Entropy-Driven Gradient Stability in Large Language Models: A Non-Equilibrium Thermodynamic Framework for Reinforcement Learning Optimization

Entropy-Driven Gradient Stability in Large Language Models: A Non-Equilibrium Thermodynamic Framework for Reinforcement Learning Optimization | 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 Entropy-Driven Gradient Stability in Large Language Models: A Non-Equilibrium Thermodynamic Framework for Reinforcement Learning Optimization Abdessamad Bourkibate This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8812364/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 T he optimization landscape of Large Language Models ( LLMs ) with extremely high parameter counts exhibits chaotic and unstable dynamics, particularly during reinforcement learning fine-tuning stages where sparse and heavy-tailed reward signals dominate. Existing approaches, such as Proximal Policy Optimization ( PPO ), rely on heuristic clipping mechanisms that impose rigid trust regions, often leading to gradient turbulence, mode collapse, and catastrophic updates. In this work, we introduce Thermodynamic Variational Optimization ( TVO ), a physics-informed framework that reformulates LLM optimization as a non-equilibrium thermodynamic process on a statistical manifold. By defining a Helmholtz free energy functional that balances reward maximization with entropy-driven dissipation, we derive a dissipative gradient flow that enforces monotonic stability without resorting to second-order curvature inversion. TVO introduces a dynamic viscosity term governed by a binary approximation of Total Variation divergence, enabling efficient, scalable control of gradient fluctuations with constant-time complexity relative to vocabulary size. We provide theoretical guarantees of stability using Lyapunov analysis and validate the framework empirically on challenging mathematical reasoning benchmarks, including MATH and AIME24. Experimental results demonstrate substantial reductions in gradient variance, elimination of training collapse, and significant improvements in sample efficiency compared to state-of-the-art proximal optimization baselines. This work positions thermodynamic principles as a foundational lens for understanding and stabilizing large-scale model optimization, offering a unifying framework that bridges reinforcement learning, information geometry, and non-equilibrium physics. Biophysics Behavioral Ecology Computational Neuroscience Mathematical Physics Computational Physics Large Language Models Reinforcement Learning from Human Feedback (RLHF) Thermodynamic Viscous Optimization (TVO) Non-Equilibrium Thermodynamics Gradient Stability AI Safety Entropy Regularization Laminar Flow Green AI Family-Safe AI Dissipative Viscosity Algorithmic Alignment Full Text Additional Declarations The authors declare no competing interests. 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. 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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-8812364","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":587302641,"identity":"9f03bc60-2b52-4196-837a-f92eeb6e74ce","order_by":0,"name":"Abdessamad Bourkibate","email":"data:image/png;base64,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","orcid":"https://orcid.org/0009-0000-6186-8071","institution":"University of the People, Computer Science Department, Pasadena, California, USA","correspondingAuthor":true,"prefix":"","firstName":"Abdessamad","middleName":"","lastName":"Bourkibate","suffix":""}],"badges":[],"createdAt":"2026-02-07 05:12:14","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-8812364/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8812364/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102280905,"identity":"841cec79-7c79-4c90-90ff-6554ebd811f5","added_by":"auto","created_at":"2026-02-10 06:58:09","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1435907,"visible":true,"origin":"","legend":"","description":"","filename":"EntropyDrivenGradientStabilityinComplexNeuralSystemsAPhysicsInformedFrameworkforLLMOptimization.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8812364/v1_covered_2603e9e5-dc77-44fe-b4bb-4a66cfa5ccd0.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eEntropy-Driven Gradient Stability in Large Language Models: A Non-Equilibrium Thermodynamic Framework for Reinforcement Learning Optimization\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"University of the People","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":"Large Language Models, Reinforcement Learning from Human Feedback (RLHF), Thermodynamic Viscous Optimization (TVO), Non-Equilibrium Thermodynamics, Gradient Stability, AI Safety, Entropy Regularization, Laminar Flow, Green AI, Family-Safe AI, Dissipative Viscosity, Algorithmic Alignment","lastPublishedDoi":"10.21203/rs.3.rs-8812364/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8812364/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cstrong\u003eT\u003c/strong\u003ehe optimization landscape of Large Language Models (\u003cstrong\u003eLLMs\u003c/strong\u003e) with extremely high parameter counts exhibits chaotic and unstable dynamics, particularly during reinforcement learning fine-tuning stages where sparse and heavy-tailed reward signals dominate. Existing approaches, such as Proximal Policy Optimization (\u003cstrong\u003ePPO\u003c/strong\u003e), rely on heuristic clipping mechanisms that impose rigid trust regions, often leading to gradient turbulence, mode collapse, and catastrophic updates.\u003cbr\u003e\nIn this work, we introduce Thermodynamic Variational Optimization (\u003cstrong\u003eTVO\u003c/strong\u003e), a physics-informed framework that reformulates LLM optimization as a non-equilibrium thermodynamic process on a statistical manifold. By defining a Helmholtz free energy functional that balances reward maximization with entropy-driven dissipation, we derive a dissipative gradient flow that enforces monotonic stability without resorting to second-order curvature inversion.\u003cbr\u003e\nTVO introduces a dynamic viscosity term governed by a binary approximation of Total Variation divergence, enabling efficient, scalable control of gradient fluctuations with constant-time complexity relative to vocabulary size. 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