STOCHASTIC EQUATIONS AND CITY GROWTH

Marc Barthelemy (Personal webpage)

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 which will be the subject of this talk. 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. 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 this law), 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. In this talk I will review the main theoretical attempts based on stochastic equations to describe these empirical facts.