CINXE.COM

<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="robots" content="max-image-preview:large"> <meta name="access" content="Yes"> <meta name="360-site-verification" content="1268d79b5e96aecf3ff2a7dac04ad990" /> <title>The Reversibility Paradox: Role of the Velocity Reversal Step | International Journal of Theoretical Physics</title> <meta name="twitter:site" content="@SpringerLink"/> <meta name="twitter:card" content="summary_large_image"/> <meta name="twitter:image:alt" content="Content cover image"/> <meta name="twitter:title" content="The Reversibility Paradox: Role of the Velocity Reversal Step"/> <meta name="twitter:description" content="International Journal of Theoretical Physics - The reversibility, or Loschmidt paradox, is a thought experiment in which microscopic reversibility is exploited to generate an apparently spontaneous..."/> <meta name="twitter:image" content="https://media.springernature.com/full/springer-static/cover-hires/journal/10773"/> <meta name="journal_id" content="10773"/> <meta name="dc.title" content="The Reversibility Paradox: Role of the Velocity Reversal Step"/> <meta name="dc.source" content="International Journal of Theoretical Physics 2023 62:9"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Springer"/> <meta name="dc.date" content="2023-09-02"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2023 The Author(s)"/> <meta name="dc.rights" content="2023 The Author(s)"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="The reversibility, or Loschmidt paradox, is a thought experiment in which microscopic reversibility is exploited to generate an apparently spontaneous trajectory in which entropy decreases, thus violating the Second Law of Thermodynamics (SLT). To achieve this, the system is allowed to evolve from a low-entropy to a high-entropy configuration, at which point all velocities of its molecules are reversed, thus producing the desired effect. In this paper we focus on the velocity reversal step. Implementing this step requires the measurement of velocities and positions, which must be stored in a memory. Even in the case of a reversible measurement, the erasure of the stored information, necessary to reset the measuring device to its original state, has an entropy cost that offsets the entropy decrease during the reverse evolution of the system. This cost is sufficient to explain why the described procedure does not violate the SLT."/> <meta name="prism.issn" content="1572-9575"/> <meta name="prism.publicationName" content="International Journal of Theoretical Physics"/> <meta name="prism.publicationDate" content="2023-09-02"/> <meta name="prism.volume" content="62"/> <meta name="prism.number" content="9"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1"/> <meta name="prism.endingPage" content="5"/> <meta name="prism.copyright" content="2023 The Author(s)"/> <meta name="prism.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="prism.url" content="https://link.springer.com/article/10.1007/s10773-023-05458-x"/> <meta name="prism.doi" content="doi:10.1007/s10773-023-05458-x"/> <meta name="citation_pdf_url" content="https://link.springer.com/content/pdf/10.1007/s10773-023-05458-x.pdf"/> <meta name="citation_fulltext_html_url" content="https://link.springer.com/article/10.1007/s10773-023-05458-x"/> <meta name="citation_journal_title" content="International Journal of Theoretical Physics"/> <meta name="citation_journal_abbrev" content="Int J Theor Phys"/> <meta name="citation_publisher" content="Springer US"/> <meta name="citation_issn" content="1572-9575"/> <meta name="citation_title" content="The Reversibility Paradox: Role of the Velocity Reversal Step"/> <meta name="citation_volume" content="62"/> <meta name="citation_issue" content="9"/> <meta name="citation_publication_date" content="2023/09"/> <meta name="citation_online_date" content="2023/09/02"/> <meta name="citation_firstpage" content="1"/> <meta name="citation_lastpage" content="5"/> <meta name="citation_article_type" content="Research"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1007/s10773-023-05458-x"/> <meta name="DOI" content="10.1007/s10773-023-05458-x"/> <meta name="size" content="47598"/> <meta name="citation_doi" content="10.1007/s10773-023-05458-x"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1007/s10773-023-05458-x&api_key="/> <meta name="description" content="The reversibility, or Loschmidt paradox, is a thought experiment in which microscopic reversibility is exploited to generate an apparently spontaneous traj"/> <meta name="dc.creator" content="Binder, P.M."/> <meta name="dc.subject" content="Physics, general"/> <meta name="dc.subject" content="Quantum Physics"/> <meta name="dc.subject" content="Elementary Particles, Quantum Field Theory"/> <meta name="dc.subject" content="Theoretical, Mathematical and Computational Physics"/> <meta name="citation_reference" content="citation_title=The Kind of Motion We Call Heat; citation_publication_date=1976; citation_id=CR1; citation_author=SG Brush; citation_publisher=North-Holland"/> <meta name="citation_reference" content="citation_title=Physics and Chance; citation_publication_date=1993; citation_id=CR2; citation_author=L Sklar; citation_publisher=Cambridge University Press"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=Resolution of the Loschmidt paradox: The origin of irreversible behavior in reversible atomistic dynamics; citation_author=BL Holian, WG Hoover, HA Posch; citation_volume=59; citation_publication_date=1987; citation_pages=10-13; citation_doi=10.1103/PhysRevLett.59.10; citation_id=CR3"/> <meta name="citation_reference" content="citation_journal_title=Adv. Phys.; citation_title=The fluctuation theorem; citation_author=DJ Evans, DJ Searles; citation_volume=51; citation_publication_date=2002; citation_pages=1529-1585; citation_doi=10.1080/00018730210155133; citation_id=CR4"/> <meta name="citation_reference" content="citation_journal_title=Am. J. Phys.; citation_title=Explaining irreversibility; citation_author=RH Swendsen; citation_volume=76; citation_publication_date=2008; citation_pages=643-648; citation_doi=10.1119/1.2894523; citation_id=CR5"/> <meta name="citation_reference" content="citation_journal_title=Phys. A; citation_title=Microscopic origins of macroscopic irreversible behavior; citation_author=JL Lebowitz; citation_volume=263; citation_publication_date=1999; citation_pages=516-527; citation_doi=10.1016/S0378-4371(98)00514-7; citation_id=CR6"/> <meta name="citation_reference" content="citation_journal_title=Found. Phys.; citation_title=Loschmidt’s and Zermelo’s paradoxes do not exist; citation_author=J Rothstein; citation_volume=4; citation_publication_date=1974; citation_pages=83-89; citation_doi=10.1007/BF00708557; citation_id=CR7"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=Reversibility and stability of information processing systems; citation_author=WH Zurek; citation_volume=53; citation_publication_date=1984; citation_pages=391-394; citation_doi=10.1103/PhysRevLett.53.391; citation_id=CR8"/> <meta name="citation_reference" content="citation_title=Philosophical Essay on Probabilities; citation_publication_date=1951; citation_id=CR9; citation_author=PS De Laplace; citation_publisher=Dover"/> <meta name="citation_reference" content="citation_title=Maxwell’s Demon; citation_publication_date=2014; citation_id=CR10; citation_publisher=Princeton University Press"/> <meta name="citation_reference" content="citation_journal_title=Int. J. Theor. Phys.; citation_title=The thermodynamics of computation – a review; citation_author=CH Bennett; citation_volume=21; citation_publication_date=1982; citation_pages=905-940; citation_doi=10.1007/BF02084158; citation_id=CR11"/> <meta name="citation_reference" content="citation_journal_title=IBM J. Res. Develop; citation_title=Reversibility and heat generation in the computing process; citation_author=R Landauer; citation_volume=5; citation_publication_date=1961; citation_pages=183-191; citation_doi=10.1147/rd.53.0183; citation_id=CR12"/> <meta name="citation_reference" content="citation_journal_title=Nature; citation_title=Experimental verification of Landauer’s principle linking information and thermodynamics; citation_author=A Bérut, A Arakelyan, A Petrosyan, S Ciliberto, R Dillenschneider, E Lutz; citation_volume=483; citation_publication_date=2012; citation_pages=187-189; citation_doi=10.1038/nature10872; citation_id=CR13"/> <meta name="citation_reference" content="citation_journal_title=Phys. Rev. Lett.; citation_title=High-precision