{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T21:10:41Z","timestamp":1787951441667,"version":"build-2784847793"},"reference-count":53,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2017,5,8]],"date-time":"2017-05-08T00:00:00Z","timestamp":1494201600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Collective organization in matter plays a significant role in its expressed physical properties. Typically, it is detected via an order parameter, appropriately defined for each given system\u2019s observed emergent patterns. Recent developments in information theory, however, suggest quantifying collective organization in a system- and phenomenon-agnostic way: decomposing the system\u2019s thermodynamic entropy density into a localized entropy, that is solely contained in the dynamics at a single location, and a bound entropy, that is stored in space as domains, clusters, excitations, or other emergent structures. As a concrete demonstration, we compute this decomposition and related quantities explicitly for the nearest-neighbor Ising model on the 1D chain, on the Bethe lattice with coordination number     k = 3    , and on the 2D square lattice, illustrating its generality and the functional insights it gives near and away from phase transitions. In particular, we consider the roles that different spin motifs play (in cluster bulk, cluster edges, and the like) and how these affect the dependencies between spins.<\/jats:p>","DOI":"10.3390\/e19050214","type":"journal-article","created":{"date-parts":[[2017,5,8]],"date-time":"2017-05-08T11:45:16Z","timestamp":1494243916000},"page":"214","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Anatomy of a Spin: The Information-Theoretic Structure of Classical Spin Systems"],"prefix":"10.3390","volume":"19","author":[{"given":"Vikram","family":"Vijayaraghavan","sequence":"first","affiliation":[{"name":"Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ryan","family":"James","sequence":"additional","affiliation":[{"name":"Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4466-5410","authenticated-orcid":false,"given":"James","family":"Crutchfield","sequence":"additional","affiliation":[{"name":"Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,5,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"298928","DOI":"10.1155\/2011\/298928","article-title":"Electronic structure of strongly correlated systems","volume":"2011","author":"Antonov","year":"2011","journal-title":"Adv. Condens. Matter Phys."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Mantegna, R.N., and Stanley, H.E. (1999). Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press.","DOI":"10.1017\/CBO9780511755767"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Raiesdana, S., Hashemi Golpayegani, M.R., and Nasrabadi, A.M. (2008, January 20\u201325). Complexity evolution in epileptic seizure. Proceedings of the IEEE 30th Annual International Conference of the Engineering in Medicine and Biology Society, Vancouver, BC, Canada.","DOI":"10.1109\/IEMBS.2008.4650113"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"M\u00e4ki-Marttunen, V., Cortes, J.M., Villarreal, M.F., and Chialvo, D.R. (2013). Disruption of transfer entropy and inter-hemispheric brain functional connectivity in patients with disorder of consciousness. BMC Neurosci., 14.","DOI":"10.1186\/1471-2202-14-S1-P83"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1038\/445715a","article-title":"Collective minds","volume":"445","author":"Couzin","year":"2007","journal-title":"Nature"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"177203","DOI":"10.1103\/PhysRevLett.111.177203","article-title":"Information flow in a kinetic Ising model peaks in the disordered phase","volume":"111","author":"Barnett","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, Wiley-Interscience. [2nd ed.]."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1038\/nphys2190","article-title":"Between order and chaos","volume":"8","author":"Crutchfield","year":"2012","journal-title":"Nat. Phys."},{"key":"ref_9","unstructured":"Shaw, R. (1984). The Dripping Faucet as a Model Chaotic System, Aerial Press."},{"key":"ref_10","first-page":"143","article-title":"Information-theoretic analysis of phase transitions","volume":"10","author":"Arnold","year":"1996","journal-title":"Complex Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"R1239","DOI":"10.1103\/PhysRevE.55.R1239","article-title":"Statistical complexity of simple one-dimensional spin systems","volume":"55","author":"Crutchfield","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"051104","DOI":"10.1103\/PhysRevE.67.051104","article-title":"Structural information in two-dimensional patterns: Entropy convergence and excess entropy","volume":"67","author":"Feldman","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"043106","DOI":"10.1063\/1.2991106","article-title":"The organization of intrinsic computation: Complexity-entropy diagrams and the diversity of natural information processing","volume":"18","author":"Feldman","year":"2008","journal-title":"Chaos"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"022128","DOI":"10.1103\/PhysRevE.87.022128","article-title":"Information theoretic aspects of the two-dimensional Ising model","volume":"87","author":"Lau","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1147\/rd.41.0066","article-title":"Information theoretical analysis of multivariate correlation","volume":"4","author":"Watanabe","year":"1960","journal-title":"IBM J. Res. Dev."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1103\/PhysRevLett.85.461","article-title":"Measuring information transfer","volume":"85","author":"Schreiber","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.physleta.2011.10.066","article-title":"A measure of statistical complexity based on predictive information with application to finite spin systems","volume":"376","author":"Abdallah","year":"2012","journal-title":"Phys. Lett. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/S0019-9958(75)80004-0","article-title":"Linear dependence structure of the entropy space","volume":"29","author":"Han","year":"1975","journal-title":"Inf. Control"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/S0019-9958(78)90275-9","article-title":"Nonnegative entropy measures of multivariate symmetric correlations","volume":"36","author":"Han","year":"1978","journal-title":"Inf. Control"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Verd\u00fa, S., and Weissman, T. (2006, January 9\u201314). Erasure entropy. Proceedings of the 2006 IEEE International Symposium on Information Theory, Seattle, WA, USA.","DOI":"10.1109\/ISIT.2006.261682"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2124","DOI":"10.1016\/j.physleta.2014.05.014","article-title":"Chaos forgets and remembers: Measuring information creation, destruction, and storage","volume":"378","author":"James","year":"2014","journal-title":"Phys. Lett. