We consider the first-passage problem for $N$ identical independent particles that are initially released uniformly in a finite domain $\Omega$ and then diffuse toward a reactive area $\Gamma$, which can be part of the outer boundary of $\Omega$ or a reaction centre in the interior of $\Omega$. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the $N$ particles reacts with $\Gamma$. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the initial condition in the scaling of the average fastest first-passage time with the particle number $N$, namely, a much stronger dependence ($1/N$ and $1/N^2$ for partially and perfectly reactive targets, respectively), in contrast to the well known inverse-logarithmic behaviour found when all particles are released from the same fixed point. We combine analytic solutions with scaling arguments and stochastic simulations to rationalise our results, which open new perspectives for studying the relevance of multiple searchers in various situations of molecular reactions, in particular, in living cells.

中文翻译：

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从单粒子随机动力学到宏观反应速率：$N$ 随机游走者的最快首次通过时间

我们考虑 $N$ 相同独立粒子的首通道问题，这些粒子最初在有限域 $\Omega$ 中均匀释放，然后向反应区域 $\Gamma$ 扩散，该区域可以是 $\ 的外边界的一部分Omega$ 或 $\Omega$ 内部的反应中心。对于完全反应和部分反应两种情况，我们获得了最快首过时间（fFPT）的前两个矩的显式公式，即第一个粒子与 $\Gamma$ 反应的时间. 此外，我们研究了 fFPT 的全概率密度。我们讨论了初始条件在粒子数 $N$ 的平均最快首次通过时间的缩放中的重要作用，即更强的依赖性（$1/N$ 和 $1/N^2$ 部分和完美反应目标，分别），与当所有粒子从同一固定点释放时发现的众所周知的逆对数行为相反。我们将分析解决方案与标度参数和随机模拟相结合，使我们的结果合理化，这为研究多个搜索者在各种分子反应情况下的相关性，特别是在活细胞中的相关性开辟了新的视角。