Bressloff Paul C
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112, USA.
Phys Rev E. 2023 Jun;107(6-1):064121. doi: 10.1103/PhysRevE.107.064121.
Encounter-based models of diffusion provide a probabilistic framework for analyzing the effects of a partially absorbing reactive surface, in which the probability of absorption depends upon the amount of surface-particle contact time. In this paper we develop a class of encounter-based models that deal with absorption at sticky boundaries. Sticky boundaries occur in a diverse range of applications, including cell biology, colloidal physics, finance, and human crowd dynamics. They also naturally arise in active matter, where confined active particles tend to spontaneously accumulate at boundaries even in the absence of any particle-particle interactions. We begin by constructing a one-dimensional encounter-based model of sticky Brownian motion (BM), which is based on the zero-range limit of nonsticky BM with a short-range attractive potential well near the origin. In this limit, the boundary-contact time is given by the amount of time (occupation time) that the particle spends at the origin. We calculate the joint probability density or propagator for the particle position and the occupation time, and then identify an absorption event as the first time that the occupation time crosses a randomly generated threshold. We illustrate the theory by considering diffusion in a finite interval with a partially absorbing sticky boundary at one end. We show how various quantities, such as the mean first passage time (MFPT) for single-particle absorption and the relaxation to steady state at the multiparticle level, depend on moments of the random threshold distribution. Finally, we determine how sticky BM can be obtained by taking a particular diffusion limit of a sticky run-and-tumble particle (RTP).
基于相遇的扩散模型提供了一个概率框架，用于分析部分吸收性反应表面的影响，其中吸收概率取决于表面与粒子的接触时间。在本文中，我们开发了一类基于相遇的模型，用于处理粘性边界处的吸收问题。粘性边界出现在各种应用中，包括细胞生物学、胶体物理学、金融学和人群动力学。它们也自然地出现在活性物质中，在那里，即使在没有任何粒子间相互作用的情况下，受限的活性粒子也倾向于在边界处自发聚集。我们首先构建一个基于相遇的一维粘性布朗运动（BM）模型，该模型基于非粘性BM的零范围极限，在原点附近有一个短程吸引势阱。在此极限下，边界接触时间由粒子在原点花费的时间（占据时间）给出。我们计算粒子位置和占据时间的联合概率密度或传播子，然后将吸收事件定义为占据时间首次超过随机生成阈值的时刻。我们通过考虑在有限区间内一端具有部分吸收性粘性边界的扩散来阐述该理论。我们展示了各种量，如单粒子吸收的平均首次通过时间（MFPT）和多粒子水平上向稳态的弛豫，如何取决于随机阈值分布的矩。最后，我们确定了如何通过对粘性随机游走粒子（RTP）取特定的扩散极限来得到粘性BM。
