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自1905年Pearson正式提出随机行走(RW)的概念以来,在数学上已经进行了广泛深入的研究,并扩充至物理、化学、生物、工程、商业和社会科学等许多领域.由于实际应用的需要,产生了对受限RW问题的研究.其中,端点起始于某一不可穿透壁,而受限于此壁的RW,可以作为最简单的受限RW问题之一.例如,高分子吸附时形成的尾形链,蛋白质受体上环链的形成,都可直接与这类受限随机行走模型相联系.然而,这一受限RW的基本问题并没有得到系统而简明的解答.作者等曾对端基附壁RW进行了研究,但仅限于最简单形状的格点,因此有必要加以扩充,推广至各种不同的格点.本工作表明,格点的几何特征容易与扩散方程中的扩散系数相联系,从而可以导出允许路径的密度函数及行走均方末端距等基本数学物理量.
Since 1905 Pearson formally proposed the concept of random walking (RW), it has undergone extensive and intensive research in mathematics and has been extended to many fields such as physics, chemistry, biology, engineering, commerce and social science, etc. Due to the practical application , Resulting in a study of the constrained RW problem where the endpoint starts at some impervious wall and the RW limited to this wall can be one of the simplest constrained RW problems.For example, polymer adsorption The formation of the tail-shaped chain, the formation of the protein chain on the receptor, can be directly related to this type of constrained stochastic walking model. However, the basic problem of this limited RW has not been systematically and succinctly answered. However, it is necessary to expand and extend it to various lattice points.This work shows that the geometric characteristics of lattice points are easy to be related to the diffusion equation Of the diffusion coefficient, so that we can derive the allowable path density function and walking MSF and other basic mathematical physical quantities.