Exact evaluation of high order statistical moments in self-organizing phenomena

Self-organizing phenomena (SOP) are ordered collective behavior that arises from small fluctuations and develops positive feedback. These collective effects are macroscopically observable and are governed by higher order statistical moments and correlation functions. An interesting example of the SOP role as the underlying Physics is the atomic superradiant emission of photon in cavities. This effect emerges from the interaction of a single quantum electromagnetic field with N emitters, each modeled as a two-level system, trapped inside an optical cavity. When the typical distance between atoms is smaller than the emitted photon wavelength, the assumption of weak interaction no longer holds. As a result, the iconic spontaneous emission of photons by atoms cannot occur as N independent events, even for N = 2, as verified in recent experiments. The quantum interference among the ensemble shapes the emission in cavities. SOP are also relevant in a large variety of classical problems, namely, efficient brochure dissemination in advertising and disease spreading by aerial vectors, such as dengue fever. Here, we present a new technique for self-organizing systems to evaluate high order statis- tics employing a classical toy model. The technique is valid only for SOP where small fluctuations originates a rapid growth rate. The toy model considered is composed by an urn and N distinguishable balls, uniquely identifiable by their labels. Each time, a given ball is randomly selected, its label is then recorded in a list and then the ball is returned back to the urn. However, the samples are formed by counting the number of unique labels found in a given list. This measure process introduces correlations in time for each sample, which in turn triggers a collective behavior, as also reported for pulsed photon emission in the superradiant models. In Probability Theory, this problem is also referred as the clas- sical coupon-collector?s problem, whose solution has been unveiled only recently, using the poissonization technique. This technique hides the interesting underlying finite sym- metries, which are relevant for physical properties. Instead, the statistics in the toy model are derived here from the geometrical properties of multidimensional Pascal triangle and are closely related to counting problems. Our findings contemplate both small and large N limit. In addition, the mapping between the classical toy model and superradiance permits the formulation of new simple classical Monte Carlo methods to investigate the cooperative regime in cavity QED models.