Complexity, contingency, and criticality.

Complexity originates from the tendency of large dynamical systems to organize themselves into a critical state, with avalanches or "punctuations" of all sizes. In the critical state, events which would otherwise be uncoupled become correlated. The apparent, historical contingency in many sciences, including geology, biology, and economics, finds a natural interpretation as a self-organized critical phenomenon. These ideas are discussed in the context of simple mathematical models of sandpiles and biological evolution. Insights are gained not only from numerical simulations but also from rigorous mathematical analysis. IMAGES:

复杂性源于大型动力学系统自发组织至临界状态的固有倾向，伴随各尺度的雪崩事件与各类“间断（punctuations）”现象。在临界状态下，原本互不关联的事件会产生关联。包括地质学、生物学与经济学在内的诸多学科中，看似存在的历史偶然性现象，可被自然地诠释为自组织临界（self-organized critical）现象。这些观点可依托沙堆与生物演化的简易数学模型展开探讨。相关认知不仅可通过数值模拟获得，亦能借助严谨的数学分析得以深化。图片：