Fluctuations of a swarm of Brownian bees

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 reaction-diffusion equation. At long times the particle density approaches a spherically symmetric steady-state 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 one-dimensional 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 steady-state solution, we calculate the two-time 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.