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A computational mechanics approach to estimate entropy and (approximate) complexity for the dynamics of the 2D Ising Ferromagnet

We present a numerical analysis of the entropy rate and statistical complexity related to the spin flip dynamics of the 2D Ising Ferromagnet at different temperatures T. We follow an information theoretic approach and test three different entropy estimation algorithms to asses entropy rate and statistical complexity of binary sequences. The latter are obtained by monitoring the orientation of a single spin on a square lattice of side-length L=256 at a given temperature parameter over time. The different entropy estimation procedures are based on the M-block Shannon entropy (a well established method that yields results for benchmarking purposes), non-sequential recursive pair substitution (providing an elaborate and an approximate estimator) and a convenient data compression algorithm contained in the zlib-library (providing an approximate estimator only). We propose an approximate measure of statistical complexity that emphasizes on correlations within the sequence and which is easy to implement, even by means of black-box data compression algorithms. Regarding the 2D Ising Ferromagnet simulated using Metropolis dynamics and for binary sequences of finite length, the proposed approximate complexity measure is peaked close to the critical temperature. For the approximate estimators, a finite-size scaling analysis reveals that the peak approaches the critical temperature as the sequence length increases. Results obtained using different spin-flip dynamics are briefly discussed. The suggested complexity measure can be extended to non-binary sequences in a straightforward manner.