论文标题
通过随机重置通过半透明接口进行扩散
Diffusion with stochastic resetting screened by a semipermeable interface
论文作者
论文摘要
在本文中，我们考虑对有限的目标$ω\ in \ r^d $的扩散搜索，其边界$ \ partialω$完全吸收。我们假设该目标被封闭的表面$ \ partial \ calm $带有$ω\ subset \ calm \ subset \ subset \ r^d $的半渗透界面所包围。也就是说，界面完全围绕着目标，因此部分筛选了扩散搜索过程。我们还假设，根据带有重置速率$ r $的Poisson过程，扩散粒子（搜索者）的位置随机重置为$ \ x_0 $。位置$ \ x_0 $被认为是在界面外部，$ \ x_0 \ in \ call^c $，这意味着当粒子在$ \ partial \ call calm $的内部内时不会发生重置。因此，可半透明的界面还筛选出重置的效果。我们首先解决了边界价值问题（BVP），以在[0，\ infty）$的半行$ x \上扩散，并在$ x = 0 $上吸收边界，一个$ x = l $的半渗透障碍，以及随机重置为$ x_0> l $ for All $ x> l $。我们计算了要被目标吸收的平均第一个通道时间（MFPT），并探索其行为，这是该界面的渗透率$κ__0$及其空间位置$ L $的函数。然后，我们对三维（3D）球体对称界面和目标进行类似计算，并证明MFPT表现出与1D情况相同的定性行为。最后，我们基于所谓的抢购BM引入了搜索过程的随机单粒子实现。后者将连续反映布朗尼运动在界面的两侧进行缝制。主要的挑战是确定折断BM产生的概率密度满足界面处的可渗透边界条件。我们展示了如何使用更新理论来实现这一目标。
In this paper we consider the diffusive search for a bounded target $Ω\in \R^d$ with its boundary $\partial Ω$ totally absorbing. We assume that the target is surrounded by a semipermeable interface given by the closed surface $\partial \calM$ with $Ω\subset \calM\subset \R^d$. That is, the interface totally surrounds the target and thus partially screens the diffusive search process. We also assume that the position of the diffusing particle (searcher) randomly resets to its initial position $\x_0$ according to a Poisson process with a resetting rate $r$. The location $\x_0$ is taken to be outside the interface, $\x_0\in \calM^c$, which means that resetting does not occur when the particle is within the interior of $\partial \calM$. Hence, the semipermeable interface also screens out the effects of resetting. We first solve the boundary value problem (BVP) for diffusion on the half-line $x\in [0,\infty)$ with an absorbing boundary at $x=0$, a semipermeable barrier at $x=L$, and stochastic resetting to $x_0>L$ for all $x>L$. We calculate the mean first passage time (MFPT) to be absorbed by the target and explore its behavior as a function of the permeability $κ_0$ of the interface and its spatial position $L$. We then perform the analogous calculations for a three-dimensional (3D) spherically symmetric interface and target, and show that the MFPT exhibits the same qualitative behavior as the 1D case. Finally, we introduce a stochastic single-particle realization of the search process based on a generalization of so-called snapping out BM. The latter sews together successive rounds of reflecting Brownian motion on either side of the interface. The main challenge is establishing that the probability density generated by the snapping out BM satisfies the permeable boundary conditions at the interface. We show how this can be achieved using renewal theory.
