Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian bees

The "Brownian bees" model describes an ensemble of N= const independent branching Brownian particles. The conservation of N is provided by a modified branching process. When a particle branches into two particles, the particle which is farthest from the origin is eliminated simultaneously. The spatial density of the particles is governed by the solution of a free boundary problem for a reaction-diffusion equation in the limit of N≫1. At long times, the particle density approaches a spherically symmetric steady-state solution with a compact support of radius ℓ[over ¯]_{0}. However, at finite N, the radius of this support, L, fluctuates. The variance of these fluctuations appears to exhibit a logarithmic anomaly [Siboni et al., Phys. Rev. E 104, 054131 (2021)2470-004510.1103/PhysRevE.104.054131]. It is proportional to N^{-1}lnN at N→∞. We investigate here the tails of the probability density function (PDF), P(L), of the swarm radius, when the absolute value of the radius fluctuation ΔL=L-ℓ[over ¯]_{0} is sufficiently larger than the typical fluctuations' scale determined by the variance. For negative deviations the PDF can be obtained in the framework of the optimal fluctuation method. This part of the PDF displays the scaling behavior lnP∝-NΔL^{2}ln^{-1}(ΔL^{-2}), demonstrating a logarithmic anomaly at small negative ΔL. For the opposite sign of the fluctuation, ΔL>0, the PDF can be obtained with an approximation of a single particle, running away. We find that lnP∝-N^{1/2}ΔL. We consider in this paper only the case when |ΔL| is much less than the typical radius of the swarm at N≫1.

Competition in a system of Brownian particles: Encouraging achievers

We introduce and analytically and numerically study a simple model of interagent competition, where underachievement is strongly discouraged. We consider N≫1 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 sqrt[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 sqrt[t]. Finally, we extend the inter-agent competition from n=2 to an arbitrary number n of competing Brownian particles (n≪N). Our analytical predictions are supported by Monte Carlo simulations.