Statistical complexity from fluctuations in the information content

Abstract

We argue that the variance of the information content ($C$), an information-theoretic quantity, can be naturally interpreted as a measure of statistical complexity. We show that $C$ satisfies widely accepted criteria for statistical complexity measures: it vanishes for both ordered and equiprobable states, while attaining maxima in intermediate regimes, typically shifted toward order. This interpretation establishes direct connections with thermodynamics and phase transitions: for systems obeying Boltzmann--Gibbs statistics, $C$ is extensive and directly proportional to energy fluctuations and heat capacity. Moreover, unlike other statistical complexity measures, it attains a maximum at continuous phase transitions, as illustrated for the two-dimensional Ising model. Applications to chaotic maps and fractional Gaussian noise further indicate that $C$ captures nontrivial dynamical structure in different classes of correlated systems.