Non-Normal Phase Transitions: A New Universality in Complex Systems

Non-Normal Phase Transitions: A New Universality in Complex Systems | 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 Article Non-Normal Phase Transitions: A New Universality in Complex Systems Virgile Troude, Didier Sornette This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7610350/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted You are reading this latest preprint version Abstract We identify a new universality class of phase transitions that arises in non-normal systems, expanding the classical framework beyond eigenvalue instabilities. In contrast to traditional critical phenomena, where transitions occur when eigenvalues cross zero, we show that the geometry of eigenvectors alone can trigger qualitative changes in dynamics. Within a large-deviation framework, transient amplification intrinsic to non-normal operators renormalizes the effective noise amplitude, which can be interpreted as an emergent temperature. Once the non-normality index $\kappa$ exceeds a critical threshold $\kappa_c$ representing the critical balance between restoring potential curvature and non-normal shear, stable equilibria lose practical relevance: fluctuations are amplified strongly enough to induce escapes, even though spectral stability is preserved. This mechanism represents a fundamentally new route to criticality---pseudo-criticality---that generalizes Kramers' escape beyond potential barriers, and can dominate noise-driven transitions in natural and engineered systems. The implications are broad. In biology, we demonstrate that DNA methylation, a cornerstone of epigenetic regulation, naturally operates in this regime: by extending a bistable model of CpG dyads to include non-normality, we reconcile long-term epigenetic memory with rapid stochastic switching observed on minute timescales. More generally, the same mechanism applies to abrupt tipping points in climate dynamics, ecological collapses, financial crises, and failures of engineered networks. By establishing that phase transitions need not be spectrally induced, but can emerge from non-normal amplification, our work introduces a predictive and compact analytical framework for quantifying sudden transitions across disciplines. Non-normality thus defines a new universality class of phase transitions in out-of-equilibrium complex systems, reshaping our understanding of how noise and structure interact to produce abrupt change. Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Phase transitions and critical phenomena Physical sciences/Mathematics and computing/Applied mathematics Full Text Additional Declarations There is NO Competing Interest. Supplementary Files sm.pdf Supplementary Material Cite Share Download PDF Status: Under Review Version 1 posted 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-7610350","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":515788741,"identity":"eaa43c38-8e90-45c9-9594-4ee6516ac840","order_by":0,"name":"Virgile Troude","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8klEQVRIiWNgGAWjYDACZiBOKICwJRgqQCLMDURoMQCqBms5AxJhJKAFDGBaGNtAFAEtBsd5H794YGBTZ97eY3jj47zaaP52oJYfFdtwaznMbmaRYJAmIXPmjLHlzG3Hc2ccZmxg7DlzG6cWyWY2NoMEg8MSEhI5ZtK8247lNgC1MDO2EdTyH6Ll75xjufMJaeFnZmN+kGBwAKKFsaEmdwMRWtiAgZwsOYPnWLFlz7EDuRuBWg7i8wsb/zHmjz8q7Pgl2Js33vhRU5c77/zhgw9+VODWAtIlgcQ5DCYP4FMPBMwfkDh1BBSPglEwCkbBSAQA+URS7BEB37MAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0009-5134-9687","institution":"Southern University of Science and technology","correspondingAuthor":true,"prefix":"","firstName":"Virgile","middleName":"","lastName":"Troude","suffix":""},{"id":515788742,"identity":"16c56d2c-9476-4425-aece-60fd5afa3a65","order_by":1,"name":"Didier Sornette","email":"","orcid":"","institution":"SUSTech","correspondingAuthor":false,"prefix":"","firstName":"Didier","middleName":"","lastName":"Sornette","suffix":""}],"badges":[],"createdAt":"2025-09-14 04:20:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7610350/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7610350/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":91818747,"identity":"b2325779-7a80-4abd-8d98-e6ebfb210873","added_by":"auto","created_at":"2025-09-22 07:05:11","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":342525,"visible":true,"origin":"","legend":"","description":"","filename":"papernc.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7610350/v1/58aaa9b297ed590ca3caa1cf.pdf"},{"id":91819008,"identity":"60af1da3-d609-4d44-b236-4c6fb08e8a0d","added_by":"auto","created_at":"2025-09-22 07:05:49","extension":"json","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":4833,"visible":true,"origin":"","legend":"","description":"","filename":"NCOMMS2573163.json","url":"https://assets-eu.researchsquare.com/files/rs-7610350/v1/0d515bffc8ce5dd580208cf0.json"},{"id":91818675,"identity":"c002a046-f0fd-45f5-868f-20030aa49eef","added_by":"auto","created_at":"2025-09-22 07:05:04","extension":"pdf","order_by":2,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":1005592,"visible":true,"origin":"","legend":"","description":"","filename":"sm.