论文标题
两维限制中两个扩散粒子的第一次持续时间
First-encounter time of two diffusing particles in two- and three-dimensional confinement
论文作者
论文摘要
当将颗粒扩散粒子的第一个爆炸时间的统计数据被限制时会发生巨大变化。在目前的工作中,我们利用蒙特卡洛模拟来研究具有反射边界的两维域中两粒子系统的行为。根据模拟的结果,我们全面概述了生存概率$ s(t)$的行为以及相关的第一键概率密度$ h(t)$ h(t)$跨越了几十年。此外,我们还为平均第一个持续时间$ \ langle \ cal {t} \ rangle $提供数值估计和经验公式,以及衰减时间$ t $,表征了生存概率的单指数长期衰减。基于两个粒子的边界和质量中心之间的距离,我们在$ s(t)$开始显着偏离其对应方的无边界情况的时间的时间内获得了经验下限$ t_b $。令人惊讶的是,对于小型粒子,对$ t $的主要贡献仅取决于总扩散率$ d = d_1+d_2 $,与一维情况形成鲜明对比。这种贡献可能与具有扩散性$ d $的虚拟布朗粒子产生的维纳香肠有关。在两个维度中,发现对$ t $的第一个转向贡献却弱取决于$ d_1/d_2 $的比率。当$ d_2 \ ll d_1 $时,我们还会研究慢速扩散极限,并在一个粒子为固定目标时讨论过渡到极限。最后,我们给出一些预期的指示,何时可以预期$ \ langle \ cal {t} \ rangle $是一个很好的近似值。
The statistics of the first-encounter time of diffusing particles changes drastically when they are placed under confinement. In the present work, we make use of Monte Carlo simulations to study the behavior of a two-particle system in two- and three-dimensional domains with reflecting boundaries. Based on the outcome of the simulations, we give a comprehensive overview of the behavior of the survival probability $S(t)$ and the associated first-encounter time probability density $H(t)$ over a broad time range spanning several decades. In addition, we provide numerical estimates and empirical formulas for the mean first-encounter time $\langle \cal{T}\rangle $, as well as for the decay time $T$ characterizing the monoexponential long-time decay of the survival probability. Based on the distance between the boundary and the center of mass of two particles, we obtain an empirical lower bound $t_B$ for the time at which $S(t)$ starts to significantly deviate from its counterpart for the no boundary case. Surprisingly, for small-sized particles, the dominant contribution to $T$ depends only on the total diffusivity $D=D_1+D_2$, in sharp contrast to the one-dimensional case. This contribution can be related to the Wiener sausage generated by a fictitious Brownian particle with diffusivity $D$. In two dimensions, the first subleading contribution to $T$ is found to depend weakly on the ratio $D_1/D_2$. We also investigate the slow-diffusion limit when $D_2 \ll D_1$ and discuss the transition to the limit when one particle is a fixed target. Finally, we give some indications to anticipate when $T$ can be expected to be a good approximation for $\langle \cal{T}\rangle$.