标题:Cream in Coffee:从随机行走到分数维度
Cream in Coffee: From Random Walks to Fractal Dimensionalities
刘庆康 2015301020026 武汉大学物理科学与技术学院
摘要:
本文首先研究随机游走问题。作为随机系统的基础,其无需使用物理法则计算,仅通过
统计和随机原理,就能对系统给出相当好的描述。本报告使用 Python 模拟了一至三维的随
机步长游走,对其均方位移做了线性拟合,并分析了其扩散系数 D。此后,依据牛奶在咖啡
中的扩散过程,进一步研究随机游走与扩散的关系,并计算了该过程中熵的变化。最后对
Eden 和 DLA 两种聚类模型进行了简要介绍,模拟并展示了 DLA 聚类的生成过程,在对 DLA
聚类的进一步分析中,引出分数维度的概念并做了一些探索。
This paper first studies the random walk problem. As a basis for a stochastic
system, it does not need to be accurately calculated using the laws of physics
and can give a fairly good description of the system only through statistics and
randomization. We used Python to simulate one-to-three-dimensional random walk,
fitted their mean square displacements linearly, and analyzed their diffusion
coefficients. Thereafter, based on the diffusion process of milk in coffee, the
relationship between random walk and diffusion was further studied, and the
change of entropy in the process was calculated. Finally, two kinds of clustering
models of Eden and DLA are briefly introduced. The generation process of DLA
clustering is simulated and demonstrated. In the further analysis of DLA
clustering, the concept of fractional dimension is elicited and explored.
目录
Ⅰ 序
Ⅱ 正文
1 Random Walks
2 Diffusion and Entropy
3 Cluster Growth Models
Ⅲ 结论
Ⅳ 引用