The S-Curves are Everywhere

This article discusses the dynamics of the S-curve phenomenon in nature. The S-curve phenomenon and its physics principle unite the spreading flows with the collecting flows, and the animate flows with the inanimate flows. The history of the volume of heated soil versus time follows an S-shaped curve that is entirely deterministic. It is also predicted that when the invading channels are tree shaped as opposed to single pipes, the entire flow from point to volume occurs faster, more easily, along a steeper S-curve. The S-curves of nature are history records of tree-shaped spreading on areas and volumes that are eventually filled during consolidation by transversal diffusion. The prevalence of S-curve phenomena in nature rivals that of tree-shaped flows, which also unite the animate, inanimate, and human realms. This phenomenon is so common that it has generated entire fields of research that seem unrelated: the spreading of biological populations, cancer tumors, chemical reactions, contaminants, languages, news, information, innovations, technologies, infrastructure, and economic activity, and the evolution of technology performance versus cumulative R&D spending.