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 [4,5].

Some specifics of the approach we develop are: We systematically eliminate access to configurations of an otherwise elementary 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.
3) Redefinition of entropy levels as logarithms of the original ones recovers the BG scheme and the free energy Legendre transform property.

Furthermore, the sequences of actions on the entropies associated with the degrees of freedom have the following consequences:
i) Confine degrees of freedom on very few configurations of ever decreasing entropies.
ii) The reduction in numbers of configurations goes down from the initial exponential of the number of degrees of freedom to only one per micro-canonical ensemble, there being an equivalent exponential number of such ensembles for the entire system. As the thermodynamic limit is approached a set of initial configurations with nonzero measure reduces to a set of vanishing measure. But in the process a new set of numbers of configurations develops. These are given by the degeneracies of the micro-canonical ensembles.
iii) The new emerging numbers of configurations grow more slowly than exponentially with the size of the system as they are binomial coefficients.

As with any partition function, the sum of the numbers of configurations times their probabilities cannot vanish nor diverge but be unity. Initially (the BG case) the numbers grow exponentially and the weights also decrease exponentially with system size. After the actions of the attractor the numbers of new ensemble degeneracies grow slower than exponentially and the new weights must now decrease accordingly. The new “canonical” partition function acquires q-exponential weights typical of q-statistics. The precise value of the tuning parameter q and its relationship with the inverse temperature is determined.

[1] Robledo A., Moyano, L.G., “q-deformed statistical-mechanical property in the dynamics of trajectories en route to the Feigenbaum attractor”, Physical Review E 77, 032613 (2008).
[2] Robledo, A., “A dynamical model for hierarchy and modular organization: The trajectories en route to the attractor at the transition to chaos”, Journal of Physics: Conf. Ser. 394, 012007 (2012).
[3] Diaz-Ruelas, A., Robledo, A., “Emergent statistical-mechanical structure in the dynamics along the period-doubling route to chaos”, Europhysics Letters 105, 40004 (2014).
[4] Bak, P., “How Nature Works” (Copernicus, New York, 1996).
[5] Watkins, N.W., Pruessner, G., Chapman, S.C., Crosby, N.B., Jensen, J.K., “25 Years of Self-organized Criticality: Concepts and Controversies”, Space Science Reviews 198, 3 (2016).