We study the statistics of the first hitting time between a one-dimensional run-and-tumble particle and a target site that switches intermittently between visible and invisible phases. The two-state dynamics of the target is independent of the motion of the particle, which can be absorbed by the target only in its visible phase. We obtain the mean first hitting time when the motion takes place in a finite domain with reflecting boundaries. Considering the turning rate of the particle as a tuning parameter, we find that ballistic motion represents the best strategy to minimize the mean first hitting time. However, the relative fluctuations of the first hitting time are large and exhibit nonmonotonous behaviors with respect to the turning rate or the target transition rates. Paradoxically, these fluctuations can be the largest for targets that are visible most of the time, and not for those that are mostly invisible or rapidly transiting between the two states. On the infinite line, the classical asymptotic behavior $\propto t^{-3/2}$ of the first hitting time distribution is typically preceded, due to target intermittency, by an intermediate scaling regime varying as $t^{-1/2}$. The extent of this transient regime becomes very long when the target is most of the time invisible, especially at low turning rates. In both finite and infinite geometries, we draw analogies with partial absorption problems.

中文翻译：

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滚滚粒子与随机门控目标之间的首次命中时间

我们研究了一维滚动颗粒与目标位置之间的第一次撞击时间的统计数据，该目标位置在可见相和不可见相之间间歇地切换。目标的二态动力学与粒子的运动无关，粒子的运动只能在目标的可见相中被目标吸收。当运动发生在具有反射边界的有限域中时，我们获得平均第一击中时间。考虑到粒子的转动速率作为调整参数，我们发现弹道运动代表了最小化平均第一次撞击时间的最佳策略。但是，第一击球时间的相对波动较大，并且相对于转弯速率或目标过渡速率表现出非单调的行为。矛盾的是，对于大多数情况下可见的目标，这些波动可能是最大的，而对于在两个州之间大多数不可见或快速转换的目标而言，这些波动可能是最大的。在无限线上，经典渐近行为$\propto {Ť}^{-3/\mathrm{2个}}$ 由于目标间断性，通常在第一个击中时间分布中的第一个击中时间分布之前为 ${Ť}^{-\mathrm{1个}/\mathrm{2个}}$。当目标在大多数情况下是不可见的时，尤其是在低转向速度时，此瞬态状态的范围变得非常长。在有限和无限几何中，我们都得出了具有部分吸收问题的类比。