AI 中文总结
研究离散和连续概率空间中统计复杂度最大化问题，通过构建变分框架推导共同平稳方程，证明解有两级结构，表明优化归结为单个参数，全局最大值由最小核心达到，揭示共同数学结构，为广义统计复杂度建立统一基础。
AI 中文摘要
统计复杂度的最大化长期以来与介于完美有序和完全无序之间的概率分布的出现相关。此前研究表明有限离散系统中复杂度最大化分布呈现两级结构，但离散和连续概率空间的类似统一处理尚不存在。本文从香农熵和雷尼熵构建广义统计复杂度的一般变分框架，推导离散概率质量和连续概率密度的共同平稳方程，证明平稳解有两个概率水平，建立通用核心-晕圈结构。还表明优化问题归结为单个多重性参数，全局复杂度最大值由最小允许核心达到。这些结果为复杂度最大化分布提供完整分析表征，揭示离散和连续设置中复杂度优化的共同数学结构，为广义统计复杂度建立统一基础，在统计力学、信息论和复杂系统分析中有潜在应用。
英文摘要
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.
Comments67 pages, 2 figures
