[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85220-en":3,"doc-seo-85220-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85220,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Path-Dependent Entropic Lagrangian for Probability Flows: Balance–Entropy Routing and Composable Information Potentials","Probability distributions are central to information theory, statistical inference, and probabilistic learning. Maximum entropy selects a state under constraints, but it does not prescribe the pathway, transport rules, or how dissipation and external information exchange are handled. The work develops a path-dependent entropic Lagrangian calculus for probability-path evolution via restricted generators, upper-limit history terms, and explicit balance–entropy port routing. It delivers thermal state relations, conservative probability balance, and nonnegative entropy production, recovering MaxEnt/Bayesian laws as stationary no-flux states and separating internal dissipation from supplied information power, with composable potentials controlling tails and robustness through structured modules.","arXiv :2607 . 10493v1 [math-ph] 11 Jul 2026  \nPath-Dependent Entropic Lagrangian for Probability Flows: Balance–Entropy Routing and Composable Information  \nPotentials  \nHuilong Rena  \na State Key Laboratory of Disaster Reduction in Civil Engineering, College of Civil Engineering, Tongji  \nUniversity, Shanghai 200092, China  \nAbstract  \nProbability distributions are central to information theory, statistical inference, and modern probabilistic learning. Maximum entropy selects a probability state under prescribed constraints, but it does not specify how that state is reached, how probability is transported, or how dissipation and external information exchange are accounted for along the path. We develop a path-dependent entropic Lagrangian calculus that extends static state selection to probability-path evolution through restricted generators, upper-limit history terms, and explicit balance–entropy port routing. The construction yields the thermal state relation, conservative probability balance, and nonnegative production under standard mobility closure. Its KL/Shannon sector recovers maximum-entropy and Bayesian laws as stationary no-flux states, while time-dependent information potentials separate internal dissipation from supplied information power. Composable information and structural potentials control tails, sparsity, robustness, regularization, and nonlocal multimodality without changing the accounting architecture. Two numerical examples verify mass conservation, energy decomposition, and the total free-energy ledger.  \nKeywords: probability flow; maximum entropy; information dynamics; port routing; Kullback–Leibler divergence; composable information potentials; entropy production; Bayesian inference  \n1. Introduction  \nProbability distributions are a common mathematical state description for information theory, statistical inference, and much of modern probabilistic learning. Entropy, relative entropy, cross-entropy, and mutual information are functionals of probability laws [31, 20, 11]; variational inference optimizes distribution-valued objectives [19, 4]; and generative methods construct transformations between probability laws [32, 16, 33] . These settings require not only principles for selecting probability states, but also laws for their admissible evolution.  \nEmail address: [hlren@tongji.edu.cn](hlren@tongji.edu.cn) (Huilong Ren)  \nMaximum entropy provides a canonical static selection rule under prescribed information constraints [17, 11] . A positive target density can be represented in Gibbs form through a suitable information potential, but the static rule does not determine the transition path, probability balance, irreversible production, or open-system information exchange. Bayesian updating likewise specifies a posterior endpoint while leaving its dynamical realization as a separate question [21, 3] .  \nThermodynamics supplies the missing language of stored energy, flux–force relations, entropy production, and boundary exchange [23, 12, 30, 13] . Related structures occur in GENERIC-type formulations and in free-energy or Wasserstein descriptions of Fokker– Planck dynamics [14, 24, 18, 25, 1] . The present work develops a different history-channel construction in which upper-limit variations and explicit port routing separate probability balance from entropy accounting.  \nThe Path-Dependent Entropic Lagrangian (PDEL) [26, 27] is an energy-valued, historyaware functional containing stored energy, thermal pairing, accumulated channel expenditure, and boundary or information ports. Restricted generators produce the thermal state relation, conservative probability balance, and nonnegative production under standard mobility closure. Maximum entropy is recovered as the stationary closed-system sector: the KL/Shannon module gives MaxEnt and Bayesian distributions as no-flux states, while a time-dependent information potential separates internal dissipation from externally supplied information power.  \nThe f","cbCaitPRAvzTK2dc","https://ap.wps.com/l/cbCaitPRAvzTK2dc","pdf",614613,4,1,32,"English","en",105,"# Introduction\n# Thermomechanical review and thermoelastic coupling\n## A minimal thermoelastic–damping prototype\n## Upper-limit channels and coupled equations","[{\"question\":\"What problem does the path-dependent entropic Lagrangian aim to solve compared with maximum entropy or Bayesian updating?\",\"answer\":\"It adds rules for how a selected probability state evolves along a path, including probability transport, irreversible production, and explicit handling of open-system information exchange. Maximum entropy and Bayesian updating specify endpoints but not their dynamical realizations.\"},{\"question\":\"How does the framework obtain balance and entropy accounting along the probability path?\",\"answer\":\"It uses restricted generators, upper-limit history terms, and explicit balance–entropy port routing. This construction yields conservative probability balance and nonnegative production under standard mobility closure.\"},{\"question\":\"In what sense are maximum-entropy and Bayesian laws recovered in this formulation?\",\"answer\":\"The KL/Shannon sector recovers maximum-entropy and Bayesian laws as stationary no-flux states. A time-dependent information potential further distinguishes internal dissipation from externally supplied information power.\"}]",1784201815,81,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"path-dependent-entropic-lagrangian-for-probability-flows-balanceentropy-routing-and-composable-information-potentials","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/path-dependent-entropic-lagrangian-for-probability-flows-balanceentropy-routing-and-composable-information-potentials/85220/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the path-dependent entropic Lagrangian aim to solve compared with maximum entropy or Bayesian updating?","Question",{"text":75,"@type":76},"It adds rules for how a selected probability state evolves along a path, including probability transport, irreversible production, and explicit handling of open-system information exchange. Maximum entropy and Bayesian updating specify endpoints but not their dynamical realizations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the framework obtain balance and entropy accounting along the probability path?",{"text":80,"@type":76},"It uses restricted generators, upper-limit history terms, and explicit balance–entropy port routing. This construction yields conservative probability balance and nonnegative production under standard mobility closure.",{"name":82,"@type":73,"acceptedAnswer":83},"In what sense are maximum-entropy and Bayesian laws recovered in this formulation?",{"text":84,"@type":76},"The KL/Shannon sector recovers maximum-entropy and Bayesian laws as stationary no-flux states. 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