test of Landauer’s principle in a feedback trap; citation_author=Y Jun, M Gavrilov, J Bechhoeffer; citation_volume=113; citation_publication_date=2014; citation_pages=190601; citation_doi=10.1103/PhysRevLett.113.190601; citation_id=CR14"/> <meta name="citation_reference" content="Hong, J., Lambson, B., Dhuey, S., Bokor, J.: Experimental test of Landauer’s principle in single-bit operations on nanomagnetic memory bits. Sci. Adv. 2, e1501492 (2016)"/> <meta name="citation_reference" content="citation_journal_title=Zeit Phys.; citation_title=Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen; citation_author=L Szilard; citation_volume=53; citation_publication_date=1929; citation_pages=840-856; citation_doi=10.1007/BF01341281; citation_id=CR16"/> <meta name="citation_reference" content="citation_journal_title=Nat. Phys.; citation_title=Information heat engine: Converting information to energy by feedback control; citation_author=S Toyabe, T Sagawa, M Ueda, E Muneyuki, M Sano; citation_volume=6; citation_publication_date=2010; citation_pages=988-992; citation_doi=10.1038/nphys1821; citation_id=CR17"/> <meta name="citation_reference" content="citation_journal_title=Science; citation_title=Comprehensive control of atomic motion; citation_author=MG Raizen; citation_volume=324; citation_publication_date=2009; citation_pages=1403-1406; citation_doi=10.1126/science.1171506; citation_id=CR18"/> <meta name="citation_reference" content="citation_journal_title=Science; citation_title=Reflections on a wall of light; citation_author=P-M Binder; citation_volume=322; citation_publication_date=2008; citation_pages=1334-1335; citation_doi=10.1126/science.1166681; citation_id=CR19"/> <meta name="citation_author" content="Binder, P.M."/> <meta name="citation_author_email" content="pbinder@hawaii.edu"/> <meta name="citation_author_institution" content="University of Hawaii, Hilo, USA"/> <meta name="format-detection" content="telephone=no"/> <meta name="citation_cover_date" content="2023/09/01"/> <meta property="og:url" content="https://link.springer.com/article/10.1007/s10773-023-05458-x"/> <meta property="og:type" content="article"/> <meta property="og:site_name" content="SpringerLink"/> <meta property="og:title" content="The Reversibility Paradox: Role of the Velocity Reversal Step - International Journal of Theoretical Physics"/> <meta property="og:description" content="The reversibility, or Loschmidt paradox, is a thought experiment in which microscopic reversibility is exploited to generate an apparently spontaneous trajectory in which entropy decreases, thus violating the Second Law of Thermodynamics (SLT). To achieve this, the system is allowed to evolve from a low-entropy to a high-entropy configuration, at which point all velocities of its molecules are reversed, thus producing the desired effect. In this paper we focus on the velocity reversal step. Implementing this step requires the measurement of velocities and positions, which must be stored in a memory. Even in the case of a reversible measurement, the erasure of the stored information, necessary to reset the measuring device to its original state, has an entropy cost that offsets the entropy decrease during the reverse evolution of the system. This cost is sufficient to explain why the described procedure does not violate the SLT."/> <meta property="og:image" content="https://media.springernature.com/full/springer-static/cover-hires/journal/10773"/> <meta name="format-detection" content="telephone=no"> <link rel="apple-touch-icon" sizes="180x180" href=/oscar-static/img/favicons/darwin/apple-touch-icon-92e819bf8a.png> <link rel="icon" type="image/png" sizes="192x192" href=/oscar-static/img/favicons/darwin/android-chrome-192x192-6f081ca7e5.png> <link rel="icon" type="image/png" sizes="32x32" href=/oscar-static/img/favicons/darwin/favicon-32x32-1435da3e82.png> <link rel="icon" type="image/png" sizes="16x16" href=/oscar-static/img/favicons/darwin/favicon-16x16-ed57f42bd2.png> <link rel="shortcut icon" data-test="shortcut-icon" href=/oscar-static/img/favicons/darwin/favicon-c6d59aafac.ico> <meta name="theme-color" content="#e6e6e6">   <link rel="stylesheet" media="print" href=/oscar-static/app-springerlink/css/print-b8af42253b.css> <style> html{text-size-adjust:100%;line-height:1.15}body{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;line-height:1.8;margin:0}details,main{display:block}h1{font-size:2em;margin:.67em 0}a{background-color:transparent;color:#025e8d}sub{bottom:-.25em;font-size:75%;line-height:0;position:relative;vertical-align:baseline}img{border:0;height:auto;max-width:100%;vertical-align:middle}button,input{font-family:inherit;font-size:100%;line-height:1.15;margin:0;overflow:visible}button{text-transform:none}[type=button],[type=submit],button{-webkit-appearance:button}[type=search]{-webkit-appearance:textfield;outline-offset:-2px}summary{display:list-item}[hidden]{display:none}button{cursor:pointer}svg{height:1rem;width:1rem} </style> <style>@media only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark) { body{background:#fff;color:#222;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;line-height:1.8;min-height:100%}a{color:#025e8d;text-decoration:underline;text-decoration-skip-ink:auto}button{cursor:pointer}img{border:0;height:auto;max-width:100%;vertical-align:middle}html{box-sizing:border-box;font-size:100%;height:100%;overflow-y:scroll}h1{font-size:2.25rem}h2{font-size:1.75rem}h1,h2,h4{font-weight:700;line-height:1.2}h4{font-size:1.25rem}body{font-size:1.125rem}*{box-sizing:inherit}p{margin-bottom:2rem;margin-top:0}p:last-of-type{margin-bottom:0}.c-ad{text-align:center}@media only screen and (min-width:480px){.c-ad{padding:8px}}.c-ad--728x90{display:none}.c-ad--728x90 .c-ad__inner{min-height:calc(1.5em + 94px)}@media only screen and (min-width:876px){.js .c-ad--728x90{display:none}}.c-ad__label{color:#333;font-size:.875rem;font-weight:400;line-height:1.5;margin-bottom:4px}.c-ad__label,.c-status-message{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-status-message{align-items:center;box-sizing:border-box;display:flex;position:relative;width:100%}.c-status-message :last-child{margin-bottom:0}.c-status-message--boxed{background-color:#fff;border:1px solid #ccc;line-height:1.4;padding:16px}.c-status-message__heading{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;font-weight:700}.c-status-message__icon{fill:currentcolor;display:inline-block;flex:0 0 auto;height:1.5em;margin-right:8px;transform:translate(0);vertical-align:text-top;width:1.5em}.c-status-message__icon--top{align-self:flex-start}.c-status-message--info .c-status-message__icon{color:#003f8d}.c-status-message--boxed.c-status-message--info{border-bottom:4px solid #003f8d}.c-status-message--error .c-status-message__icon{color:#c40606}.c-status-message--boxed.c-status-message--error{border-bottom:4px solid #c40606}.c-status-message--success .c-status-message__icon{color:#00b8b0}.c-status-message--boxed.c-status-message--success{border-bottom:4px solid #00b8b0}.c-status-message--warning .c-status-message__icon{color:#edbc53}.c-status-message--boxed.c-status-message--warning{border-bottom:4px solid #edbc53}.eds-c-header{background-color:#fff;border-bottom:2px solid #01324b;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;line-height:1.5;padding:8px 0 0}.eds-c-header__container{align-items:center;display:flex;flex-wrap:nowrap;gap:8px 16px;justify-content:space-between;margin:0 auto 8px;max-width:1280px;padding:0 8px;position:relative}.eds-c-header__nav{border-top:2px solid #c5e0f4;padding-top:4px;position:relative}.eds-c-header__nav-container{align-items:center;display:flex;flex-wrap:wrap;margin:0 auto 4px;max-width:1280px;padding:0 8px;position:relative}.eds-c-header__nav-container>:not(:last-child){margin-right:32px}.eds-c-header__link-container{align-items:center;display:flex;flex:1 0 auto;gap:8px 16px;justify-content:space-between}.eds-c-header__list{list-style:none;margin:0;padding:0}.eds-c-header__list-item{font-weight:700;margin:0 auto;max-width:1280px;padding:8px}.eds-c-header__list-item:not(:last-child){border-bottom:2px solid #c5e0f4}.eds-c-header__item{color:inherit}@media only screen and (min-width:768px){.eds-c-header__item--menu{display:none;visibility:hidden}.eds-c-header__item--menu:first-child+*{margin-block-start:0}}.eds-c-header__item--inline-links{display:none;visibility:hidden}@media only screen and (min-width:768px){.eds-c-header__item--inline-links{display:flex;gap:16px 