Way More Than the Sum of Their Parts: From Statistical to Structural Mixtures

2025/07/10 by Crutchfield, James P.
#Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Machine Learning (cs.LG) #Statistical Mechanics (cond-mat.stat-mech) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2507.07343

Abstract

We show that mixtures comprised of multicomponent systems typically are much more structurally complex than the sum of their parts; sometimes, infinitely more complex. We contrast this with the more familiar notion of statistical mixtures, demonstrating how statistical mixtures miss key aspects of emergent hierarchical organization. This leads us to identify a new kind of structural complexity inherent in multicomponent systems and to draw out broad consequences for system ergodicity.

Citations

• Topology, Convergence, and Reconstruction of Predictive States
• Signatures of Infinity: Nonergodicity and Resource Scaling in Prediction, Complexity, and Learning
• Exact Complexity: The Spectral Decomposition of Intrinsic Computation
• Infinite Excess Entropy Processes with Countable-State Generators
• Enumerating Finitary Processes
• Automated Pattern Detection--An Algorithm for Constructing Optimally\n Synchronizing Multi-Regular Language Filters
• Regularities Unseen, Randomness Observed: Levels of Entropy Convergence
• Computational Mechanics: Pattern and Prediction, Structure and Simplicity
• The Evolutionary Unfolding of Complexity
• On the equilibrium of heterogeneous substances

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