{"paper_id":"415f9ebc-78c8-4a97-b9e6-8b89e677b596","body_text":"Discrete System Phase Space Modification by Vanishing Noise | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Discrete System Phase Space Modification by Vanishing Noise Franco Bagnoli, Michele Baia, Tommaso Matteuzzi This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6245229/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract We investigate two classes of discrete systems: the Hopfield model with deterministic update dynamics and elementary cellular automata. We obtain a sampling of the attractor space of the Hopfield model and the complete phase space of all minimal elementary cellular automata for small lattice sizes. Starting from the maximal entropy distribution (all configurations equiprobable), we show how the dynamics affects this distribution. We then investigate how a vanishing noise alters this phase space, connecting attractors and modifying the asymptotic probability distribution. For both the Hopfield model and Cellular Automata, this modification does not always decrease the entropy. Discrete dynamical systems Elementary cellular automata Attractors Noise-induced transitions Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 09 Jun, 2025 Reviews received at journal 19 May, 2025 Reviews received at journal 04 May, 2025 Reviewers agreed at journal 01 Apr, 2025 Reviewers agreed at journal 01 Apr, 2025 Reviewers invited by journal 26 Mar, 2025 Editor assigned by journal 20 Mar, 2025 Submission checks completed at journal 18 Mar, 2025 First submitted to journal 17 Mar, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {\"props\":{\"pageProps\":{\"initialData\":{\"identity\":\"rs-6245229\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":false,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":437069560,\"identity\":\"5f45e385-94dc-4aa9-8726-1d5add49ceae\",\"order_by\":0,\"name\":\"Franco Bagnoli\",\"email\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA70lEQVRIiWNgGAWjYBAC9gYInQAmHxjYMDAwMz44wGCAWwvPAWQtCQZpQC3MBkAtuPWgaWE4DCSZQcrxaJE+/OzBjwqGPP7+tQcfJBScT9zOzsx4gKHgD24tfGnmhj1nGIolbrxLNkgwuJ24s5mZAa/D7HkYzKQZ2xgSG26cMZMAadlwmP8Afr/wsH8Da5kP0XIOqIWALTw8PBBbNpzvAWk5QJSWMsmeMxLFhjd4jIF+STYGa0kwMMbnsG0SPyps8uTOnzF88OGPneyG84eZP3z4I4dTCxRIAFECEj8BhzpUwH+AKGWjYBSMglEwAgEAR61RS1pA8gsAAAAASUVORK5CYII=\",\"orcid\":\"\",\"institution\":\"University of Florence\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"Franco\",\"middleName\":\"\",\"lastName\":\"Bagnoli\",\"suffix\":\"\"},{\"id\":437069562,\"identity\":\"904be418-9508-43eb-9f8c-dc083e68a068\",\"order_by\":1,\"name\":\"Michele Baia\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"University of Florence\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Michele\",\"middleName\":\"\",\"lastName\":\"Baia\",\"suffix\":\"\"},{\"id\":437069564,\"identity\":\"6221edfb-54a4-49f6-8382-ce0ea837a7f5\",\"order_by\":2,\"name\":\"Tommaso Matteuzzi\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"University of Florence\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Tommaso\",\"middleName\":\"\",\"lastName\":\"Matteuzzi\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2025-03-17 13:53:23\",\"currentVersionCode\":1,\"declarations\":\"\",\"doi\":\"10.21203/rs.3.rs-6245229/v1\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-6245229/v1\",\"draftVersion\":[],\"editorialEvents\":[],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":80341436,\"identity\":\"5aab7883-56d7-47ba-b833-187ca6013e94\",\"added_by\":\"auto\",\"created_at\":\"2025-04-10 17:56:42\",\"extension\":\"pdf\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":1151760,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"Vanishingnoise.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-6245229/v1_covered_e9e256ba-b08a-467a-b705-7a9e8fabf376.pdf\"}],\"financialInterests\":\"No competing interests reported.\",\"formattedTitle\":\"Discrete System Phase Space Modification by Vanishing Noise\",\"fulltext\":[],\"fulltextSource\":\"\",\"fullText\":\"\",\"funders\":[],\"hasAdminPriorityOnWorkflow\":false,\"hasManuscriptDocX\":false,\"hasOptedInToPreprint\":true,\"hasPassedJournalQc\":\"\",\"hasAnyPriority\":false,\"hideJournal\":false,\"highlight\":\"\",\"institution\":\"\",\"isAcceptedByJournal\":true,\"isAuthorSuppliedPdf\":true,\"isDeskRejected\":\"\",\"isHiddenFromSearch\":false,\"isInQc\":false,\"isInWorkflow\":false,\"isPdf\":true,\"isPdfUpToDate\":true,\"isWithdrawnOrRetracted\":false,\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"natural-computing\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":false,\"externalIdentity\":\"naco\",\"sideBox\":\"Learn more about [Natural Computing](http://link.springer.com/journal/11047)\",\"snPcode\":\"11047\",\"submissionUrl\":\"https://submission.nature.com/new-submission/11047/3\",\"title\":\"Natural Computing\",\"twitterHandle\":\"\",\"acdcEnabled\":true,\"dfaEnabled\":true,\"editorialSystem\":\"em\",\"reportingPortfolio\":\"Springer Hybrid\",\"inReviewEnabled\":true,\"inReviewRevisionsEnabled\":false},\"keywords\":\"Discrete dynamical systems, Elementary cellular automata, Attractors, Noise-induced transitions\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-6245229/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-6245229/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"We investigate two classes of discrete systems: the Hopfield model with deterministic update dynamics and elementary cellular automata. 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