16px;visibility:visible}}.eds-c-header__item--divider:before{border-left:2px solid #c5e0f4;content:"";height:calc(100% - 16px);margin-left:-15px;position:absolute;top:8px}.eds-c-header__brand{padding:16px 8px}.eds-c-header__brand a{display:block;line-height:1;text-decoration:none}.eds-c-header__brand img{height:1.5rem;width:auto}.eds-c-header__link{color:inherit;display:inline-block;font-weight:700;padding:16px 8px;position:relative;text-decoration-color:transparent;white-space:nowrap;word-break:normal}.eds-c-header__icon{fill:currentcolor;display:inline-block;font-size:1.5rem;height:1em;transform:translate(0);vertical-align:bottom;width:1em}.eds-c-header__icon+*{margin-left:8px}.eds-c-header__expander{background-color:#f0f7fc}.eds-c-header__search{display:block;padding:24px 0}@media only screen and (min-width:768px){.eds-c-header__search{max-width:70%}}.eds-c-header__search-container{position:relative}.eds-c-header__search-label{color:inherit;display:inline-block;font-weight:700;margin-bottom:8px}.eds-c-header__search-input{background-color:#fff;border:1px solid #000;padding:8px 48px 8px 8px;width:100%}.eds-c-header__search-button{background-color:transparent;border:0;color:inherit;height:100%;padding:0 8px;position:absolute;right:0}.has-tethered.eds-c-header__expander{border-bottom:2px solid #01324b;left:0;margin-top:-2px;top:100%;width:100%;z-index:10}@media only screen and (min-width:768px){.has-tethered.eds-c-header__expander--menu{display:none;visibility:hidden}}.has-tethered .eds-c-header__heading{display:none;visibility:hidden}.has-tethered .eds-c-header__heading:first-child+*{margin-block-start:0}.has-tethered .eds-c-header__search{margin:auto}.eds-c-header__heading{margin:0 auto;max-width:1280px;padding:16px 16px 0}.eds-c-pagination{align-items:center;display:flex;flex-wrap:wrap;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;gap:16px 0;justify-content:center;line-height:1.4;list-style:none;margin:0;padding:32px 0}@media only screen and (min-width:480px){.eds-c-pagination{padding:32px 16px}}.eds-c-pagination__item{margin-right:8px}.eds-c-pagination__item--prev{margin-right:16px}.eds-c-pagination__item--next .eds-c-pagination__link,.eds-c-pagination__item--prev .eds-c-pagination__link{padding:16px 8px}.eds-c-pagination__item--next{margin-left:8px}.eds-c-pagination__item:last-child{margin-right:0}.eds-c-pagination__link{align-items:center;color:#222;cursor:pointer;display:inline-block;font-size:1rem;margin:0;padding:16px 24px;position:relative;text-align:center;transition:all .2s ease 0s}.eds-c-pagination__link:visited{color:#222}.eds-c-pagination__link--disabled{border-color:#555;color:#555;cursor:default}.eds-c-pagination__link--active{background-color:#01324b;background-image:none;border-radius:8px;color:#fff}.eds-c-pagination__link--active:focus,.eds-c-pagination__link--active:hover,.eds-c-pagination__link--active:visited{color:#fff}.eds-c-pagination__link-container{align-items:center;display:flex}.eds-c-pagination__icon{fill:#222;height:1.5rem;width:1.5rem}.eds-c-pagination__icon--disabled{fill:#555}.eds-c-pagination__visually-hidden{clip:rect(0,0,0,0);border:0;clip-path:inset(50%);height:1px;overflow:hidden;padding:0;position:absolute!important;white-space:nowrap;width:1px}.c-breadcrumbs{color:#333;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;list-style:none;margin:0;padding:0}.c-breadcrumbs>li{display:inline}svg.c-breadcrumbs__chevron{fill:#333;height:10px;margin:0 .25rem;width:10px}.c-breadcrumbs--contrast,.c-breadcrumbs--contrast .c-breadcrumbs__link{color:#fff}.c-breadcrumbs--contrast svg.c-breadcrumbs__chevron{fill:#fff}@media only screen and (max-width:479px){.c-breadcrumbs .c-breadcrumbs__item{display:none}.c-breadcrumbs .c-breadcrumbs__item:last-child,.c-breadcrumbs .c-breadcrumbs__item:nth-last-child(2){display:inline}}.c-skip-link{background:#01324b;bottom:auto;color:#fff;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;padding:8px;position:absolute;text-align:center;transform:translateY(-100%);width:100%;z-index:9999}@media (prefers-reduced-motion:reduce){.c-skip-link{transition:top .3s ease-in-out 0s}}@media print{.c-skip-link{display:none}}.c-skip-link:active,.c-skip-link:hover,.c-skip-link:link,.c-skip-link:visited{color:#fff}.c-skip-link:focus{transform:translateY(0)}.l-with-sidebar{display:flex;flex-wrap:wrap}.l-with-sidebar>*{margin:0}.l-with-sidebar__sidebar{flex-basis:var(--with-sidebar--basis,400px);flex-grow:1}.l-with-sidebar>:not(.l-with-sidebar__sidebar){flex-basis:0px;flex-grow:999;min-width:var(--with-sidebar--min,53%)}.l-with-sidebar>:first-child{padding-right:4rem}@supports (gap:1em){.l-with-sidebar>:first-child{padding-right:0}.l-with-sidebar{gap:var(--with-sidebar--gap,4rem)}}.c-header__link{color:inherit;display:inline-block;font-weight:700;padding:16px 8px;position:relative;text-decoration-color:transparent;white-space:nowrap;word-break:normal}.app-masthead__colour-4{--background-color:#ff9500;--gradient-light:rgba(0,0,0,.5);--gradient-dark:rgba(0,0,0,.8)}.app-masthead{background:var(--background-color,#0070a8);position:relative}.app-masthead:after{background:radial-gradient(circle at top right,var(--gradient-light,rgba(0,0,0,.4)),var(--gradient-dark,rgba(0,0,0,.7)));bottom:0;content:"";left:0;position:absolute;right:0;top:0}@media only screen and (max-width:479px){.app-masthead:after{background:linear-gradient(225deg,var(--gradient-light,rgba(0,0,0,.4)),var(--gradient-dark,rgba(0,0,0,.7)))}}.app-masthead__container{color:var(--masthead-color,#fff);margin:0 auto;max-width:1280px;padding:0 16px;position:relative;z-index:1}.u-button{align-items:center;background-color:#01324b;background-image:none;border:4px solid transparent;border-radius:32px;cursor:pointer;display:inline-flex;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;font-weight:700;justify-content:center;line-height:1.3;margin:0;padding:16px 32px;position:relative;transition:all .2s ease 0s;width:auto}.u-button svg,.u-button--contrast svg,.u-button--primary svg,.u-button--secondary svg,.u-button--tertiary svg{fill:currentcolor}.u-button,.u-button:visited{color:#fff}.u-button,.u-button:hover{box-shadow:0 0 0 1px #01324b;text-decoration:none}.u-button:hover{border:4px solid #fff}.u-button:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.u-button:focus,.u-button:hover{background-color:#fff;background-image:none;color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--primary:focus svg path,.app-masthead--pastel .c-pdf-download .u-button--primary:hover svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover svg path,.u-button--primary:focus svg path,.u-button--primary:hover svg path,.u-button:focus svg path,.u-button:hover svg path{fill:#01324b}.u-button--primary{background-color:#01324b;background-image:none;border:4px solid transparent;box-shadow:0 0 0 1px #01324b;color:#fff;font-weight:700}.u-button--primary:visited{color:#fff}.u-button--primary:hover{border:4px solid #fff;box-shadow:0 0 0 1px #01324b;text-decoration:none}.u-button--primary:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.u-button--primary:focus,.u-button--primary:hover{background-color:#fff;background-image:none;color:#01324b}.u-button--secondary{background-color:#fff;border:4px solid #fff;color:#01324b;font-weight:700}.u-button--secondary:visited{color:#01324b}.u-button--secondary:hover{border:4px solid #01324b;box-shadow:none}.u-button--secondary:focus,.u-button--secondary:hover{background-color:#01324b;color:#fff}.app-masthead--pastel .c-pdf-download .u-button--secondary:focus svg path,.app-masthead--pastel .c-pdf-download .u-button--secondary:hover svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:focus svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:hover svg path,.u-button--secondary:focus svg path,.u-button--secondary:hover svg path,.u-button--tertiary:focus svg path,.u-button--tertiary:hover svg path{fill:#fff}.u-button--tertiary{background-color:#ebf1f5;border:4px solid transparent;box-shadow:none;color:#666;font-weight:700}.u-button--tertiary:visited{color:#666}.u-button--tertiary:hover{border:4px solid #01324b;box-shadow:none}.u-button--tertiary:focus,.u-button--tertiary:hover{background-color:#01324b;color:#fff}.u-button--contrast{background-color:transparent;background-image:none;color:#fff;font-weight:400}.u-button--contrast:visited{color:#fff}.u-button--contrast,.u-button--contrast:focus,.u-button--contrast:hover{border:4px solid #fff}.u-button--contrast:focus,.u-button--contrast:hover{background-color:#fff;background-image:none;color:#000}.u-button--contrast:focus