm center of mass X (t ) and the swarm radius ℓ (t ) . Linearizing a pertinent Langevin equation around the deterministic steadystate solution, we calculate the twotime covariances of X (t ) and ℓ (t ) . The variance of X (t ) directly follows from the covariance of X (t ) , and it scales as 1 /N as to be expected from the law of large numbers. The variance of ℓ (t ) behaves differently: It exhibits an anomalous scaling (1 /N )lnN . This anomaly appears because all spatial scales, including a narrow region near the edges of the swarm where only a few particles are present, give a significant contribution to the variance. We argue that the variance of ℓ (t ) can be obtained from the covariance of ℓ (t ) by introducing a cutoff at the microscopic time 1 /N where the continuum Langevin description breaks down. Our theoretical predictions are in good agreement with Monte Carlo simulations of the microscopic model. Generalizations to higher dimensions are briefly discussed.
 Publication:

Physical Review E
 Pub Date:
 November 2021
 DOI:
 10.1103/PhysRevE.104.054131
 arXiv:
 arXiv:2106.11619
 Bibcode:
 2021PhRvE.104e4131S
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Mathematics  Probability
 EPrint:
 8 pages, 4 figures