Self-organization conducted by the dynamics towards the attractor at the onset of chaos

We construct an all-inclusive statistical-mechanical model for self-organization based on the hierarchical properties of the nonlinear dynamics towards the attractors that define the period-doubling route to chaos [1-3]. The aforementioned dynamics imprints a sequential assemblage of the model that privileges progressively lower entropies, while a new set of configurations emerges due to the collective partitioning of the original system into secluded portions. The initial canonical balance between numbers of configurations and Boltzmann-Gibbs (BG) statistical weights is drastically altered and ultimately eliminated by the sequential actions of the attractor. However the emerging set of configurations implies a different and novel entropy growth process that eventually upsets the original loss and has the capability of locking the system into a self-organized state with characteristics of criticality, therefore reminiscent in spirit to the so-called self-organized criticality.
Some specifics of the approach we develop are: We systematically eliminate access to configurations of an otherwise conventional thermal system model by progressively partitioning it into isolated portions until only remains a subset of configurations of vanishing measure. Each isolated portion becomes essentially a micro-canonical ensemble. The thermal system consists of a large number of (effective) degrees of freedom, each occupying entropy levels with the form of inverse powers of two. The sequential process replaces the original configurations by an emerging discrete scale invariant set of ensemble configurations with allowed entropies that are necessarily inverse powers of two. In doing this we achieve the following results: 1) The constrained thermal system becomes a close analogue of the dynamics towards the multifractal attractor at the period-doubling onset of chaos. 2) The statistical-mechanical properties of the thermal system depart from those of the ordinary Boltzmann-Gibbs form and acquire features from q-statistics.