Complexity and Unpredictable Scaling of Hierarchical Structures

The problem of characterizing complexity is formulated in the framework of a hierarchical modelling of physical systems. Complexity is related to the task of predicting the asymptotic scaling behaviour of system’s observables from the available finite-resolution measurements. The analysis is performed on one-dimensional stationary symbolic patterns such as those encountered in nonlinear dynamics, cellular automata, spin systems, making use of suitable coding methods in a statistical-mechanical environment. The asymptotic limit of thermodynamic sums is estimated within a grand-canonical formalism, with the help of transfer-matrix and renormalization techniques: the convergence of the associated scaling functions is directly related to the previously defined complexity measure.

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