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

Abstract

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 $N \gg 1$. At long times, the particle density approaches a spherically symmetric steady state solution with a compact support of radius $\bar{\ell}_0$. However, at finite $N$, the radius of this support, $L$, fluctuates. The variance of these fluctuations appears to exhibit a logarithmic anomaly [Siboni {\em et al}., Phys. Rev. E. {\bf104}, 054131 (2021)]. It is proportional to $N^{-1}\ln N$ at $N\to\infty$. 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 $\Delta L=L-\bar{\ell}_0$ 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 (OFM). This part of the PDF displays the scaling behavior: $\ln P\propto - N \Delta L^2\, \ln^{-1}(\Delta L^{-2})$, demonstrating a logarithmic anomaly at small negative $\Delta L$. For the opposite sign of the fluctuation, $\Delta L > 0$, the PDF can be obtained with an approximation of a single particle, running away. We find that $\ln P \propto -N^{1/2}\Delta L$. We consider in this paper only the case, when $|\Delta L|$ is much less than the typical radius of the swarm at $N\gg 1$.