Statistical Measures of Complexity: Why?

Abstract

We review several statistical complexity measures proposed over the last decade and a half as general indicators of structure or correlation. Recently, Lopez-Ruiz, Mancini, and Calbet [Phys. Lett. A 209 (1995) 321] introduced another measure of statistical complexity C_{LMC} that, like others, satisfies the ``boundary conditions'' of vanishing in the extreme ordered and disordered limits. We examine some properties of C_{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_{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.