论文标题
阈值表面反应和局部时间重置
Threshold surface reactions and local time resetting
论文作者
论文摘要
在本文中，我们考虑了粒子在包含单个目标$ \ calu $的域中扩散的阈值表面吸收机制。目标边界$ \ partial \ calu $被视为一个反应性表面，在与time $ t $接触时，可以修改粒子的内部状态$ z_t $，其中$ z_0 = h $。状态$ z_t $被认为是所谓边界当地时间的单调降低功能，并且在$ z_t $达到零之后，吸收就会发生。 （当地时间的边界是一个布朗的功能，它决定了粒子在$ \ partial \ calu $的附近花费的时间。）我们首先展示如何根据概率密度或广义传播器$ p_0（\ x，\ el，\ ell，t | \ x_0）$（for a for a for a p o \ x____________________t），以$ p_0（\ e x $ \ \ x t），\ \ \ x t的概率密度或一般性传播器$（其中$ \ x_t $和$ \ ell_t $分别表示粒子位置和当地时间$ t $，而$ \ x_0 $是初始位置。然后，我们引入了一个广义的随机重置协议，其中$ \ x_t $和内部状态$ z_t $均重置为其初始值，$ \ x_t \ rightArow \ rightArrow \ rightArrow \ x_0 $和$ z_t \ rightarrow h $，以Poisson Rate $ r $ $ R $。后者在数学上等同于重置边界本地时间，$ \ ell_t \ rightarrow 0 $。由于重置受续订过程的控制，因此可以通过生存概率表示复位的生存概率而无需重置，这意味着可以通过计算$ p_0（\ x，\ ell，t | \ x_0）$的laplace变换来确定吸收的统计数据。我们将其与仅重置粒子位置的情况进行对比，而粒子位置不受更新过程的控制。我们使用在球形域中的半线和球形靶标的扩散的简单示例说明了理论。
In this paper we consider a threshold surface absorption mechanism for a particle diffusing in a domain containing a single target $\calU $. The target boundary $\partial \calU$ is taken to be a reactive surface that modifies an internal state $Z_t$ of the particle when in contact with the surface at time $t$, with $Z_0=h$. The state $Z_t$ is taken to be a monotonically decreasing function of the so-called boundary local time, and absorption occurs as soon as $Z_t$ reaches zero. (The boundary local time is a Brownian functional that determines the amount of time that the particle spends in a neighborhood of $\partial \calU$.) We first show how to analyze threshold surface absorption in terms of the joint probability density or generalized propagator $P_0(\x,\ell,t|\x_0)$ for the pair $(\X_t,\ell_t)$ in the case of a perfectly reflecting surface, where $\X_t$ and $\ell_t$ denote the particle position and local time at time $t$, respectively, and $\x_0$ is the initial position. We then introduce a generalized stochastic resetting protocol in which both the position $\X_t$ and the internal state $Z_t$ are reset to their initial values, $\X_t\rightarrow \x_0$ and $Z_t\rightarrow h$, at a Poisson rate $r$. The latter is mathematically equivalent to resetting the boundary local time, $\ell_t\rightarrow 0$. Since resetting is governed by a renewal process, the survival probability with resetting can be expressed in terms of the survival probability without resetting, which means that the statistics of absorption can be determined by calculating the Laplace transform of $P_0(\x,\ell,t|\x_0)$ with respect to $t$. We contrast this with the case where only particle position is reset, which is not governed by a renewal process. We illustrate the theory using the simple examples of diffusion on the half-line and a spherical target in a spherical domain.