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7610350/v1/2f40c3266ed6e2e772bd8c8f.pdf"},{"id":92178611,"identity":"e5f48e88-3197-42d9-a18a-0844051c0621","added_by":"auto","created_at":"2025-09-25 13:14:42","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":358481,"visible":true,"origin":"","legend":"Article File","description":"","filename":"papernc.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7610350/v1_covered_628f32e8-776a-4c4a-9100-1e07eee9c365.pdf"},{"id":91818731,"identity":"ef47e99f-9769-41a4-9c93-bf5c4d8ca1c4","added_by":"auto","created_at":"2025-09-22 07:05:07","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":1005592,"visible":true,"origin":"","legend":"Supplementary Material","description":"","filename":"sm.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7610350/v1/8d74ccf7bf4734eb8ed08b7c.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Non-Normal Phase Transitions: A New Universality in Complex Systems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-7610350/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7610350/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We identify a new universality class of phase transitions that arises in non-normal systems, expanding the classical framework beyond eigenvalue instabilities. In contrast to traditional critical phenomena, where transitions occur when eigenvalues cross zero, we show that the geometry of eigenvectors alone can trigger qualitative changes in dynamics. Within a large-deviation framework, transient amplification intrinsic to non-normal operators renormalizes the effective noise amplitude, which can be interpreted as an emergent temperature. Once the non-normality index $\\kappa$\r\n exceeds a critical threshold $\\kappa_c$ representing the critical balance between restoring potential curvature and non-normal shear, \r\n stable equilibria lose practical relevance: fluctuations are amplified strongly enough to induce escapes, even though spectral stability is preserved. This mechanism represents a fundamentally new route to criticality---pseudo-criticality---that generalizes Kramers' escape beyond potential barriers, and can dominate noise-driven transitions in natural and engineered systems.\r\n The implications are broad. In biology, we demonstrate that DNA methylation, a cornerstone of epigenetic regulation, naturally operates in this regime: by extending a bistable model of CpG dyads to include non-normality, we reconcile long-term epigenetic memory with rapid stochastic switching observed on minute timescales. More generally, the same mechanism applies to abrupt tipping points in climate dynamics, ecological collapses, financial crises, and failures of engineered networks. By establishing that phase transitions need not be spectrally induced, but can emerge from non-normal amplification, our work introduces a predictive and compact analytical framework for quantifying sudden transitions across disciplines. Non-normality thus defines a new universality class of phase transitions in out-of-equilibrium complex systems, reshaping our understanding of how noise and structure interact to produce abrupt change.","manuscriptTitle":"Non-Normal Phase Transitions: A New Universality in Complex Systems","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-09-22 06:55:12","doi":"10.21203/rs.3.rs-7610350/v1","editorialEvents":[],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"communications-physics","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"commsphys","sideBox":"Learn more about [Communications Physics](http://www.nature.com/commsphys/)","snPcode":"","submissionUrl":"","title":"Communications Physics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"Communications Series","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"6b5e6e3d-5b0c-4246-bfbf-0f4a8e988235","owner":[],"postedDate":"September 22nd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":54792245,"name":"Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Phase transitions and critical phenomena"},{"id":54792246,"name":"Physical sciences/Mathematics and computing/Applied mathematics"}],"tags":[],"updatedAt":"2026-04-03T03:20:59+00:00","versionOfRecord":[],"versionCreatedAt":"2025-09-22 06:55:12","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7610350","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7610350","identity":"rs-7610350","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}