Berezhkovskii Alexander M, Dagdug Leonardo, Bezrukov Sergey M
Program in Physical Biology, Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, Bethesda, Maryland 20892, USA.
J Chem Phys. 2015 Aug 28;143(8):084103. doi: 10.1063/1.4928741.
This paper is devoted to bulk-mediated surface diffusion of a particle which can diffuse both on a flat surface and in the bulk layer above the surface. It is assumed that the particle is on the surface initially (at t = 0) and at time t, while in between it may escape from the surface and come back any number of times. We propose a new approach to the problem, which reduces its solution to that of a two-state problem of the particle transitions between the surface and the bulk layer, focusing on the cumulative residence times spent by the particle in the two states. These times are random variables, the sum of which is equal to the total observation time t. The advantage of the proposed approach is that it allows for a simple exact analytical solution for the double Laplace transform of the conditional probability density of the cumulative residence time spent on the surface by the particle observed for time t. This solution is used to find the Laplace transform of the particle mean square displacement and to analyze the peculiarities of its time behavior over the entire range of time. We also establish a relation between the double Laplace transform of the conditional probability density and the Fourier-Laplace transform of the particle propagator over the surface. The proposed approach treats the cases of both finite and infinite bulk layer thicknesses (where bulk-mediated surface diffusion is normal and anomalous at asymptotically long times, respectively) on equal footing.
本文致力于研究粒子的体介导表面扩散，该粒子既能在平面表面扩散，也能在表面上方的体层中扩散。假设粒子最初（在(t = 0)时）在表面上，在时刻(t)，粒子可能会从表面逸出并返回任意次数。我们提出了一种解决该问题的新方法，该方法将其解简化为粒子在表面和体层之间跃迁的双态问题的解，重点关注粒子在这两个状态下花费的累积停留时间。这些时间是随机变量，它们的总和等于总观测时间(t)。所提出方法的优点是，对于在时间(t)内观测到的粒子在表面上花费的累积停留时间的条件概率密度的双拉普拉斯变换，它允许有一个简单精确的解析解。该解用于求粒子均方位移的拉普拉斯变换，并分析其在整个时间范围内时间行为的特点。我们还建立了条件概率密度的双拉普拉斯变换与粒子在表面上的传播子的傅里叶 - 拉普拉斯变换之间的关系。所提出的方法在同等基础上处理了有限和无限体层厚度的情况（在渐近长的时间下，体介导表面扩散分别是正常的和反常的）。
