Many chemical reactions can be formulated in terms of particle diffusion in a complex energy landscape. Transition path theory (TPT) is a theoretical framework for describing the direct (reaction) pathways from reactant to product states within this energy landscape, and calculating the effective reaction rate. It is now the standard method for analyzing rare events between long lived states. In this paper, we consider a completely different application of TPT, namely, a dual-aspect diffusive search process in which a particle alternates between collecting cargo from a source domain A and then delivering it to a target domain B. The rate of resource accumulation at the target, k _AB , is determined by the statistics of direct (reactive or transport) paths from A to B. Rather than considering diffusion in a complex energy landscape, we focus on pure diffusion with stochastic resetting. Resetting introduces two non-trivial problems in the application of TPT. First, the process is not time-reversal invariant, which is reflected by the fact that there exists a unique non-equilibrium stationary state (NESS). Second, calculating k _AB involves determining the total probability flux of direct transport paths across a dividing surface S between A and B. This requires taking into account discontinuous jumps across S due to resetting. We derive a general expression for k _AB and show that it is independent of the choice of dividing surface. Finally, using the example of diffusion in a finite interval, we show that there exists an optimal resetting rate at which k _AB is maximized. We explore how this feature depends on model parameters.

中文翻译：

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随机重置扩散搜索的转移路径理论

许多化学反应可以用复杂能量景观中的粒子扩散来表达。过渡路径理论（TPT）是一个理论框架，用于描述该能量景观中从反应物到产物状态的直接（反应）路径，并计算有效反应速率。它现在是分析长寿状态之间罕见事件的标准方法。在本文中，我们考虑 TPT 的一种完全不同的应用，即双方面扩散搜索过程，其中粒子交替从源域收集货物A然后将其传递到目标域乙。目标资源积累率，k _AB ，由从 A 到 B 的直接（反应或传输）路径的统计数据确定。我们不考虑复杂能源景观中的扩散，而是关注具有随机重置的纯扩散。重置在 TPT 的应用中引入了两个重要的问题。首先，该过程不是时间反演不变的，这体现在存在唯一的非平衡稳态（NESS）。二、计算k _AB 涉及确定穿过分割表面的直接传输路径的总概率通量S之间A和乙。这需要考虑到不连续的跳跃S由于重置。我们推导出一个通用表达式k _AB 并表明它与分割表面的选择无关。最后，使用有限区间内的扩散示例，我们证明存在一个最佳重置率，此时k _AB 被最大化。我们探讨这个特征如何依赖于模型参数。