{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T14:39:48Z","timestamp":1764859188599,"version":"build-2065373602"},"reference-count":78,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,5]],"date-time":"2020-02-05T00:00:00Z","timestamp":1580860800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100010198","name":"Ministerio de Econom\u00eda, Industria y Competitividad, Gobierno de Espa\u00f1a","doi-asserted-by":"publisher","award":["FIS2015-71851-P","Mar\\'{\\i}a de Maeztu Program for Units of Excellence in R\\&D, MDM-2014-0445"],"award-info":[{"award-number":["FIS2015-71851-P","Mar\\'{\\i}a de Maeztu Program for Units of Excellence in R\\&D, MDM-2014-0445"]}],"id":[{"id":"10.13039\/501100010198","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PGC-FIS2018-099629-B-I00"],"award-info":[{"award-number":["PGC-FIS2018-099629-B-I00"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The word-frequency distribution provides the fundamental building blocks that generate discourse in natural language. It is well known, from empirical evidence, that the word-frequency distribution of almost any text is described by Zipf\u2019s law, at least approximately. Following Stephens and Bialek (2010), we interpret the frequency of any word as arising from the interaction potentials between its constituent letters. Indeed, Jaynes\u2019 maximum-entropy principle, with the constrains given by every empirical two-letter marginal distribution, leads to a Boltzmann distribution for word probabilities, with an energy-like function given by the sum of the all-to-all pairwise (two-letter) potentials. The so-called improved iterative-scaling algorithm allows us finding the potentials from the empirical two-letter marginals. We considerably extend Stephens and Bialek\u2019s results, applying this formalism to words with length of up to six letters from the English subset of the recently created Standardized Project Gutenberg Corpus. We find that the model is able to reproduce Zipf\u2019s law, but with some limitations: the general Zipf\u2019s power-law regime is obtained, but the probability of individual words shows considerable scattering. In this way, a pure statistical-physics framework is used to describe the probabilities of words. As a by-product, we find that both the empirical two-letter marginal distributions and the interaction-potential distributions follow well-defined statistical laws.<\/jats:p>","DOI":"10.3390\/e22020179","type":"journal-article","created":{"date-parts":[[2020,2,6]],"date-time":"2020-02-06T02:59:18Z","timestamp":1580957958000},"page":"179","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["From Boltzmann to Zipf through Shannon and Jaynes"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5280-2692","authenticated-orcid":false,"given":"\u00c1lvaro","family":"Corral","sequence":"first","affiliation":[{"name":"Centre de Recerca Matem\u00e0tica, Edifici C, Campus Bellaterra, E-08193 Barcelona, Spain"},{"name":"Departament de Matem\u00e0tiques, Facultat de Ci\u00e8ncies, Universitat Aut\u00f2noma de Barcelona, E-08193 Barcelona, Spain"},{"name":"Barcelona Graduate School of Mathematics, Edifici C, Campus Bellaterra, E-08193 Barcelona, Spain"},{"name":"Complexity Science Hub Vienna, Josefst\u00e4dter Stra<i>\u03b2<\/i>e 39, 1080 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Montserrat","family":"Garc\u00eda del Muro","sequence":"additional","affiliation":[{"name":"Departament de F\u00edsica de la Mat\u00e8ria Condensada, Universitat de Barcelona, Mart\u00ed i Franqu\u00e8s 1, E-08028 Barcelona, Spain"},{"name":"IN2UB, Universitat de Barcelona, Mart\u00ed i Franqu\u00e8s 1, E-08028 Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,5]]},"reference":[{"key":"ref_1","first-page":"14","article-title":"Zipf\u2019s law everywhere","volume":"5","author":"Li","year":"2002","journal-title":"Glottometrics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"036111","DOI":"10.1103\/PhysRevE.83.036111","article-title":"Testing the Pareto against the lognormal distributions with the uniformly most powerful unbiased test applied to the distribution of cities","volume":"83","author":"Malevergne","year":"2011","journal-title":"Phys. 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