svg path,.u-button--contrast:hover svg path{fill:#000}.u-button--disabled,.u-button:disabled{background-color:transparent;background-image:none;border:4px solid #ccc;color:#000;cursor:default;font-weight:400;opacity:.7}.u-button--disabled svg,.u-button:disabled svg{fill:currentcolor}.u-button--disabled:visited,.u-button:disabled:visited{color:#000}.u-button--disabled:focus,.u-button--disabled:hover,.u-button:disabled:focus,.u-button:disabled:hover{border:4px solid #ccc;text-decoration:none}.u-button--disabled:focus,.u-button--disabled:hover,.u-button:disabled:focus,.u-button:disabled:hover{background-color:transparent;background-image:none;color:#000}.u-button--disabled:focus svg path,.u-button--disabled:hover svg path,.u-button:disabled:focus svg path,.u-button:disabled:hover svg path{fill:#000}.u-button--small,.u-button--xsmall{font-size:.875rem;padding:2px 8px}.u-button--small{padding:8px 16px}.u-button--large{font-size:1.125rem;padding:10px 35px}.u-button--full-width{display:flex;width:100%}.u-button--icon-left svg{margin-right:8px}.u-button--icon-right svg{margin-left:8px}.u-clear-both{clear:both}.u-container{margin:0 auto;max-width:1280px;padding:0 16px}.u-justify-content-space-between{justify-content:space-between}.u-display-none{display:none}.js .u-js-hide,.u-hide{display:none;visibility:hidden}.u-visually-hidden{clip:rect(0,0,0,0);border:0;clip-path:inset(50%);height:1px;overflow:hidden;padding:0;position:absolute!important;white-space:nowrap;width:1px}.u-icon{fill:currentcolor;display:inline-block;height:1em;transform:translate(0);vertical-align:text-top;width:1em}.u-list-reset{list-style:none;margin:0;padding:0}.u-ma-16{margin:16px}.u-mt-0{margin-top:0}.u-mt-24{margin-top:24px}.u-mt-32{margin-top:32px}.u-mb-8{margin-bottom:8px}.u-mb-32{margin-bottom:32px}.u-button-reset{background-color:transparent;border:0;padding:0}.u-sans-serif{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.u-serif{font-family:Merriweather,serif}h1,h2,h4{-webkit-font-smoothing:antialiased}p{overflow-wrap:break-word;word-break:break-word}.u-h4{font-size:1.25rem;font-weight:700;line-height:1.2}.u-mbs-0{margin-block-start:0!important}.c-article-header{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-article-identifiers{color:#6f6f6f;display:flex;flex-wrap:wrap;font-size:1rem;line-height:1.3;list-style:none;margin:0 0 8px;padding:0}.c-article-identifiers__item{border-right:1px solid #6f6f6f;list-style:none;margin-right:8px;padding-right:8px}.c-article-identifiers__item:last-child{border-right:0;margin-right:0;padding-right:0}@media only screen and (min-width:876px){.c-article-title{font-size:1.875rem;line-height:1.2}}.c-article-author-list{display:inline;font-size:1rem;list-style:none;margin:0 8px 0 0;padding:0;width:100%}.c-article-author-list__item{display:inline;padding-right:0}.c-article-author-list__show-more{display:none;margin-right:4px}.c-article-author-list__button,.js .c-article-author-list__item--hide,.js .c-article-author-list__show-more{display:none}.js .c-article-author-list--long .c-article-author-list__show-more,.js .c-article-author-list--long+.c-article-author-list__button{display:inline}@media only screen and (max-width:767px){.js .c-article-author-list__item--hide-small-screen{display:none}.js .c-article-author-list--short .c-article-author-list__show-more,.js .c-article-author-list--short+.c-article-author-list__button{display:inline}}#uptodate-client,.js .c-article-author-list--expanded .c-article-author-list__show-more{display:none!important}.js .c-article-author-list--expanded .c-article-author-list__item--hide-small-screen{display:inline!important}.c-article-author-list__button,.c-button-author-list{background:#ebf1f5;border:4px solid #ebf1f5;border-radius:20px;color:#666;font-size:.875rem;line-height:1.4;padding:2px 11px 2px 8px;text-decoration:none}.c-article-author-list__button svg,.c-button-author-list svg{margin:1px 4px 0 0}.c-article-author-list__button:hover,.c-button-author-list:hover{background:#025e8d;border-color:transparent;color:#fff}.c-article-body .c-article-access-provider{padding:8px 16px}.c-article-body .c-article-access-provider,.c-notes{border:1px solid #d5d5d5;border-image:initial;border-left:none;border-right:none;margin:24px 0}.c-article-body .c-article-access-provider__text{color:#555}.c-article-body .c-article-access-provider__text,.c-notes__text{font-size:1rem;margin-bottom:0;padding-bottom:2px;padding-top:2px;text-align:center}.c-article-body .c-article-author-affiliation__address{color:inherit;font-weight:700;margin:0}.c-article-body .c-article-author-affiliation__authors-list{list-style:none;margin:0;padding:0}.c-article-body .c-article-author-affiliation__authors-item{display:inline;margin-left:0}.c-article-authors-search{margin-bottom:24px;margin-top:0}.c-article-authors-search__item,.c-article-authors-search__title{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-article-authors-search__title{color:#626262;font-size:1.05rem;font-weight:700;margin:0;padding:0}.c-article-authors-search__item{font-size:1rem}.c-article-authors-search__text{margin:0}.c-code-block{border:1px solid #fff;font-family:monospace;margin:0 0 24px;padding:20px}.c-code-block__heading{font-weight:400;margin-bottom:16px}.c-code-block__line{display:block;overflow-wrap:break-word;white-space:pre-wrap}.c-article-share-box{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;margin-bottom:24px}.c-article-share-box__description{font-size:1rem;margin-bottom:8px}.c-article-share-box__no-sharelink-info{font-size:.813rem;font-weight:700;margin-bottom:24px;padding-top:4px}.c-article-share-box__only-read-input{border:1px solid #d5d5d5;box-sizing:content-box;display:inline-block;font-size:.875rem;font-weight:700;height:24px;margin-bottom:8px;padding:8px 10px}.c-article-share-box__additional-info{color:#626262;font-size:.813rem}.c-article-share-box__button{background:#fff;box-sizing:content-box;text-align:center}.c-article-share-box__button--link-like{background-color:transparent;border:0;color:#025e8d;cursor:pointer;font-size:.875rem;margin-bottom:8px;margin-left:10px}.c-article-associated-content__container .c-article-associated-content__collection-label{font-size:.875rem;line-height:1.4}.c-article-associated-content__container .c-article-associated-content__collection-title{line-height:1.3}.c-reading-companion{clear:both;min-height:389px}.c-reading-companion__figures-list,.c-reading-companion__references-list{list-style:none;min-height:389px;padding:0}.c-reading-companion__references-list--numeric{list-style:decimal inside}.c-reading-companion__figure-item{border-top:1px solid #d5d5d5;font-size:1rem;padding:16px 8px 16px 0}.c-reading-companion__figure-item:first-child{border-top:none;padding-top:8px}.c-reading-companion__reference-item{font-size:1rem}.c-reading-companion__reference-item:first-child{border-top:none}.c-reading-companion__reference-item a{word-break:break-word}.c-reading-companion__reference-citation{display:inline}.c-reading-companion__reference-links{font-size:.813rem;font-weight:700;list-style:none;margin:8px 0 0;padding:0;text-align:right}.c-reading-companion__reference-links>a{display:inline-block;padding-left:8px}.c-reading-companion__reference-links>a:first-child{display:inline-block;padding-left:0}.c-reading-companion__figure-title{display:block;font-size:1.25rem;font-weight:700;line-height:1.2;margin:0 0 8px}.c-reading-companion__figure-links{display:flex;justify-content:space-between;margin:8px 0 0}.c-reading-companion__figure-links>a{align-items:center;display:flex}.c-article-section__figure-caption{display:block;margin-bottom:8px;word-break:break-word}.c-article-section__figure .video,p.app-article-masthead__access--above-download{margin:0 0 16px}.c-article-section__figure-description{font-size:1rem}.c-article-section__figure-description>*{margin-bottom:0}.c-cod{display:block;font-size:1rem;width:100%}.c-cod__form{background:#ebf0f3}.c-cod__prompt{font-size:1.125rem;line-height:1.3;margin:0 0 24px}.c-cod__label{display:block;margin:0 0 4px}.c-cod__row{display:flex;margin:0 0 16px}.c-cod__row:last-child{margin:0}.c-cod__input{border:1px solid #d5d5d5;border-radius:2px;flex-shrink:0;margin:0;padding:13px}.c-cod__input--submit{background-color:#025e8d;border:1px solid #025e8d;color:#fff;flex-shrink:1;margin-left:8px;transition:background-color .2s ease-out 0s,color .2s ease-out 0s}.c-cod__input--submit-single{flex-basis:100%;flex-shrink:0;margin:0}.c-cod__input--submit:focus,.c-cod__input--submit:hover{background-color:#fff;color:#025e8d}.save-data .c-article-author-institutional-author__sub-division,.save-data .c-article-equation__number,.save-data .c-article-figure-description,.save-data .c-article-fullwidth-content,.save-data .c-article-main-column,.save-data .c-article-satellite-article-link,.save-data .c-article-satellite-subtitle,.save-data .c-article-table-container,.save-data .c-blockquote__body,.save-data .c-code-block__heading,.save-data .c-reading-companion__figure-title,.save-data .c-reading-companion__reference-citation,.save-data .c-site-messages--nature-briefing-email-variant .serif,.save-data .c-site-messages--nature-briefing-email-variant.serif,.save-data .serif,.save-data .u-serif,.save-data h1,.save-data h2,.save-data h3{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-pdf-download__link{display:flex;flex:1 1 0%;padding:13px 24px}.c-pdf-download__link:hover{text-decoration:none}@media only screen and (min-width:768px){.c-context-bar--sticky .c-pdf-download__link{align-items:center;flex:1 1 183px}}@media only screen and (max-width:320px){.c-context-bar--sticky .c-pdf-download__link{padding:16px}}.c-article-body .c-article-recommendations-list,.c-book-body .c-article-recommendations-list{display:flex;flex-direction:row;gap:16px 16px;margin:0;max-width:100%;padding:16px 0 0}.c-article-body .c-article-recommendations-list__item,.c-book-body .c-article-recommendations-list__item{flex:1 1 0%}@media only screen and (max-width:767px){.c-article-body .c-article-recommendations-list,.c-book-body .c-article-recommendations-list{flex-direction:column}}.c-article-body .c-article-recommendations-card__authors{display:none;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;line-height:1.5;margin:0 0 8px}@media only screen and (max-width:767px){.c-article-body .c-article-recommendations-card__authors{display:block;margin:0}}.c-article-body .c-article-history{margin-top:24px}.app-article-metrics-bar p{margin:0}.app-article-masthead{display:flex;flex-direction:column;gap:16px 16px;padding:16px 0 24px}.app-article-masthead__info{display:flex;flex-direction:column;flex-grow:1}.app-article-masthead__brand{border-top:1px solid hsla(0,0%,100%,.8);display:flex;flex-direction:column;flex-shrink:0;gap:8px 8px;min-height:96px;padding:16px 0 0}.app-article-masthead__brand img{border:1px solid #fff;border-radius:8px;box-shadow:0 4px 15px 0 hsla(0,0%,50%,.25);height:auto;left:0;position:absolute;width:72px}.app-article-masthead__journal-link{display:block;font-size:1.125rem;font-weight:700;margin:0 0 8px;max-width:400px;padding:0 0 0 88px;position:relative}.app-article-masthead__journal-title{-webkit-box-orient:vertical;-webkit-line-clamp:3;display:-webkit-box;overflow:hidden}.app-article-masthead__submission-link{align-items:center;display:flex;font-size:1rem;gap:4px 4px;margin:0 0 0 88px}.app-article-masthead__access{align-items:center;display:flex;flex-wrap:wrap;font-size:.875rem;font-weight:300;gap:4px 4px;margin:0}.app-article-masthead__buttons{display:flex;flex-flow:column wrap;gap:16px 16px}.app-article-masthead__access svg,.app-masthead--pastel .c-pdf-download .u-button--primary svg,.app-masthead--pastel .c-pdf-download .u-button--secondary svg,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary svg,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary svg{fill:currentcolor}.app-article-masthead a{color:#fff}.app-masthead--pastel .c-pdf-download .u-button--primary,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary{background-color:#025e8d;background-image:none;border:2px solid transparent;box-shadow:none;color:#fff;font-weight:700}.app-masthead--pastel .c-pdf-download .u-button--primary:visited,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:visited{color:#fff}.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{text-decoration:none}.app-masthead--pastel .c-pdf-download .u-button--primary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.app-masthead--pastel .c-pdf-download .u-button--primary:focus,.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{background-color:#fff;background-image:none;color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{background:0 0;border:2px solid #025e8d;box-shadow:none;color:#025e8d}.app-masthead--pastel .c-pdf-download .u-button--secondary,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary{background:0 0;border:2px solid #025e8d;color:#025e8d;font-weight:700}.app-masthead--pastel .c-pdf-download .u-button--secondary:visited,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:visited{color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--secondary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:hover{background-color:#01324b;background-color:#025e8d;border:2px solid transparent;box-shadow:none;color:#fff}.app-masthead--pastel .c-pdf-download .u-button--secondary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:focus{background-color:#fff;background-image:none;border:4px solid #fc0;color:#01324b}@media only screen and (min-width:768px){.app-article-masthead{flex-direction:row;gap:64px 64px;padding:24px 0}.app-article-masthead__brand{border:0;padding:0}.app-article-masthead__brand img{height:auto;position:static;width:auto}.app-article-masthead__buttons{align-items:center;flex-direction:row;margin-top:auto}.app-article-masthead__journal-link{display:flex;flex-direction:column;gap:24px 24px;margin:0 0 8px;padding:0}.app-article-masthead__submission-link{margin:0}}@media only screen and (min-width:1024px){.app-article-masthead__brand{flex-basis:400px}}.app-article-masthead .c-article-identifiers{font-size:.875rem;font-weight:300;line-height:1;margin:0 0 8px;overflow:hidden;padding:0}.app-article-masthead .c-article-identifiers--cite-list{margin:0 0 16px}.app-article-masthead .c-article-identifiers *{color:#fff}.app-article-masthead .c-cod{display:none}.app-article-masthead .c-article-identifiers__item{border-left:1px solid #fff;border-right:0;margin:0 17px 8px -9px;padding:0 0 0 8px}.app-article-masthead .c-article-identifiers__item--cite{border-left:0}.app-article-metrics-bar{display:flex;flex-wrap:wrap;font-size:1rem;padding:16px 0 0;row-gap:24px}.app-article-metrics-bar__item{padding:0 16px 0 0}.app-article-metrics-bar__count{font-weight:700}.app-article-metrics-bar__label{font-weight:400;padding-left:4px}.app-article-metrics-bar__icon{height:auto;margin-right:4px;margin-top:-4px;width:auto}.app-article-metrics-bar__arrow-icon{margin:4px 0 0 4px}.app-article-metrics-bar a{color:#000}.app-article-metrics-bar .app-article-metrics-bar__item--metrics{padding-right:0}.app-overview-section .c-article-author-list,.app-overview-section__authors{line-height:2}.app-article-metrics-bar{margin-top:8px}.c-book-toc-pagination+.c-book-section__back-to-top{margin-top:0}.c-article-body .c-article-access-provider__text--chapter{color:#222;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;padding:20px 0}.c-article-body .c-article-access-provider__text--chapter svg.c-status-message__icon{fill:#003f8d;vertical-align:middle}.c-article-body-section__content--separator{padding-top:40px}.c-pdf-download__link{max-height:44px}.app-article-access .u-button--primary,.app-article-access .u-button--primary:visited{color:#fff}.c-article-sidebar{display:none}@media only screen and (min-width:1024px){.c-article-sidebar{display:block}}.c-cod__form{border-radius:12px}.c-cod__label{font-size:.875rem}.c-cod .c-status-message{align-items:center;justify-content:center;margin-bottom:16px;padding-bottom:16px}@media only screen and (min-width:1024px){.c-cod .c-status-message{align-items:inherit}}.c-cod .c-status-message__icon{margin-top:4px}.c-cod .c-cod__prompt{font-size:1rem;margin-bottom:16px}.c-article-body .app-article-access,.c-book-body .app-article-access{display:block}@media only screen and (min-width:1024px){.c-article-body .app-article-access,.c-book-body .app-article-access{display:none}}.c-article-body .app-card-service{margin-bottom:32px}@media only screen and (min-width:1024px){.c-article-body .app-card-service{display:none}}.app-article-access .buybox__buy .u-button--secondary,.app-article-access .u-button--primary,.c-cod__row .u-button--primary{background-color:#025e8d;border:2px solid #025e8d;box-shadow:none;font-size:1rem;font-weight:700;gap:8px 8px;justify-content:center;line-height:1.5;padding:8px 24px}.app-article-access .buybox__buy .u-button--secondary,.app-article-access .u-button--primary:hover,.c-cod__row .u-button--primary:hover{background-color:#fff;color:#025e8d}.app-article-access .buybox__buy .u-button--secondary:hover{background-color:#025e8d;color:#fff}.buybox__buy .c-notes__text{color:#666;font-size:.875rem;padding:0 16px 8px}.c-cod__input{flex-basis:auto;width:100%}.c-article-title{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:2.25rem;font-weight:700;line-height:1.2;margin:12px 0}.c-reading-companion__figure-item figure{margin:0}@media only screen and (min-width:768px){.c-article-title{margin:16px 0}}.app-article-access{border:1px solid #c5e0f4;border-radius:12px}.app-article-access__heading{border-bottom:1px solid #c5e0f4;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1.125rem;font-weight:700;margin:0;padding:16px;text-align:center}.app-article-access .buybox__info svg{vertical-align:middle}.c-article-body .app-article-access p{margin-bottom:0}.app-article-access .buybox__info{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;margin:0}.app-article-access{margin:0 0 32px}@media only screen and (min-width:1024px){.app-article-access{margin:0 0 24px}}.c-status-message{font-size:1rem}.c-article-body{font-size:1.125rem}.c-article-body dl,.c-article-body ol,.c-article-body p,.c-article-body ul{margin-bottom:32px;margin-top:0}.c-article-access-provider__text:last-of-type,.c-article-body .c-notes__text:last-of-type{margin-bottom:0}.c-article-body ol p,.c-article-body ul p{margin-bottom:16px}.c-article-section__figure-caption{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-reading-companion__figure-item{border-top-color:#c5e0f4}.c-reading-companion__sticky{max-width:400px}.c-article-section .c-article-section__figure-description>*{font-size:1rem;margin-bottom:16px}.c-reading-companion__reference-item{border-top:1px solid #d5d5d5;padding:16px 0}.c-reading-companion__reference-item:first-child{padding-top:0}.c-article-share-box__button,.js .c-article-authors-search__item .c-article-button{background:0 0;border:2px solid #025e8d;border-radius:32px;box-shadow:none;color:#025e8d;font-size:1rem;font-weight:700;line-height:1.5;margin:0;padding:8px 24px;transition:all .2s ease 0s}.c-article-authors-search__item .c-article-button{width:100%}.c-pdf-download .u-button{background-color:#fff;border:2px solid #fff;color:#01324b;justify-content:center}.c-context-bar__container .c-pdf-download .u-button svg,.c-pdf-download .u-button svg{fill:currentcolor}.c-pdf-download .u-button:visited{color:#01324b}.c-pdf-download .u-button:hover{border:4px solid #01324b;box-shadow:none}.c-pdf-download .u-button:focus,.c-pdf-download .u-button:hover{background-color:#01324b}.c-pdf-download .u-button:focus svg path,.c-pdf-download .u-button:hover svg path{fill:#fff}.c-context-bar__container .c-pdf-download .u-button{background-image:none;border:2px solid;color:#fff}.c-context-bar__container .c-pdf-download .u-button:visited{color:#fff}.c-context-bar__container .c-pdf-download .u-button:hover{text-decoration:none}.c-context-bar__container .c-pdf-download .u-button:focus{box-shadow:none;outline:0;text-decoration:none}.c-context-bar__container .c-pdf-download .u-button:focus,.c-context-bar__container .c-pdf-download .u-button:hover{background-color:#fff;background-image:none;color:#01324b}.c-context-bar__container .c-pdf-download .u-button:focus svg path,.c-context-bar__container .c-pdf-download .u-button:hover svg path{fill:#01324b}.c-context-bar__container .c-pdf-download .u-button,.c-pdf-download .u-button{box-shadow:none;font-size:1rem;font-weight:700;line-height:1.5;padding:8px 24px}.c-context-bar__container .c-pdf-download .u-button{background-color:#025e8d}.c-pdf-download .u-button:hover{border:2px solid #fff}.c-pdf-download .u-button:focus,.c-pdf-download .u-button:hover{background:0 0;box-shadow:none;color:#fff}.c-context-bar__container .c-pdf-download .u-button:hover{border:2px solid #025e8d;box-shadow:none;color:#025e8d}.c-context-bar__container .c-pdf-download .u-button:focus,.c-pdf-download .u-button:focus{border:2px solid #025e8d}.c-article-share-box__button:focus:focus,.c-article__pill-button:focus:focus,.c-context-bar__container .c-pdf-download .u-button:focus:focus,.c-pdf-download .u-button:focus:focus{outline:3px solid #08c;will-change:transform}.c-pdf-download__link .u-icon{padding-top:0}.c-bibliographic-information__column button{margin-bottom:16px}.c-article-body .c-article-author-affiliation__list p,.c-article-body .c-article-author-information__list p,figure{margin:0}.c-article-share-box__button{margin-right:16px}.c-status-message--boxed{border-radius:12px}.c-article-associated-content__collection-title{font-size:1rem}.app-card-service__description,.c-article-body .app-card-service__description{color:#222;margin-bottom:0;margin-top:8px}.app-article-access__subscriptions a,.app-article-access__subscriptions a:visited,.app-book-series-listing__item a,.app-book-series-listing__item a:hover,.app-book-series-listing__item a:visited,.c-article-author-list a,.c-article-author-list a:visited,.c-article-buy-box a,.c-article-buy-box a:visited,.c-article-peer-review a,.c-article-peer-review a:visited,.c-article-satellite-subtitle a,.c-article-satellite-subtitle a:visited,.c-breadcrumbs__link,.c-breadcrumbs__link:hover,.c-breadcrumbs__link:visited{color:#000}.c-article-author-list svg{height:24px;margin:0 0 0 6px;width:24px}.c-article-header{margin-bottom:32px}@media only screen and (min-width:876px){.js .c-ad--conditional{display:block}}.u-lazy-ad-wrapper{background-color:#fff;display:none;min-height:149px}@media only screen and (min-width:876px){.u-lazy-ad-wrapper{display:block}}p.c-ad__label{margin-bottom:4px}.c-ad--728x90{background-color:#fff;border-bottom:2px solid #cedbe0} } </style> <style>@media only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark) { .eds-c-header__brand img{height:24px;width:203px}.app-article-masthead__journal-link img{height:93px;width:72px}@media only screen and (min-width:769px){.app-article-masthead__journal-link img{height:161px;width:122px}} } </style> <link rel="stylesheet" data-test="critical-css-handler" data-inline-css-source="critical-css" href=/oscar-static/app-springerlink/css/core-darwin-3c86549cfc.css media="print" onload="this.media='all';this.onload=null"> <link rel="stylesheet" data-test="critical-css-handler" data-inline-css-source="critical-css" href="/oscar-static/app-springerlink/css/enhanced-darwin-article-72ba046d97.css" media="print" onload="this.media='only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark)';this.onload=null"> <script type="text/javascript"> config = { env: 'live', site: '10773.springer.com', siteWithPath: '10773.springer.com' + window.location.pathname, twitterHashtag: '10773', cmsPrefix: 'https://studio-cms.springernature.com/studio/', publisherBrand: 'Springer', mustardcut: false }; </script> <script> window.dataLayer = [{"GA Key":"UA-26408784-1","DOI":"10.1007/s10773-023-05458-x","Page":"article","springerJournal":true,"Publishing Model":"Hybrid Access","page":{"attributes":{"environment":"live"}},"Country":"HK","japan":false,"doi":"10.1007-s10773-023-05458-x","Journal Id":10773,"Journal Title":"International Journal of Theoretical Physics","imprint":"Springer","Keywords":"Foundations of physics, History of science, Loschmidt paradox, Landauer’s principle, Maxwell’s demon, Second Law of Thermodynamics","kwrd":["Foundations_of_physics","History_of_science","Loschmidt_paradox","Landauer’s_principle","Maxwell’s_demon","Second_Law_of_Thermodynamics"],"Labs":"Y","ksg":"Krux.segments","kuid":"Krux.uid","Has Body":"Y","Features":[],"Open Access":"Y","hasAccess":"Y","bypassPaywall":"N","user":{"license":{"businessPartnerID":[],"businessPartnerIDString":""}},"Access Type":"open","Bpids":"","Bpnames":"","BPID":["1"],"VG Wort Identifier":"vgzm.415900-10.1007-s10773-023-05458-x","Full HTML":"Y","Subject Codes":["SCP","SCP00002","SCP19080","SCP23029","SCP19005"],"pmc":["P","P00002","P19080","P23029","P19005"],"session":{"authentication":{"loginStatus":"N"},"attributes":{"edition":"academic"}},"content":{"serial":{"eissn":"1572-9575"},"type":"Article","category":{"pmc":{"primarySubject":"Physics","primarySubjectCode":"P","secondarySubjects":{"1":"Physics, general","2":"Quantum Physics","3":"Elementary Particles, Quantum Field Theory","4":"Theoretical, Mathematical and Computational Physics"},"secondarySubjectCodes":{"1":"P00002","2":"P19080","3":"P23029","4":"P19005"}},"sucode":"SC12","articleType":"Research"},"attributes":{"deliveryPlatform":"oscar"}},"Event Category":"Article"}]; </script> <script data-test="springer-link-article-datalayer"> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ ga4MeasurementId: 'G-B3E4QL2TPR', ga360TrackingId: 'UA-26408784-1', twitterId: 'o47a7', baiduId: 'aef3043f025ccf2305af8a194652d70b', ga4ServerUrl: 'https://collect.springer.com', imprint: 'springerlink', page: { attributes:{ featureFlags: [{ name: 'darwin-orion', active: true }, { name: 'chapter-books-recs', active: true } ], darwinAvailable: true } } }); </script> <script> (function(w, d) { w.config = w.config || {}; w.config.mustardcut = false; if (w.matchMedia && w.matchMedia('only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark)').matches) { w.config.mustardcut = true; d.classList.add('js'); d.classList.remove('grade-c'); d.classList.remove('no-js'); } })(window, document.documentElement); </script> <script class="js-entry"> if (window.config.mustardcut) { (function(w, d) { window.Component = {}; window.suppressShareButton = false; window.onArticlePage = true; var currentScript = d.currentScript || d.head.querySelector('script.js-entry'); function catchNoModuleSupport() { var scriptEl = d.createElement('script'); return (!