Cities are complex systems, where micro-scale phenomena lead to emergent, macro-scale patterns. Scaling in cities – the study of how characteristics of cities change with urban population – has been explained based on social interactions and networks, but cities are also governed by the laws of physics, such as thermodynamics and conservation of mass. Here we explore scaling laws in the context of the city as a far-from-equilibrium thermodynamic system, showing how scaling phenomena are consistent with ideas of cities as dissipative systems, the emergence of higher-order structures, and the Maximum Power Principle. We theorize that as cities grow, they use an increasing amount of energy and resources to produce an incremental change in useful work done. Furthermore, they require additional energy and materials to build higher-order structures, such as transportation systems, to overcome increasing density and congestion. We support our theoretical approach with new empirically observed scaling relationships.

城市是复杂的系统，微观现象导致出现宏观宏观格局。基于社会互动和网络，已经解释了城市规模化-研究城市的特征如何随着城市人口的变化而进行的研究，但是城市也受诸如热力学和质量守恒等物理定律支配。在这里，我们探讨了城市作为远离平衡热力学系统的城市中的尺度定律，显示了尺度现象与城市作为耗散系统的思想，高阶结构的出现以及最大功率原理的一致性。我们认为，随着城市的发展，城市将使用越来越多的能源和资源，从而在完成的有用工作中产生逐渐的变化。此外，它们需要更多的能量和材料来构建高阶结构，例如运输系统，以克服日益增加的密度和拥堵。我们通过新的经验观察到的比例关系来支持我们的理论方法。