[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118606-en":3,"doc-seo-118606-105":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":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},118606,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Heat flux for semilocal machine-learning potentials - Letter","Green-Kubo (GK) provides a rigorous route to simulate heat transport, but it depends on an accurate potential-energy surface and well-converged equilibrium statistics. Machine-learning potentials can reproduce first-principles accuracy at far lower cost, enabling access beyond conventional simulation time and length scales. This Letter formulates how to apply GK with message-passing semilocal machine-learning potentials, deriving an adapted heat-flux expression implementable via automatic differentiation without sacrificing efficiency. Thermal conductivity of zirconium dioxide is computed and validated over temperatures.","Letter  \nHeat flux for semilocal machine-learning potentials  \nMarcel F. Langer  , 1, 2, 3 , * Florian Knoop  ,4, 3 ,† Christian Carbogno  ,3 Matthias Scheffler,3 and Matthias Rupp 3, 5, 6  \n1 Machine Learning Group, Technische Universität Berlin, 10587 Berlin, Germany  \n2 Berlin Institute for the Foundations of Learning and Data, 10623 Berlin, Germany  \n3 The NOMAD Laboratory at the FHI of the Max-Planck-Gesellschaft and IRIS Adlershof of the Humboldt Universität zu Berlin, 14195 Berlin, Germany  \n4 Theoretical Physics Division, Department of Physics, Chemistry and Biology (IFM), Linköping University, 581 83 Linköping, Sweden  \n5 Department of Computer and Information Science, University of Konstanz, 78464 Konstanz, Germany  \n6 Materials Research and Technology Department, Luxembourg Institute of Science and Technology, Belvaux, Luxembourg  (Received 30 March 2023; accepted 14 July 2023; published 13 September 2023)  \nThe Green-Kubo (GK) method is a rigorous framework for heat transport simulations in materials. However, it requires an accurate description of the potential-energy surface and carefully converged statistics. Machinelearning potentials can achieve the accuracy of first-principles simulations while allowing to reach well beyond their simulation time and length scales at a fraction of the cost. In this Letter, we explain how to apply the GK approach to the recent class of message-passing machine-learning potentials, which iteratively consider semilocal interactions beyond the initial interaction cutoff. We derive an adapted heat flux formulation that can be implemented using automatic differentiation without compromising computational efficiency. The approach is demonstrated and validated by calculating the thermal conductivity of zirconium dioxide across temperatures.  \nDOI: 10.1103/PhysRevB.108.L100302  \nThe thermal conductivity tensor κ describes the ability of a material to conduct heat when exposed to a temperature gradient. Its computational prediction is of great interest for the design of novel high-performance materials which are needed, for example, as thermal barrier coatings in engines [1] or thermoelectrics for waste heat recovery [2] . Such materials often feature complex structure and strongly anharmonic potential-energy surfaces (PESs) [3,4] . This implies the need to evaluate κ with a non-perturbative method such as the Green-Kubo (GK) method [5–9] .  \nIn the GK approach, κ is expressed in terms of the integral of the autocorrelation function of the instantaneous heat flux J (t ) as observed in equilibrium molecular dynamics (MD) simulations,  \nκ (T , p) = kB~~ ~~1T2~~ ~~V t 􀀂0 t dτ 􀀄J(τ) ⊗ J(0)􀀆 T ,p , (1) where kB is the Boltzmann constant, T is the temperature, Vis the simulation cell volume, and 􀀄·􀀆 T ,p denotes an ensemble average at temperature T and pressure p.  \nHigh-accuracy MD simulations can be performed using density-functional theory (DFT) when the exchangecorrelation approximation is reliable [10] . For the evaluation  \n*[Corresponding author: mail@marcel.science](Corresponding author: mail@marcel.science)[ ](Corresponding author: mail@marcel.science)†Corresponding author: fl[orian.knoop@liu.se](orian.knoop@liu.se)  \nPublished by the American Physical Society under the terms of the Creative Commons Attribution 4 .0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.  \nof Eq. (1), this approach [11, 12] suffers from its numerical cost, which limits the system sizes and timescales that can be treated, and therefore requires additional denoising and extrapolation approaches [12–14] . The alternative, so far, has been the use of semiempirical force fields (FFs) [15] . Here, the interatomic interactions are described by a physically motivated analytical equation that includes free parameters which are fitted to experimental or ab in","cbCaijyeElqS1t1M","https://ap.wps.com/l/cbCaijyeElqS1t1M","pdf",1104316,1,7,"English","en",105,"# Heat transport and Green-Kubo framework\n## Thermal conductivity via heat-flux autocorrelation\n## Machine-learning potentials: local vs semilocal\n## Extending GK to semilocal message-passing models\n## Application to zirconium dioxide","[{\"question\":\"What is the Green-Kubo method used for in this work?\",\"answer\":\"The Green-Kubo method expresses the thermal conductivity through the time integral of the autocorrelation of the instantaneous heat flux measured in equilibrium MD simulations.\"},{\"question\":\"Why are semilocal machine-learning potentials important here?\",\"answer\":\"Semilocal machine-learning potentials build longer-range correlations iteratively from local interactions using message-passing mechanisms, offering improved flexibility and accuracy over strictly local models.\"},{\"question\":\"How does the paper make heat-flux evaluation compatible with semilocal potentials?\",\"answer\":\"It derives an adapted heat-flux formulation that explicitly accounts for semilocal interactions and can be implemented using automatic differentiation while preserving computational efficiency.\"}]","Heat flux for semilocal machine-learning potentials - Letter | PDF",1785684481,18,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"heat-flux-for-semilocal-machine-learning-potentials-letter","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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":53},"https://docshare.wps.com/document/heat-flux-for-semilocal-machine-learning-potentials-letter/118606/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the Green-Kubo method used for in this work?","Question",{"text":75,"@type":76},"The Green-Kubo method expresses the thermal conductivity through the time integral of the autocorrelation of the instantaneous heat flux measured in equilibrium MD simulations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are semilocal machine-learning potentials important here?",{"text":80,"@type":76},"Semilocal machine-learning potentials build longer-range correlations iteratively from local interactions using message-passing mechanisms, offering improved flexibility and accuracy over strictly local models.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper make heat-flux evaluation compatible with semilocal potentials?",{"text":84,"@type":76},"It derives an adapted heat-flux formulation that explicitly accounts for semilocal interactions and can be implemented using automatic differentiation while preserving computational efficiency.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]