{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:46:03Z","timestamp":1760147163245,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T00:00:00Z","timestamp":1673568000000},"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>The Theory of the Adjacent Possible (TAP) equation has been proposed as an appropriate description of super-exponential growth phenomena, where a phase of slow growth is followed by a rapid increase, leading to a \u201chockey stick\u201d curve. This equation, initially conceived to describe the growth in time of the number of new types of artifacts, has also been applied to several natural phenomena. A possible drawback is that it may overestimate the number of new artifact types, since it does not take into account the fact that interactions, among existing types, may produce types which have already been previously discovered. We introduce here a Binary String World (BSW) where new string types can be generated by interactions among (at most two) already existing types. We introduce a continuous limit of the TAP equation for the BSW; we solve it analytically and show that it leads to divergence in finite time. We also introduce a criterion to distinguish this type of behavior from the familiar exponential growth, which diverges only as t \u2192 \u221d. In the BSW, it is possible to directly model the generation of new types, and to check whether the newborns are actually novel types, thus discarding the rediscoveries of already existing types. We show that the type of growth is still TAP-like, rather than exponential, although of course in simulations one never can observes true divergence. We also show that this property is robust with respect to some changes in the model, as long as it deals with types (and not with individuals).<\/jats:p>","DOI":"10.3390\/e25010168","type":"journal-article","created":{"date-parts":[[2023,1,16]],"date-time":"2023-01-16T02:29:55Z","timestamp":1673836195000},"page":"168","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Super-Exponential Growth in Models of a Binary String World"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5991-5470","authenticated-orcid":false,"given":"Marco","family":"Villani","sequence":"first","affiliation":[{"name":"Department of Physics, Informatics and Mathematics, Modena and Reggio Emilia University, 41121 Modena, Italy"},{"name":"European Centre for Living Technology, 30123 Venice, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1417-5106","authenticated-orcid":false,"given":"Roberto","family":"Serra","sequence":"additional","affiliation":[{"name":"Department of Physics, Informatics and Mathematics, Modena and Reggio Emilia University, 41121 Modena, Italy"},{"name":"European Centre for Living Technology, 30123 Venice, Italy"},{"name":"Institute of Advanced Studies, University of Amsterdam, 1012 WX Amsterdam, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1126\/science.7618100","article-title":"Population growth and earth\u2019s human carrying capacity","volume":"269","author":"Cohen","year":"1995","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1016\/S0378-4371(01)00105-4","article-title":"Finite-time singularity in the dynamics of the world population, economic and financial indices","volume":"294","author":"Johansen","year":"2001","journal-title":"Phys. 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