论文标题

第一组重置的优化和增长

Optimization and Growth in First-Passage Resetting

论文作者

De Bruyne, B., Randon-Furling, J., Redner, S.

论文摘要

我们结合了重置和第一步的过程,以定义\ emph {第一通道重置},其中随机步行到固定位置的重置是由步行本身的第一张音频事件触发的。在无限域中,各向同性扩散的第一步重置是非固定的,随着时间的推移,重置事件的数量为$ \ sqrt {t} $。我们通过分析粒子计算所得的空间概率分布,并通过几何路径分解获得此分布。在有限的间隔中,我们定义了通过第一组重置控制的优化问题;这种情况是由可靠性理论激励的。目的是操作一个接近其最大容量的系统,而不会经历太多故障。但是,当发生故障时,系统将重置为最小的工作点。我们定义并优化了一个目标函数,以最大化奖励(接近最大操作)减去每个故障的罚款。我们还研究了该基本模型的扩展,以包括每个重置和两个维度后的延迟。最后,我们研究了一个域的生长动力学,其中域边界每当散射粒子到达边界之后,在该域发生重置事件时,域边界会逐渐降低。我们确定半无限线和有限间隔的结构域的生长速率,并找到广泛的行为,这些行为取决于粒子撞击边界时经济衰退发生的程度。

We combine the processes of resetting and first-passage to define \emph{first-passage resetting}, where the resetting of a random walk to a fixed position is triggered by a first-passage event of the walk itself. In an infinite domain, first-passage resetting of isotropic diffusion is non-stationary, with the number of resetting events growing with time as $\sqrt{t}$. We calculate the resulting spatial probability distribution of the particle analytically, and also obtain this distribution by a geometric path decomposition. In a finite interval, we define an optimization problem that is controlled by first-passage resetting; this scenario is motivated by reliability theory. The goal is to operate a system close to its maximum capacity without experiencing too many breakdowns. However, when a breakdown occurs the system is reset to its minimal operating point. We define and optimize an objective function that maximizes the reward (being close to maximum operation) minus a penalty for each breakdown. We also investigate extensions of this basic model to include delay after each reset and to two dimensions. Finally, we study the growth dynamics of a domain in which the domain boundary recedes by a specified amount whenever the diffusing particle reaches the boundary after which a resetting event occurs. We determine the growth rate of the domain for the semi-infinite line and the finite interval and find a wide range of behaviors that depend on how much the recession occurs when the particle hits the boundary.

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