渗流理论、方法、进展及存在问题

渗流理论是从随机扩散现象(如流体粒子通过孔隙介质逐步扩散并形成随机路径的过程)中抽象出的一种广泛的数学模型,主要研究无序体系随机几何结构形成过程中的演化规律、行为特征及各种临界现象。渗流理论涉及概率统计、图论、统计物理、随机过程、拓扑、几何、代数、动力系统分析等众多领域,是数学的一个重要分支。渗流理论不仅具有重要的理论意义,而且应用背景广阔。过去的50年中,渗流理论因表述简单、内涵丰富,获得了广泛应用,为化学、生态学、物理学、材料科学、传染病学、复杂网络等领域中的问题带来了新视角,提供了新的理论方法。目前渗流研究处于发展的关键时期,渗流中相变临界条件的确定、渗流团簇的复杂结构及其在临界点附近几何特征的理解是渗流研究的核心问题。文章介绍渗流理论研究的主要问题、方法和结果,简要回顾渗流理论发展的标志性成果,最后提出一些值得思考和研究的问题。

Percolation theory,method, progress,and existing problems

Percolation theory is an extensive mathematical model abstracted from random diffusion phenomena (such as the process of fluid particles gradually diffusing through porous media and forming random paths)﹒ It mainly focuses on the evolution law, behavior characteristics and various critical phenomena during the formation process of random geometric structures in disordered systems﹒Percolation is an important branch of mathematics, which involves Graph theory, statistical physics, stochastic processes, topology, geometry, algebra and dynamic systems analysis etc﹒ Percolation theory is not only of great theoretical significance but also has a broad application background﹒ It has brought new perspectives and provided new theoretical methods for problems in the fields of ecology, physics, material science, infectious diseases, and complex networks, etc﹒ Nowadays, percolation is in its critical period﹒ It is the key problems of percolation to determine the critical conditions for percolating transition, and to understand the intricate structure of percolation clusters and their geometric characteristics near the critical point﹒ This paper introduces the main problems, methods and results of percolation theory, and briefly reviews its landmark achievements﹒ Finally, some questions worthy of consideration and research are put forward﹒