We investigate how confinement may drastically change both the probability density of the first-encounter time and the associated survival probability in the case of two diffusing particles. To obtain analytical insights into this problem, we focus on two one-dimensional settings: a half-line and an interval. We first consider the case with equal particle diffusivities, for which exact results can be obtained for the survival probability and the associated first-encounter time density valid over the full time domain. We also evaluate the moments of the first-encounter time when they exist. We then turn to the case with unequal diffusivities and focus on the long-time behavior of the survival probability. Our results highlight the great impact of boundary effects in diffusion-controlled kinetics even for simple one-dimensional settings, as well as the difficulty of obtaining analytic results as soon as the translational invariance of such systems is broken.

中文翻译：

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限制中两个扩散粒子的首次相遇时间

我们研究了在两个扩散粒子的情况下，限制如何极大地改变首次遇到时间的概率密度和相关的生存概率。为了获得对该问题的分析见解，我们集中于两个一维设置：半线和间隔。我们首先考虑具有相同粒子扩散率的情况，对于这种情况，可以在整个时域上获得有效的生存概率和相关的首次遇到时间密度的精确结果。我们还会评估首次遇到的时刻（如果存在）。然后，我们转向扩散率不相等的情况，并关注生存概率的长期行为。我们的结果强调了边界效应对扩散控制动力学的巨大影响，即使对于简单的一维设置，