Kozak John J, Balakrishnan V
Department of Chemistry, Iowa State University, Ames, Iowa 50011-3111, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Feb;65(2 Pt 1):021105. doi: 10.1103/PhysRevE.65.021105. Epub 2002 Jan 15.
The exact analytic expression for the mean time to absorption (or mean walk length) for a particle performing a random walk on a finite Sierpinski gasket with a trap at one vertex is found to be T((n))=[3(n)5(n+1)+4(5(n))-3(n)]/(3(n+1)+1) where n denotes the generation index of the gasket, and the mean is over a set of starting points of the walk distributed uniformly over all the other sites of the gasket. In terms of the number N(n) of sites on the gasket and the spectral dimension d of the gasket, the precise asymptotic behavior for large N(n) is T((n))-->1/3(2N(n))(2/d)-N1.464. This serves as a partial check on our result, as it is (a) intermediate between the known results T-N2 (d=1) and T-N ln N (d=2) for random walks on d-dimensional Euclidean lattices and (b) consistent with the known result for the asymptotic behavior of the mean number of distinct sites visited in a random walk on a fractal lattice.
对于在有限谢尔宾斯基垫片上进行随机游走且在一个顶点处有陷阱的粒子，其吸收平均时间（或平均游走长度）的精确解析表达式为(T((n)) = [3(n)5(n + 1) + 4(5(n)) - 3(n)] / (3(n + 1) + 1))，其中(n)表示垫片的生成指数，且该平均值是对在垫片所有其他位置上均匀分布的一组游走起点进行计算的。就垫片上的位置数(N(n))和垫片的谱维度(d)而言，对于大(N(n))时的精确渐近行为是(T((n)) \to 1/3(2N(n))(2/d) - N1.464)。这可作为对我们结果的部分检验，因为它（a）介于(d)维欧几里得晶格上随机游走的已知结果(T - N^2)（(d = 1)）和(T - N \ln N)（(d = 2)）之间，并且（b）与分形晶格上随机游走中访问的不同位置平均数量的渐近行为的已知结果一致。
