Generating Discrete Power-Law Distributions from a Death- Multiple Immigration Population Process
J O Matthews · AIP conference proceedings · 2003
We consider the evolution of a simple population process governed by deaths and multiple immigrations that arrive with rates particular to their order. For a particular choice of rates, the equilibrium solution has a discrete power‐law form. The model is a generalization of a process investigated previously where immigrants arrived in pairs [1]. The general properties of this model are discussed in a companion paper. The population is initiated with precisely M individuals present and evolves to an equilibrium distribution with a power‐law tail. However the power‐law tails of the equilibrium distribution are established immediately, so that moments and correlation properties of the population are undefined for any non‐zero time. The technique we develop to characterize this process utilizes external monitoring that counts the emigrants leaving the population in specified time intervals. This counting distribution also possesses a power‐law tail for all sampling times and the resulting time series exhibits two features worthy of note, a large variation in the strength of the signal, reflecting the power‐law PDF; and secondly, intermittency of the emissions. We show that counting with a detector of finite dynamic range regularizes naturally the fluctuations, in effect ‘clipping’ the events. All previously undefined characteristics such as the mean, autocorrelation and probabilities to the first event and time between events are well defined and derived. These properties, although obtained by discarding much data, nevertheless possess embedded power‐law regimes that characterize the population in a way that is analogous to box averaging determination of fractal‐dimension.