  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.
