Coherence–Flux Conservation with an Internal Parameter

Coherence–Flux Conservation with an Internal Parameter | 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Article Coherence–Flux Conservation with an Internal Parameter Miguel Jorge Díaz Luna This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8379999/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract I introduce a generalized conservation–balance framework for scalar fields by extending the standard continuity equation with an internal variable T representing a hidden structural degree of freedom. For the Lagrangian density L(phi, d_t phi, grad phi, d_T phi) = 1/2 (d_t phi)^2 − c^2/2 |grad phi|^2 − alpha/2 (d_T phi)^2 − V(phi), the associated variational structure yields an extended continuity equation that reduces to an exact conservation law only in the shift-invariant case V′(phi) = 0. For a general potential V(phi), the symmetry is explicitly broken and the equation becomes a balance law with a source term d_t J0 + div J + d_T Js = − V′(phi), where J0 = d_t phi is the canonical momentum density, J = c^2 grad phi the spatial flux, and Js = alpha d_T phi the structural flux. This formulation unifies apparently non-conservative dynamics in observable spatial domains by interpreting dissipation-like behaviour as flux redistribution along the internal direction T, rather than as a violation of conservation. I establish well-posedness of the resulting wave equation posed on an extended spatial domain (x, T), derive the corresponding extended-domain balance relation, and show how the structural flux accounts for apparent mass or energy leakage, amplitude attenuation, or effective damping without violating the global balance. Numerical experiments based on finite-difference schemes confirm that, in the shift-invariant case, the extended integral invariant is preserved within a relative error below approximately 2.5% under perturbations and mesh refinements. This variational extension provides a compact and physically motivated framework for modeling apparent non-conservation phenomena in systems with hidden internal variables, including wave propagation in complex media, transport in composite or microstructured materials, and effective models in optics or mathematical biology. Conservation laws Internal variables Noether theorem Extended continuity equation Wave equations Structural flux Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted 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-8379999","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":575282245,"identity":"51c129a4-853b-4952-88c2-3bd82ba8ecf9","order_by":0,"name":"Miguel Jorge Díaz Luna","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0klEQVRIiWNgGAWjYFCCA4YHEir+1/OD2AkFxGkxOPDgDHOCZANIiwFx1hgcfNjGnGBwAMwmQr1u4+ENBxLb2PKMz69O/PDAgEGeX+wAfi1mB44VHEg4x1NsduPtZgmgwwxnzk4gpOWMwYGEMgnGbTfObgBpSTC4TZQWNgPGzTPObv5Bgpa2hMQN/L3biLUF5JczB4wlbvBus0gwkCDCLzcOb3z4o+KAHH//2c03f1TYyPNLE9DCIHEAxgCrlCCgHAT4G2CMA3hUjYJRMApGwYgGALOHUyYrLK0zAAAAAElFTkSuQmCC","orcid":"","institution":"Independent Researcher","correspondingAuthor":true,"prefix":"","firstName":"Miguel","middleName":"Jorge Díaz","lastName":"Luna","suffix":""}],"badges":[],"createdAt":"2025-12-16 22:23:18","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8379999/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8379999/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102558349,"identity":"996716ac-0d7b-4982-b959-9b70b7fc7a52","added_by":"auto","created_at":"2026-02-13 03:10:19","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":200685,"visible":true,"origin":"","legend":"","description":"","filename":"CoherenceFluxConservationwithanInternalParameter.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8379999/v1_covered_20c29402-7df5-4be0-b0d3-fa8968525dfc.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Coherence–Flux Conservation with an Internal Parameter","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"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":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Conservation laws, Internal variables, Noether theorem, Extended continuity equation, Wave equations, Structural flux","lastPublishedDoi":"10.21203/rs.3.rs-8379999/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8379999/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"I introduce a generalized conservation–balance framework for scalar fields by extending the standard continuity equation with an internal variable T representing a hidden structural degree of freedom. For the Lagrangian density\nL(phi, d_t phi, grad phi, d_T phi) = 1/2 (d_t phi)^2 − c^2/2 |grad phi|^2 − alpha/2 (d_T phi)^2 − V(phi),\nthe associated variational structure yields an extended continuity equation that reduces to an exact conservation law only in the shift-invariant case V′(phi) = 0. For a general potential V(phi), the symmetry is explicitly broken and the equation becomes a balance law with a source term\nd_t J0 + div J + d_T Js = − V′(phi),\nwhere J0 = d_t phi is the canonical momentum density, J = c^2 grad phi the spatial flux, and Js = alpha d_T phi the structural flux. This formulation unifies apparently non-conservative dynamics in observable spatial domains by interpreting dissipation-like behaviour as flux redistribution along the internal direction T, rather than as a violation of conservation.\nI establish well-posedness of the resulting wave equation posed on an extended spatial domain (x, T), derive the corresponding extended-domain balance relation, and show how the structural flux accounts for apparent mass or energy leakage, amplitude attenuation, or effective damping without violating the global balance. Numerical experiments based on finite-difference schemes confirm that, in the shift-invariant case, the extended integral invariant is preserved within a relative error below approximately 2.5% under perturbations and mesh refinements.\nThis variational extension provides a compact and physically motivated framework for modeling apparent non-conservation phenomena in systems with hidden internal variables, including wave propagation in complex media, transport in composite or microstructured materials, and effective models in optics or mathematical biology.","manuscriptTitle":"Coherence–Flux Conservation with an Internal Parameter","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-28 18:26:05","doi":"10.21203/rs.3.rs-8379999/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"cbc2aa79-b41c-4d88-be2d-cc916eb4cb9e","owner":[],"postedDate":"January 28th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-02-13T03:09:51+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-28 18:26:05","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8379999","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8379999","identity":"rs-8379999","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}