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"037109","DOI":"10.1063\/1.3637494","article-title":"Anatomy of a bit: Information in a time series observation","volume":"21","author":"James","year":"2011","journal-title":"Chaos"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4713","DOI":"10.3390\/e16094713","article-title":"Information anatomy of stochastic equilibria","volume":"16","author":"Marzen","year":"2014","journal-title":"Entropy"},{"key":"ref_24","unstructured":"Pikovsky, A., Rosenblum, M., and Kurths, J. (2003). Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1119\/1.1971557","article-title":"Gibbs versus Boltzmann entropies","volume":"33","author":"Jaynes","year":"1965","journal-title":"Am. J. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Grandy, W.T. (2008). Entropy and the Time Evolution of Macroscopic Systems, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199546176.001.0001"},{"key":"ref_28","unstructured":"Ruelle, D. (1969). Statistical Mechanics: Rigorous Results, World Scientific."},{"key":"ref_29","unstructured":"Grimett, G.R. (1999). Percolation, Springer. [2nd ed.]."},{"key":"ref_30","unstructured":"Reza, F.M. (1961). An Introduction to Information Theory, Courier Corporation."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1007\/BF02125696","article-title":"Entropy and global Markov properties","volume":"126","author":"Goldstein","year":"1990","journal-title":"Commun. Math. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/BF01044436","article-title":"Two-sided bounds on the free energy from local states in Monte Carlo simulations","volume":"56","author":"Schlijper","year":"1989","journal-title":"J. Stat. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Sch\u00fcrmann, T. (arXiv, 2015). A Note on Entropy Estimation, arXiv.","DOI":"10.1162\/NECO_a_00775"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1103\/PhysRevLett.62.361","article-title":"Collective Monte Carlo updating for spin systems","volume":"62","author":"Wolff","year":"1989","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1088\/1367-2630\/5\/1\/127","article-title":"A one-dimensional Ising model for C 70 molecular ordering in C 70-peapods","volume":"5","author":"Yutaka","year":"2003","journal-title":"New J. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"618","DOI":"10.1103\/PhysRevLett.85.618","article-title":"Interaction of H2 with Si(001) \u2212 (2\u00d71): Solution of the barrier puzzle","volume":"85","author":"Zimmermann","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1349","DOI":"10.1063\/1.1731411","article-title":"Theory of \u201cmelting\u201d of the helical form in double chains of the DNA type","volume":"33","author":"Zimm","year":"1960","journal-title":"J. Chem. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/0370-1573(85)90060-2","article-title":"Thermal denaturation of DNA molecules: A comparison of theory with experiment","volume":"126","author":"Wartell","year":"1985","journal-title":"Phys. Rep."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"10582","DOI":"10.1073\/pnas.96.19.10582","article-title":"How can statistical mechanics contribute to social science?","volume":"96","author":"Durlauf","year":"1999","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_40","unstructured":"Pathria, R.K., and Beale, P.D. (1996). Statistical Mechanics, Elsevier Science."},{"key":"ref_41","unstructured":"Baxter, R.J. (1982). Exactly Solved Models in Statistical Mechanics, Academic Press."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/0375-9601(79)90017-3","article-title":"An exact result for the 1D random Ising model in a transverse field","volume":"72","author":"Pfeuty","year":"1979","journal-title":"Phys. Lett. A"},{"key":"ref_43","unstructured":"Feldman, D.P. (1998). Computational Mechanics of Classical Spin Systems. [Ph.D. Thesis, University of California]."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"026127","DOI":"10.1103\/PhysRevE.71.026127","article-title":"Exact cluster size distribution in the one-dimensional Ising model","volume":"71","author":"Yilmaz","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Bollob\u00e1s, B. (2001). Random Graphs Vol. 73, Cambridge Studies in Advanced Mathematics, Cambridge University Press.","DOI":"10.1017\/CBO9780511814068"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1544","DOI":"10.1137\/120887382","article-title":"Computing bounds for entropy of stationary Zd Markov random fields","volume":"27","author":"Marcus","year":"2013","journal-title":"SIAM J. Discret. Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"037109","DOI":"10.1063\/1.3486801","article-title":"Information modification and particle collisions in distributed computation","volume":"20","author":"Lizier","year":"2010","journal-title":"Chaos"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1137\/0222066","article-title":"Polynomial-time approximation algorithms for the Ising model","volume":"22","author":"Jerrum","year":"1993","journal-title":"SIAM J. Comput."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"238701","DOI":"10.1103\/PhysRevLett.116.238701","article-title":"Information flows? A critique of transfer entropies","volume":"116","author":"James","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"022143","DOI":"10.1103\/PhysRevE.93.022143","article-title":"The elusive present: Hidden past and future dependence and why we build models","volume":"93","author":"Ara","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1142\/S021952591000258X","article-title":"Quantifying emergence in terms of persistent mutual information","volume":"13","author":"Ball","year":"2010","journal-title":"Adv. Complex Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1017\/S0305004100027419","article-title":"Some generalized order-disorder transformations","volume":"Volume 48","author":"Potts","year":"1952","journal-title":"Mathematical Proceedings of the Cambridge Philosophical Society"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"037114","DOI":"10.1063\/1.3608120","article-title":"Local entropy and structure in a two-dimensional frustrated system","volume":"21","author":"Robinson","year":"2011","journal-title":"Chaos"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/5\/214\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:35:05Z","timestamp":1760207705000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/5\/214"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,5,8]]},"references-count":53,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2017,5]]}},"alternative-id":["e19050214"],"URL":"https:\/\/doi.org\/10.3390\/e19050214","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,5,8]]}}}