AI summaryⓘ
The authors study how to find probability distributions that balance order and randomness by maximizing a measure called statistical complexity. They create a unified method that works for both discrete cases (like dice rolls) and continuous cases (like measuring temperatures), based on combinations of Shannon and Renyi entropies. Their main finding is that the best complexity distributions always have exactly two levels of probability, forming a "core-halo" pattern. They also show that the most complex distribution has a very small core with one dominant state in discrete systems or a tiny core in continuous ones. This work provides a clear and general mathematical description of these complexity-maximizing distributions.
Statistical ComplexityShannon EntropyRenyi EntropyProbability DistributionsDiscrete ProbabilityContinuous ProbabilityVariational FrameworkCore-Halo StructureOptimizationStationary Solutions
Abstract
The maximization of statistical complexity has long been associated with the emergence of probability distributions lying between perfect order and complete disorder. While previous studies have shown that complexity-maximizing distributions exhibit a two-level structure in finite discrete systems, an analogous unified treatment for both discrete and continuous probability spaces has remained unavailable. In this work, we develop a general variational framework for a generalized statistical complexity constructed from Shannon and Renyi entropies. We derive a common stationary equation governing both discrete probability masses and continuous probability densities and prove that every stationary solution necessarily possesses exactly two probability levels, establishing a universal core-halo structure. We further demonstrate that the optimization problem reduces to a single multiplicity parameter and prove that the global complexity maximum is attained by the smallest admissible core, corresponding to a single dominant state in the discrete case and an infinitesimal core in the continuous limit. These results provide a complete analytical characterization of the complexity-maximizing distributions and reveal a common mathematical structure underlying complexity optimization in both discrete and continuous settings. The framework establishes a unified foundation for generalized statistical complexity with potential applications in statistical mechanics, information theory, and the analysis of complex systems.