Péclet数决定漂移扩散中向加速再启动的转变
Péclet number governs transition to acceleratory restart in drift-diffusion
摘要：
第一通道过程可以分为两类：一类是通过引入重新启动而加速的，另一类则显示出相反的反应。在物理系统中，两类之间的过渡可能会发生，因为控制参数会发生变化，以跨越一个普遍的临界点。然而，仍然缺乏一个完全易于处理的模型系统来教我们如何进行这种转变。为了弥合这一差距，我们量化的随机重新启动的漂移扩散过程的吸收边界的第一次通过时间的影响。在那里，我们发现，过渡是由Péclet数（$Pe$）--平流和扩散传输的速率之间的比率。当$Pe>1$的过程是漂移控制和重新启动只能阻碍其完成。相反，当$0\leq~Pe<1$时，该过程是扩散控制的，并且重新启动可以将其完成加速$\sim1/Pe$的因子。当以最佳速率$r^{\star}\simeq r_0^{\star}\left(1-Pe\right)$重新启动过程时，会出现这种加速，其中$r_0^{\star}$代表纯扩散极限中的最佳重新启动速率。本文考虑的过渡站在重新启动现象的核心，是相关的各种各样的过程中，被驱动到完成噪声的存在。这些过程中的每一个都有独特的特征，但我们的分析表明，重新启动过渡类似于其他相变-它的一些中心特征是完全通用的。
英文摘要：
First-passage processes can be divided in two classes: those that are accelerated by the introduction of restart and those that display an opposite response. In physical systems, a transition between the two classes may occur as governing parameters are varied to cross a universal tipping point. However, a fully tractable model system to teach us how this transition unfolds is still lacking. To bridge this gap, we quantify the effect of stochastic restart on the first-passage time of a drift-diffusion process to an absorbing boundary. There, we find that the transition is governed by the Péclet number ($Pe$) --- the ratio between the rates of advective and diffusive transport. When $Pe>1$ the process is drift-controlled and restart can only hinder its completion. In contrast, when $0\leq~Pe<1$ the process is diffusion-controlled and restart can speed-up its completion by a factor of $\sim1/Pe$. Such speedup occurs when the process is restarted at an optimal rate $r^{\star}\simeq r_0^{\star}\left(1-Pe\right)$, where $r_0^{\star}$ stands for the optimal restart rate in the pure-diffusion limit. The transition considered herein stands at the core of restart phenomena and is relevant to a large variety of processes that are driven to completion in the presence of noise. Each of these processes has unique characteristics, but our analysis reveals that the restart transition resembles other phase transitions --- some of its central features are completely generic.
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