Exact spatiotemporal dynamics of lattice random walks in hexagonal and honeycomb domains
Exact spatiotemporal dynamics of lattice random walks in hexagonal and honeycomb domains
复制标题
六边形和蜂窝域中晶格随机游走的精确时空动力学
DOI：
10.1103/physreve.107.054139
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发表时间：
2023
影响因子：
2.4
通讯作者：
Marris D
中科院分区：
文献类型：
--
作者：
Marris D
自然和人造系统中的各种输运过程本质上是随机的。为了模拟它们的随机性，晶格随机游动已经被采用了很长一段时间，主要是考虑笛卡尔晶格。然而，在有界空间中的许多应用中，域的几何形状可能对动力学产生深远的影响，应该加以考虑。我们认为在这里的情况下，六个邻居（六边形）和三个邻居（蜂窝状）晶格，这是利用模型，从吸附原子扩散金属和激发扩散单壁碳纳米管动物觅食策略和形成的领土的气味标记生物。在这些和其他例子中，研究六边形几何中晶格随机行走动力学的主要理论工具是通过模拟。在大多数情况下，解析表示法是不可接近的，特别是在有界六边形中，给出了步行者受到的复杂的“之”字形边界条件。在这里，我们推广的图像的方法，六边形的几何形状，并获得封闭形式的表达式的占用概率，所谓的传播子，晶格随机行走的六边形和蜂窝晶格周期，反射和吸收边界条件。在周期性的情况下，我们确定了两种可能的选择的图像放置和相应的传播。使用它们，我们构建了其他边界条件的精确传播子，并推导出与传输相关的统计量，如一个或多个目标的首次通过概率及其方法，阐明了边界条件对传输特性的影响。
A variety of transport processes in natural and man-made systems are intrinsically random. To model their stochasticity, lattice random walks have been employed for a long time, mainly by considering Cartesian lattices. However, in many applications in bounded space the geometry of the domain may have profound effects on the dynamics and ought to be accounted for. We consider here the cases of the six-neighbor (hexagonal) and three-neighbor (honeycomb) lattices, which are utilized in models ranging from adatoms diffusing in metals and excitations diffusing on single-walled carbon nanotubes to animal foraging strategy and the formation of territories in scent-marking organisms. In these and other examples, the main theoretical tool to study the dynamics of lattice random walks in hexagonal geometries has been via simulations. Analytic representations have in most cases been inaccessible, in particular in bounded hexagons, given the complicated “zigzag” boundary conditions that a walker is subject to. Here we generalize the method of images to hexagonal geometries and obtain closed-form expressions for the occupation probability, the so-called propagator, for lattice random walks both on hexagonal and honeycomb lattices with periodic, reflective, and absorbing boundary conditions. In the periodic case, we identify two possible choices of image placement and their corresponding propagators. Using them, we construct the exact propagators for the other boundary conditions, and we derive transport-related statistical quantities such as first-passage probabilities to one or multiple targets and their means, elucidating the effect of the boundary condition on transport properties.