('noModule' in scriptEl) && 'onbeforeload' in scriptEl) } var headScripts = [ {'src': '/oscar-static/js/polyfill-es5-bundle-572d4fec60.js', 'async': false} ]; var bodyScripts = [ {'src': '/oscar-static/js/global-article-es5-bundle-dad1690b0d.js', 'async': false, 'module': false}, {'src': '/oscar-static/js/global-article-es6-bundle-e7d03c4cb3.js', 'async': false, 'module': true} ]; function createScript(script) { var scriptEl = d.createElement('script'); scriptEl.src = script.src; scriptEl.async = script.async; if (script.module === true) { scriptEl.type = "module"; if (catchNoModuleSupport()) { scriptEl.src = ''; } } else if (script.module === false) { scriptEl.setAttribute('nomodule', true) } if (script.charset) { scriptEl.setAttribute('charset', script.charset); } return scriptEl; } for (var i = 0; i < headScripts.length; ++i) { var scriptEl = createScript(headScripts[i]); currentScript.parentNode.insertBefore(scriptEl, currentScript.nextSibling); } d.addEventListener('DOMContentLoaded', function() { for (var i = 0; i < bodyScripts.length; ++i) { var scriptEl = createScript(bodyScripts[i]); d.body.appendChild(scriptEl); } }); // Webfont repeat view var config = w.config; if (config && config.publisherBrand && sessionStorage.fontsLoaded === 'true') { d.documentElement.className += ' webfonts-loaded'; } })(window, document); } </script> <script data-src="https://cdn.optimizely.com/js/27195530232.js" data-cc-script="C03"></script> <script data-test="gtm-head"> window.initGTM = function() { if (window.config.mustardcut) { (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-MRVXSHQ'); } } </script> <script> (function (w, d, t) { function cc() { var h = w.location.hostname; var e = d.createElement(t), s = d.getElementsByTagName(t)[0]; if (h.indexOf('springer.com') > -1 && h.indexOf('biomedcentral.com') === -1 && h.indexOf('springeropen.com') === -1) { if (h.indexOf('link-qa.springer.com') > -1 || h.indexOf('test-www.springer.com') > -1) { e.src = 'https://cmp.springer.com/production_live/en/consent-bundle-17-52.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.springer.com/production_live/en/consent-bundle-17-52.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('biomedcentral.com') > -1) { if (h.indexOf('biomedcentral.com.qa') > -1) { e.src = 'https://cmp.biomedcentral.com/production_live/en/consent-bundle-15-36.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.biomedcentral.com/production_live/en/consent-bundle-15-36.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('springeropen.com') > -1) { if (h.indexOf('springeropen.com.qa') > -1) { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-16-34.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-16-34.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('springernature.com') > -1) { if (h.indexOf('beta-qa.springernature.com') > -1) { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-49-43.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-NK22KLS')"); } else { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-49-43.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-NK22KLS')"); } } else { e.src = '/oscar-static/js/cookie-consent-es5-bundle-cb57c2c98a.js'; e.setAttribute('data-consent', h); } s.insertAdjacentElement('afterend', e); } cc(); })(window, document, 'script'); </script> <link rel="canonical" href="https://link.springer.com/article/10.1007/s10773-023-05458-x"/> <script type="application/ld+json">{"mainEntity":{"headline":"The Reversibility Paradox: Role of the Velocity Reversal Step","description":"The reversibility, or Loschmidt paradox, is a thought experiment in which microscopic reversibility is exploited to generate an apparently spontaneous trajectory in which entropy decreases, thus violating the Second Law of Thermodynamics (SLT). To achieve this, the system is allowed to evolve from a low-entropy to a high-entropy configuration, at which point all velocities of its molecules are reversed, thus producing the desired effect. In this paper we focus on the velocity reversal step. Implementing this step requires the measurement of velocities and positions, which must be stored in a memory. Even in the case of a reversible measurement, the erasure of the stored information, necessary to reset the measuring device to its original state, has an entropy cost that offsets the entropy decrease during the reverse evolution of the system. This cost is sufficient to explain why the described procedure does not violate the SLT.","datePublished":"2023-09-02T00:00:00Z","dateModified":"2023-09-02T00:00:00Z","pageStart":"1","pageEnd":"5","license":"http://creativecommons.org/licenses/by/4.0/","sameAs":"https://doi.org/10.1007/s10773-023-05458-x","keywords":["Foundations of physics","History of science","Loschmidt paradox","Landauer’s principle","Maxwell’s demon","Second Law of Thermodynamics","Physics","general","Quantum Physics","Elementary Particles","Quantum Field Theory","Theoretical","Mathematical and Computational Physics"],"image":[],"isPartOf":{"name":"International Journal of Theoretical Physics","issn":["1572-9575"],"volumeNumber":"62","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Springer US","logo":{"url":"https://www.springernature.com/app-sn/public/images/logo-springernature.png","@type":"ImageObject"},"@type":"Organization"},"author":[{"name":"P.M. Binder","url":"http://orcid.org/0000-0001-5242-6319","affiliation":[{"name":"University of Hawaii","address":{"name":"University of Hawaii, Hilo, USA","@type":"PostalAddress"},"@type":"Organization"}],"email":"pbinder@hawaii.edu","@type":"Person"}],"isAccessibleForFree":true,"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> </head> <body class="" >  <noscript> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>   <noscript data-test="gtm-body"> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>  <div class="u-visually-hidden" aria-hidden="true" data-test="darwin-icons"> <?xml version="1.0" encoding="UTF-8"?><!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><symbol id="icon-eds-i-accesses-medium" viewBox="0 0 24 24"><path d="M15.59 1a1 1 0 0 1 .706.291l5.41 5.385a1 1 0 0 1 .294.709v13.077c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742H15a1 1 0 0 1 0-2h4.455a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.8L15.178 3H5.545a.543.543 0 0 0-.538.451L5 3.538v8.607a1 1 0 0 1-2 0V3.538A2.542 2.542 0 0 1 5.545 1h10.046ZM8 13c2.052 0 4.66 1.61 6.36 3.4l.124.141c.333.41.516.925.516 1.459 0 .6-.232 1.178-.64 1.599C12.666 21.388 10.054 23 8 23c-2.052 0-4.66-1.61-6.353-3.393A2.31 2.31 0 0 1 1 18c0-.6.232-1.178.64-1.6C3.34 14.61 5.948 13 8 13Zm0 2c-1.369 0-3.552 1.348-4.917 2.785A.31.31 0 0 0 3 18c0 .083.031.161.09.222C4.447 19.652 6.631 21 8 21c1.37 0 3.556-1.35 4.917-2.785A.31.31 0 0 0 13 18a.32.32 0 0 0-.048-.17l-.042-.052C11.553 16.348 9.369 15 8 15Zm0 1a2 2 0 1 1 0 4 2 2 0 0 1 0-4Z"/></symbol><symbol id="icon-eds-i-altmetric-medium" viewBox="0 0 24 24"><path d="M12 1c5.978 0 10.843 4.77 10.996 10.712l.004.306-.002.022-.002.248C22.843 18.23 17.978 23 12 23 5.925 23 1 18.075 1 12S5.925 1 12 1Zm-1.726 9.246L8.848 12.53a1 1 0 0 1-.718.461L8.003 13l-4.947.014a9.001 9.001 0 0 0 17.887-.001L16.553 13l-2.205 3.53a1 1 0 0 1-1.735-.068l-.05-.11-2.289-6.106ZM12 3a9.001 9.001 0 0 0-8.947 8.013l4.391-.012L9.652 7.47a1 1 0 0 1 1.784.179l2.288 6.104 1.428-2.283a1 1 0 0 1 .722-.462l.129-.008 4.943.012A9.001 9.001 0 0 0 12 3Z"/></symbol><symbol id="icon-eds-i-arrow-bend-down-medium" viewBox="0 0 24 24"><path