论文标题
根据二维马尔可夫链的分析描述与Margolus邻里的细胞自动机的扩散
Analytical description of the diffusion in a cellular automaton with the Margolus neighbourhood in terms of the two-dimensional Markov chain
论文作者
论文摘要
根据考虑单个粒子的随机运动的投影，分析了具有Margolus邻域的一参数二维细胞自动机。引入与运动方向相关的辅助随机变量，我们将正在考虑的问题减少到二维马尔可夫链的研究中。使用概率生成函数方法得出并精确地求解了概率分布的主方程。概率分布在雅各比多项式方面进行了分析。获得的解决方案的矩使我们得出了与Margolus邻域相对于二维细胞自动机在二维细胞自动机中扩散系数的参数依赖性的精确分析公式。我们的分析结果与其他作者的早期经验结果一致，并改进了它们。对于使用细胞自动机进行建模二维扩散，特别是对于多组分问题而言，结果令人感兴趣。
The one-parameter two-dimensional cellular automaton with the Margolus neighbourhood is analyzed based on the considering the projection of the stochastic movements of a single particle. Introducing the auxiliary random variable associated with the direction of the movement, we reduce the problem under consideration to the study of a two-dimensional Markov chain. The master equation for the probability distribution is derived and solved exactly using the probability generating function method. The probability distribution is expressed analytically in terms of Jacobi polynomials. The moments of the obtained solution allowed us to derive the exact analytical formula for the parametric dependence of the diffusion coefficient in the two-dimensional cellular automaton with the Margolus neighbourhood. Our analytic results agree with earlier empirical results of other authors and refine them. The results are of interest for the modelling two-dimensional diffusion using cellular automata especially for the multicomponent problem.
