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Open AccessArticle
Way More Than the Sum of Their Parts: From Statistical to Structural Mixtures
by
James P. Crutchfield
James P. Crutchfield
Complexity Sciences Center and Physics and Astronomy Department, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA
Entropy 2026, 28(1), 111; https://doi.org/10.3390/e28010111 (registering DOI)
Submission received: 26 November 2025
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Revised: 12 January 2026
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Accepted: 13 January 2026
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Published: 16 January 2026
Abstract
We show that mixtures comprising 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.
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MDPI and ACS Style
Crutchfield, J.P.
Way More Than the Sum of Their Parts: From Statistical to Structural Mixtures. Entropy 2026, 28, 111.
https://doi.org/10.3390/e28010111
AMA Style
Crutchfield JP.
Way More Than the Sum of Their Parts: From Statistical to Structural Mixtures. Entropy. 2026; 28(1):111.
https://doi.org/10.3390/e28010111
Chicago/Turabian Style
Crutchfield, James P.
2026. "Way More Than the Sum of Their Parts: From Statistical to Structural Mixtures" Entropy 28, no. 1: 111.
https://doi.org/10.3390/e28010111
APA Style
Crutchfield, J. P.
(2026). Way More Than the Sum of Their Parts: From Statistical to Structural Mixtures. Entropy, 28(1), 111.
https://doi.org/10.3390/e28010111
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