Stochastic equations constitute a major ingredient in many branches of science, from physics to biology and engineering. Not surprisingly, they appear in many quantitative studies of complex systems. In particular, this type of equation is useful for understanding the dynamics of urban population. Empirically, the population of cities follows a seemingly universal law—called Zipf’s law—which was discovered about a century ago and states that when sorted in decreasing order, the population of a city varies as the inverse of its rank. Recent data however showed that this law is only approximate and in some cases not even verified. In addition, the ranks of cities follow a turbulent dynamics: some cities rise while other fall and disappear. Both these aspects—Zipf’s law (and deviations around it), and the turbulent dynamics of ranks—need to be explained by the same theoretical framework and it is natural to look for the equation that governs the evolution of urban populations. We will review here the main theoretical attempts based on stochastic equations to describe these empirical facts. We start with the simple Gibrat model that introduces random growth rates, and we will then discuss the Gabaix model that adds friction for allowing the existence of a stationary distribution. Concerning the dynamics of ranks, we will discuss a phenomenological stochastic equation that describes rank variations in many systems—including cities—and displays a noise-induced transition. We then illustrate the importance of exchanges between the constituents of the system with the diffusion with noise equation. We will explicit this in the case of cities where a stochastic equation for populations can be derived from first principles and confirms the crucial importance of inter-urban migrations shocks for explaining the statistics and the dynamics of the population of cities.

中文翻译：

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随机方程和城市


随机方程构成了从物理学到生物学和工程学的许多科学分支的主要组成部分。毫不奇怪，它们出现在许多复杂系统的定量研究中。特别是，此类方程对于了解城市人口的动态非常有用。从经验上看，城市人口遵循一个看似普遍的定律——称为齐普夫定律，该定律是在大约一个世纪前发现的，该定律指出，当按降序排序时，城市人口的变化与其排名成反比。然而，最近的数据表明，该定律只是近似的，在某些情况下甚至未经验证。此外，城市的排名呈现出一种动荡的动态：一些城市上升，另一些城市下降甚至消失。这两个方面——齐普夫定律（及其周围的偏差）和等级的动荡动态——都需要用相同的理论框架来解释，并且很自然地寻找控制城市人口演变的方程。我们将在这里回顾基于随机方程来描述这些经验事实的主要理论尝试。我们从引入随机增长率的简单 Gibrat 模型开始，然后我们将讨论 Gabaix 模型，该模型增加了摩擦以允许平稳分布的存在。关于排名的动态，我们将讨论一个唯象随机方程，该方程描述了许多系统（包括城市）中的排名变化，并显示了噪声引起的转变。然后，我们用噪声扩散方程说明系统组成部分之间交换的重要性。 我们将在城市的情况下明确这一点，其中人口的随机方程可以从第一原理推导出来，并确认城市间移民冲击对于解释城市人口的统计数据和动态的至关重要性。