Physical complexity and Zipf's law

This article deals with a measure of the complexity of a physical system recently proposed by Schapiro and puts it into the context of other recently discussed measures of complexity. We discuss this new measure in terms of a simple Markovian evolution model, extending and specifying the model given by Schapiro, which has the advantage of being analyically tractable. We find that the proposed complexity measure leads to interesting results: there exists a kind of phase transition in this system with a vanishing value of the probabilityc of generating a new species. This phase transition is related to a specific complexity of about 3 bits. By investigating decreasingc (c ∼N−q,N the total number of individuals), we find that the complexity per species grows monotonically withq, diverging logarithmically withN as q goes to infinity.