  The rank-size plots of a large number of different physical and
socio-economic systems are usually said to follow Zipf's law, but a unique
framework for the comprehension of this ubiquitous scaling law is still
lacking. Here we show that a dynamical approach is crucial: during their
evolution, some systems are attracted towards Zipf's law, while others presents
Zipf's law only temporarily and, therefore, spuriously. A truly Zipfian
dynamics is characterized by a dynamical constraint, or coherence, among the
parameters of the generating PDF, and the number of elements in the system. A
clear-cut example of such coherence is natural language. Our framework allows
us to derive some quantitative results that go well beyond the usual Zipf's
law: i) earthquakes can evolve only incoherently and thus show Zipf's law
spuriously; this allows an assessment of the largest possible magnitude of an
earthquake occurring in a geographical region. ii) We prove that Zipfian
dynamics are not additive, explaining analytically why US cities evolve
coherently, while world cities do not. iii) Our concept of coherence can be
used for model selection, for example, the Yule-Simon process can describe the
dynamics of world countries' GDP. iv) World cities present spurious Zipf's law
and we use this property for estimating the maximal population of an urban
agglomeration.
