Power Laws and Cities Population

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We study a few dynamical systems composed of many components whose sizes evolve according to multiplicative stochastic rules. We compare them with respect to the emergence of power laws in the size distribution of their components. We show that the details specifying and enforcing the smallest size of the components are crucial as well as the rules for creating new components. In the systems where the power law holds, one obtains for the exponent alpha the experimentally measured value 1 for a wide range of parameters.