论文标题
蜂群大小的中等典型波动的概率
Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian Bees
论文作者
论文摘要
``布朗蜜蜂''模型描述了$ n = $ 〜const独立的布朗颗粒的合奏。 $ n $的保护由修改后的分支过程提供。当粒子分支成两个粒子时，与原点最远的粒子同时消除。颗粒的空间密度受反应扩散方程的自由边界问题的解决方案的控制，以$ n \ gg 1 $的极限。长期以来，粒子密度接近球形对称的稳态解，并用半径$ \ bar {\ ell} _0 $紧凑地支撑。但是，在有限$ n $的情况下，这种支持的半径，$ l $，波动。这些波动的方差似乎表现出对数异常[siboni {\ em等人}。 Rev. E. {\ bf104}，054131（2021）]。它与$ n^{ - 1} \ ln n $ at $ n \ to \ infty $成比例。我们在这里调查了群半径的概率密度函数（PDF），$ p（l）$的尾部，当半径波动的绝对值$Δl= l- \ bar {\ ell} _0 $比通过方差确定的典型波动尺度大得多。对于负偏差，可以在最佳波动方法（OFM）的框架中获得PDF。 PDF的这一部分显示了缩放行为：$ \ ln p \ propto-nΔl^2 \，\ ln^{ - 1}（Δl^{ - 2}）$，在小负$Δl$上展示了对数异常。对于波动的相反符号，$Δl> 0 $，可以通过单个粒子的近似值获得PDF，逃走。我们发现$ \ ln p \ propto -n^{1/2}Δl$。我们仅在本文中考虑，当$ |Δl| $远低于$ n \ gg 1 $的群体半径少得多。
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 $ΔL=L-\bar{\ell}_0$ 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: $\ln P\propto - N ΔL^2\, \ln^{-1}(ΔL^{-2})$, 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 $\ln P \propto -N^{1/2}ΔL$. We consider in this paper only the case, when $|ΔL|$ is much less than the typical radius of the swarm at $N\gg 1$.
