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 $N\gg 1$. At long times, the particle density approaches a spherically symmetric steady-state solution with a compact support of radius ${\overline{\ell }}_0$. However, at finite $N$, the radius of this support, $L$, fluctuates. The variance of these fluctuations appears to exhibit a logarithmic anomaly [Siboni et al., Phys. Rev. E 104, 054131 (2021)]. It is proportional to $N^{-1}lnN$ at $N\to \infty$. We investigate here the tails of the probability density function (PDF), $P\left(L\right)$, of the swarm radius, when the absolute value of the radius fluctuation $\mathrm{\Delta }L=L-{\overline{\ell }}_0$ 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. This part of the PDF displays the scaling behavior $lnP\propto -N\mathrm{\Delta }{L}^2{ln}^{-1}\left(\mathrm{\Delta }{L}^{-2}\right)$, demonstrating a logarithmic anomaly at small negative $\mathrm{\Delta }L$. For the opposite sign of the fluctuation, $\mathrm{\Delta }L>0$, the PDF can be obtained with an approximation of a single particle, running away. We find that $lnP\propto -N^{1/2}\mathrm{\Delta }L$. We consider in this paper only the case when $\left|\mathrm{\Delta }L\right|$ is much less than the typical radius of the swarm at $N\gg 1$.

Figure 1

A schematic plot of the distribution of the swarm radius, $P\left(L\right)$. The peak of the distribution is at $x=\pi /2$ which is the edge of the mean-field swarm density $U\left(x\right)$. Typical fluctuations are Gaussian with variance (5) [4], while the two, very asymmetric large deviation tails of $P\left(L\right)$ are described by Eqs. (6) and (7). The dotted line corresponds to a conjectured crossover regime between the typical fluctuations and the atypical positive fluctuations regime.

Figure 2

Shown is $\tilde{p}\left(\xi \right)$ defined in Eq. (34) vs $\xi$.

Figure 3

Shown is $\tilde{q}\left(\xi \right)$ defined in Eq. (46) vs $\xi$.

Figure 4

Numerical solution of the OFM problem with $t_{\lambda }={10}^{-5}$ at $t=-15.7t_{\lambda }$. The solution in the bulk, in the domain $\overline{\mathrm{\Omega }}$ is shown here. The left panel shows $p_s(x,t)$. Compare it with Eq. (30), which gives $p(x\in \overline{\mathrm{\Omega }},t)\simeq 0.122$ at this time. The right panel shows $q_s(x,t)$ by the solid (blue) line and $U\left(x\right)$ by the red (dashed) line; see Eq. (35). The difference between the latter two lines is almost invisible, excluding the region near $x={\overline{\ell }}_0$ (the domain $\mathrm{\Omega }$)

Figure 5

Shown is a comparison of the (theoretical) self-similar and numerical solutions. The left panel shows the momentum in the domain $\mathrm{\Omega }$. The right panel shows the normalized bees' density $q$. The solid (blue) curves are obtained from numerical results for $t_{\lambda }={10}^{-5}$ and $t=-15.7t_{\lambda }$ in accordance to Eqs. (61) and (60), respectively. The dashed (red) lines are the self-similar solutions (33) and (46), respectively.

Figure 6

Simulated relationship between the edge displacement $\mathrm{\Delta }L$ and $t_{\lambda }$. The markers represent results of the simulations. The (blue) line shows the fit (62). See also Eq. (54).

Figure 7

Simulated relationship between the action $S$ and $t_{\lambda }$. The markers represent results of the simulations. The (blue) line shows the fit (63). See also Eq. (55).

Figure 8

The solid (blue) line is a curve defined parametrically by Eqs. (62) and (63). The markers are obtained from results of the numerical solutions. The dashed (black) line corresponds to $S=(\pi /4)\mathrm{\Delta }{L}^2/|ln\left(0.011\mathrm{\Delta }{L}^2\right)|$. Existence of subleading terms, considered in the text, is seen apparently.

Figure 9

Shown is a sketch for $q(x,0)$ in the case of single time scale $\lambda \left(t\right)$, when $p(0,0)$ and the timescale $t_0\ll 1$. It corresponds to the solid line. The region ${\overline{\ell }}_0-x\ll 1$ is shown only. The dashed line presents $U\left(x\right)$. The area between the dashed and solid lines can be estimated as $\sim ({\overline{\ell }}_0-L_0)\sqrt{t_0}$.

Figure 10

Atypical positive fluctuations of $\mathrm{\Delta }L$ are dominated by the “runaway particle” scenario, in which a single particle quickly travels from the edge support of $U\left(x\right)$, at $x=\pi /2$, to the position $x=L$, while the rest of the particles do not display any unusual behavior. During the creation of this fluctuation, the branching process is entirely suppressed.