Zipf's law and the effect of ranking on probability distributions

Ranking procedures are widely used in the description of many different types of complex systems. Zipf's law is one of the most remarkable frequency-rank relationships and has been observed independently in physics, linguistics, biology, demography, etc. We show that ranking plays a crucial role in making it possible to detect empirical relationships in systems that exist in one realization only, even when the statistical ensemble to which the systems belong has a very broad probability distribution. Analytical results and numerical simulations are presented which clarify the relations between the probability distributions and the behavior of expected values for unranked and ranked random variables. This analysis is performed, in particular, for the evolutionary model presented in our previous papers which leads to Zipf's law and reveals the underlying mechanism of this phenomenon in terms of a system with interdependent and interacting components as opposed to the ''ideal gas'' models suggested by previous researchers. The ranking procedure applied to this model leads to a new, unexpected phenomenon: a characteristic ''staircase'' behavior of the mean values of the ranked variables (ranked occupation numbers). This result is due to the broadness of the probability distributions for the occupation numbers and does not follow from the ''ideal gas'' model. Thus, it provides an opportunity by comparison with empirical data, to obtain evidence as to which model relates to reality.