Power law behavior associated with a Fibonacci Lucas model and generalized statistical models

Abstract

A Fibonacci-Lucas based statistical model and several other related models are studied. The canonical and grand canonical partition functions for these models are developed.Partition structure such as the distribution of sizes as in a cluster distribution is explored.Ensemble averaging over all partitions leads to a scale invariant power law behavior at a particular critical like point. The canonical ensemble of the Fibonacci-Lucas case involves the Gegenbauer polynomial.The model has a hyperbolic power law behavior, a feature linked to the golden mean ratio of two adjacent Fibonacci numbers and also the connection of Lucas numbers to the golden mean. The relation to other power law behavior, such as Zipf and Pareto laws, is mentioned. For the cases considered, the grand canonical ensemble involves the Gauss hypergeometric function F(a,b,c,z) with specific values for a,b,c. The general case has a variable power law behavior with tau exponent equal to1+c-a-b. An application with a=1/2, b=1, c=3 and thus tau=5/2 very closely approximates Bose-Einstein condensation. The zeta function zeta(3/2)=2.61 of the exact theory is replaced with 8/3 and zeta(5/2)= 1.34 with 4/3. At the condensation point the number of cycles of length falls as a scale invariant power law. The cycles, which arise from permutation symmetries associated with Bose-Einstein statistics, can be viewed as links in a complex network of connections between particles. This scale invariant power law parallels that seen in complex networks. The growth of the network is developed using recurrence properties of the model. Constraints imposed by the canonical ensemble and associates correlations lead to some number theoretic connections between Fibonacci and Lucas numbers as an incidental consequence of this approach.