The figures show the temporal solution of the demographic dynamics (Eq. (13)) when rotational symmetry is restored through stochasticity averaged over long times. a Shows the population structure at different times (dashed lines) and the distribution with smallest divergence from Zipf’s law over a long run (red), measured by the smallest observed DKL(P∣Pz). b shows the trajectory of the Kullback-Leibler divergence away from Zipf’s law over time. After a long time period, the population structure approximates Zipf’s law and fluctuates around it. In this regime, at any single time only samples showing some deviations from Zipf’s distribution are observable: it is only on the average over many structure vectors at long times that Zipf’s law emerges as a good characterization of the city size distribution. Fig. 4 shows that this was indeed the case for the US urban system until recently.