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Measures of statistical complexity: why? (English) Zbl 1026.82505
Summary: We review several statistical complexity measures proposed over the last decade and a half as general indicators of structure or correlation. Recently, López-Ruiz, Mancini, and Calbet (1995) introduced another measure of statistical complexity $$C_{\text{LMC}}$$ that, like others, satisfies the “boundary conditions” of vanishing in the extreme ordered and disordered limits. We examine some properties of $$C_{\text{LMC}}$$ and find that it is neither an intensive nor an extensive thermodynamic variable and that it vanishes exponentially in the thermodynamic limit for all one-dimensional finite-range spin systems. We propose a simple alteration of $$C_{\text{LMC}}$$ that renders it extensive. However, this remedy results in a quantity that is a trivial function of the entropy density and hence of no use as a measure of structure or memory. We conclude by suggesting that a useful “statistical complexity” must not only obey the ordered-random boundary conditions of vanishing, it must also be defined in a setting that gives a clear interpretation to what structures are quantified.

##### MSC:
 82B03 Foundations of equilibrium statistical mechanics
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