{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T22:22:14Z","timestamp":1779315734403,"version":"3.51.4"},"reference-count":64,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,3,7]],"date-time":"2017-03-07T00:00:00Z","timestamp":1488844800000},"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>We propose a Cellular Automata (CA) model in which three ubiquitous and relevant processes in nature are present, namely, spatial competition, distinction between dynamically stronger and weaker agents and the existence of an inner resistance to changes in the actual state     S n     (=\u22121,0,+1) of each CA lattice cell n (which we call inertia). Considering ensembles of initial lattices, we study the average properties of the CA final stationary configuration structures resulting from the system time evolution. Assuming the inertia a (proper) control parameter, we identify qualitative changes in the CA spatial patterns resembling usual phase transitions. Interestingly, some of the observed features may be associated with continuous transitions (critical phenomena). However, certain quantities seem to present jumps, typical of discontinuous transitions. We argue that these apparent contradictory findings can be attributed to the inertia parameter\u2019s discrete character. Along the work, we also briefly discuss a few potential applications for the present CA formulation.<\/jats:p>","DOI":"10.3390\/e19030102","type":"journal-article","created":{"date-parts":[[2017,3,7]],"date-time":"2017-03-07T11:12:32Z","timestamp":1488885152000},"page":"102","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Emergence of Distinct Spatial Patterns in Cellular Automata with Inertia: A Phase Transition-Like Behavior"],"prefix":"10.3390","volume":"19","author":[{"given":"Klaus","family":"Kramer","sequence":"first","affiliation":[{"name":"CEPLAN, Universidade do Estado de Santa Catarina, 89283-081 S\u00e3o Bento do Sul-SC, Brazil"},{"name":"Departamento de F\u00edsica, Universidade Federal do Paran\u00e1, C.P. 19044, 81531-980 Curitiba-PR, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marlus","family":"Koehler","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal do Paran\u00e1, C.P. 19044, 81531-980 Curitiba-PR, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carlos","family":"Fiore","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica, Universidade de S\u00e3o Paulo, C.P. 66318, 05315-970 S\u00e3o Paulo-SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marcos","family":"Da Luz","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal do Paran\u00e1, C.P. 19044, 81531-980 Curitiba-PR, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,3,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mandelbrot, B.B. 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