d="m11.852 20.989.058.007L12 21l.075-.003.126-.017.111-.03.111-.044.098-.052.104-.074.082-.073 6-6a1 1 0 0 0-1.414-1.414L13 17.585v-12.2C13 4.075 11.964 3 10.667 3H4a1 1 0 1 0 0 2h6.667c.175 0 .333.164.333.385v12.2l-4.293-4.292a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414l6 6c.035.036.073.068.112.097l.11.071.114.054.105.035.118.025Z"/></symbol><symbol id="icon-eds-i-arrow-bend-down-small" viewBox="0 0 16 16"><path d="M1 2a1 1 0 0 0 1 1h5v8.585L3.707 8.293a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414l5 5 .063.059.093.069.081.048.105.048.104.035.105.022.096.01h.136l.122-.018.113-.03.103-.04.1-.053.102-.07.052-.043 5.04-5.037a1 1 0 1 0-1.415-1.414L9 11.583V3a2 2 0 0 0-2-2H2a1 1 0 0 0-1 1Z"/></symbol><symbol id="icon-eds-i-arrow-bend-up-medium" viewBox="0 0 24 24"><path d="m11.852 3.011.058-.007L12 3l.075.003.126.017.111.03.111.044.098.052.104.074.082.073 6 6a1 1 0 1 1-1.414 1.414L13 6.415v12.2C13 19.925 11.964 21 10.667 21H4a1 1 0 0 1 0-2h6.667c.175 0 .333-.164.333-.385v-12.2l-4.293 4.292a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l6-6c.035-.036.073-.068.112-.097l.11-.071.114-.054.105-.035.118-.025Z"/></symbol><symbol id="icon-eds-i-arrow-bend-up-small" viewBox="0 0 16 16"><path d="M1 13.998a1 1 0 0 1 1-1h5V4.413L3.707 7.705a1 1 0 0 1-1.32.084l-.094-.084a1 1 0 0 1 0-1.414l5-5 .063-.059.093-.068.081-.05.105-.047.104-.035.105-.022L7.94 1l.136.001.122.017.113.03.103.04.1.053.102.07.052.043 5.04 5.037a1 1 0 1 1-1.415 1.414L9 4.415v8.583a2 2 0 0 1-2 2H2a1 1 0 0 1-1-1Z"/></symbol><symbol id="icon-eds-i-arrow-diagonal-medium" viewBox="0 0 24 24"><path d="M14 3h6l.075.003.126.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.054.114.035.105.03.148L21 4v6a1 1 0 0 1-2 0V6.414l-4.293 4.293a1 1 0 0 1-1.414-1.414L17.584 5H14a1 1 0 0 1-.993-.883L13 4a1 1 0 0 1 1-1ZM4 13a1 1 0 0 1 1 1v3.584l4.293-4.291a1 1 0 1 1 1.414 1.414L6.414 19H10a1 1 0 0 1 .993.883L11 20a1 1 0 0 1-1 1l-6.075-.003-.126-.017-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08a1.01 1.01 0 0 1-.097-.112l-.071-.11-.054-.114-.035-.105-.025-.118-.007-.058L3 20v-6a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-arrow-diagonal-small" viewBox="0 0 16 16"><path d="m2 15-.082-.004-.119-.016-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08a1.008 1.008 0 0 1-.097-.112l-.071-.11-.031-.062-.034-.081-.024-.076-.025-.118-.007-.058L1 14.02V9a1 1 0 1 1 2 0v2.584l2.793-2.791a1 1 0 1 1 1.414 1.414L4.414 13H7a1 1 0 0 1 .993.883L8 14a1 1 0 0 1-1 1H2ZM14 1l.081.003.12.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.031.062.034.081.024.076.03.148L15 2v5a1 1 0 0 1-2 0V4.414l-2.96 2.96A1 1 0 1 1 8.626 5.96L11.584 3H9a1 1 0 0 1-.993-.883L8 2a1 1 0 0 1 1-1h5Z"/></symbol><symbol id="icon-eds-i-arrow-down-medium" viewBox="0 0 24 24"><path d="m20.707 12.728-7.99 7.98a.996.996 0 0 1-.561.281l-.157.011a.998.998 0 0 1-.788-.384l-7.918-7.908a1 1 0 0 1 1.414-1.416L11 17.576V4a1 1 0 0 1 2 0v13.598l6.293-6.285a1 1 0 0 1 1.32-.082l.095.083a1 1 0 0 1-.001 1.414Z"/></symbol><symbol id="icon-eds-i-arrow-down-small" viewBox="0 0 16 16"><path d="m1.293 8.707 6 6 .063.059.093.069.081.048.105.049.104.034.056.013.118.017L8 15l.076-.003.122-.017.113-.03.085-.032.063-.03.098-.058.06-.043.05-.043 6.04-6.037a1 1 0 0 0-1.414-1.414L9 11.583V2a1 1 0 1 0-2 0v9.585L2.707 7.293a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414Z"/></symbol><symbol id="icon-eds-i-arrow-left-medium" viewBox="0 0 24 24"><path d="m11.272 3.293-7.98 7.99a.996.996 0 0 0-.281.561L3 12.001c0 .32.15.605.384.788l7.908 7.918a1 1 0 0 0 1.416-1.414L6.424 13H20a1 1 0 0 0 0-2H6.402l6.285-6.293a1 1 0 0 0 .082-1.32l-.083-.095a1 1 0 0 0-1.414.001Z"/></symbol><symbol id="icon-eds-i-arrow-left-small" viewBox="0 0 16 16"><path d="m7.293 1.293-6 6-.059.063-.069.093-.048.081-.049.105-.034.104-.013.056-.017.118L1 8l.003.076.017.122.03.113.032.085.03.063.058.098.043.06.043.05 6.037 6.04a1 1 0 0 0 1.414-1.414L4.417 9H14a1 1 0 0 0 0-2H4.415l4.292-4.293a1 1 0 0 0 .083-1.32l-.083-.094a1 1 0 0 0-1.414 0Z"/></symbol><symbol id="icon-eds-i-arrow-right-medium" viewBox="0 0 24 24"><path d="m12.728 3.293 7.98 7.99a.996.996 0 0 1 .281.561l.011.157c0 .32-.15.605-.384.788l-7.908 7.918a1 1 0 0 1-1.416-1.414L17.576 13H4a1 1 0 0 1 0-2h13.598l-6.285-6.293a1 1 0 0 1-.082-1.32l.083-.095a1 1 0 0 1 1.414.001Z"/></symbol><symbol id="icon-eds-i-arrow-right-small" viewBox="0 0 16 16"><path d="m8.707 1.293 6 6 .059.063.069.093.048.081.049.105.034.104.013.056.017.118L15 8l-.003.076-.017.122-.03.113-.032.085-.03.063-.058.098-.043.06-.043.05-6.037 6.04a1 1 0 0 1-1.414-1.414L11.583 9H2a1 1 0 1 1 0-2h9.585L7.293 2.707a1 1 0 0 1-.083-1.32l.083-.094a1 1 0 0 1 1.414 0Z"/></symbol><symbol id="icon-eds-i-arrow-up-medium" viewBox="0 0 24 24"><path d="m3.293 11.272 7.99-7.98a.996.996 0 0 1 .561-.281L12.001 3c.32 0 .605.15.788.384l7.918 7.908a1 1 0 0 1-1.414 1.416L13 6.424V20a1 1 0 0 1-2 0V6.402l-6.293 6.285a1 1 0 0 1-1.32.082l-.095-.083a1 1 0 0 1 .001-1.414Z"/></symbol><symbol id="icon-eds-i-arrow-up-small" viewBox="0 0 16 16"><path d="m1.293 7.293 6-6 .063-.059.093-.069.081-.048.105-.049.104-.034.056-.013.118-.017L8 1l.076.003.122.017.113.03.085.032.063.03.098.058.06.043.05.043 6.04 6.037a1 1 0 0 1-1.414 1.414L9 4.417V14a1 1 0 0 1-2 0V4.415L2.707 8.707a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414Z"/></symbol><symbol id="icon-eds-i-article-medium" viewBox="0 0 24 24"><path d="M8 7a1 1 0 0 0 0 2h4a1 1 0 1 0 0-2H8ZM8 11a1 1 0 1 0 0 2h8a1 1 0 1 0 0-2H8ZM7 16a1 1 0 0 1 1-1h8a1 1 0 1 1 0 2H8a1 1 0 0 1-1-1Z"/><path d="M5.545 1A2.542 2.542 0 0 0 3 3.538v16.924A2.542 2.542 0 0 0 5.545 23h12.91A2.542 2.542 0 0 0 21 20.462V3.5A2.5 2.5 0 0 0 18.5 1H5.545ZM5 3.538C5 3.245 5.24 3 5.545 3H18.5a.5.5 0 0 1 .5.5v16.962c0 .293-.24.538-.546.538H5.545A.542.542 0 0 1 5 20.462V3.538Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-book-medium" viewBox="0 0 24 24"><path d="M18.5 1A2.5 2.5 0 0 1 21 3.5v12c0 1.16-.79 2.135-1.86 2.418l-.14.031V21h1a1 1 0 0 1 .993.883L21 22a1 1 0 0 1-1 1H6.5A3.5 3.5 0 0 1 3 19.5v-15A3.5 3.5 0 0 1 6.5 1h12ZM17 18H6.5a1.5 1.5 0 0 0-1.493 1.356L5 19.5A1.5 1.5 0 0 0 6.5 21H17v-3Zm1.5-15h-12A1.5 1.5 0 0 0 5 4.5v11.837l.054-.025a3.481 3.481 0 0 1 1.254-.307L6.5 16h12a.5.5 0 0 0 .492-.41L19 15.5v-12a.5.5 0 0 0-.5-.5ZM15 6a1 1 0 0 1 0 2H9a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-book-series-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M1 3.786C1 2.759 1.857 2 2.82 2H6.18c.964 0 1.82.759 1.82 1.786V4h3.168c.668 0 1.298.364 1.616.938.158-.109.333-.195.523-.252l3.216-.965c.923-.277 1.962.204 2.257 1.187l4.146 13.82c.296.984-.307 1.957-1.23 2.234l-3.217.965c-.923.277-1.962-.203-2.257-1.187L13 10.005v10.21c0 1.04-.878 1.785-1.834 1.785H7.833c-.291 0-.575-.07-.83-.195A1.849 1.849 0 0 1 6.18 22H2.821C1.857 22 1 21.241 1 20.214V3.786ZM3 4v11h3V4H3Zm0 16v-3h3v3H3Zm15.075-.04-.814-2.712 2.874-.862.813 2.712-2.873.862Zm1.485-5.49-2.874.862-2.634-8.782 2.873-.862 2.635 8.782ZM8 20V6h3v14H8Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-calendar-acceptance-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-.534 7.747a1 1 0 0 1 .094 1.412l-4.846 5.538a1 1 0 0 1-1.352.141l-2.77-2.076a1 1 0 0 1 1.2-1.6l2.027 1.519 4.236-4.84a1 1 0 0 1 1.411-.094ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-date-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1ZM8 15a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm-4-4a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-decision-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-2.935 8.246 2.686 2.645c.34.335.34.883 0 1.218l-2.686 2.645a.858.858 0 0 1-1.213-.009.854.854 0 0 1 .009-1.21l1.05-1.035H7.984a.992.992 0 0 1-.984-1c0-.552.44-1 .984-1h5.928l-1.051-1.036a.854.854 0 0 1-.085-1.121l.076-.088a.858.858 0 0 1 1.213-.009ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-impact-factor-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-3.2 6.924a.48.48 0 0 1 .125.544l-1.52 3.283h2.304c.27 0 .491.215.491.483a.477.477 0 0 1-.13.327l-4.18 4.484a.498.498 0 0 1-.69.031.48.48 0 0 1-.125-.544l1.52-3.284H9.291a.487.487 0 0 1-.491-.482c0-.121.047-.238.13-.327l4.18-4.484a.498.498 0 0 1 .69-.031ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-call-papers-medium" viewBox="0 0 24 24"><g><path d="m20.707 2.883-1.414 1.414a1 1 0 0 0 1.414 1.414l1.414-1.414a1 1 0 0 0-1.414-1.414Z"/><path d="M6 16.054c0 2.026 1.052 2.943 3 2.943a1 1 0 1 1 0 2c-2.996 0-5-1.746-5-4.943v-1.227a4.068 4.068 0 0 1-1.83-1.189 4.553 4.553 0 0 1-.87-1.455 4.868 4.868 0 0 1-.3-1.686c0-1.17.417-2.298 1.17-3.14.38-.426.834-.767 1.338-1 .51-.237 1.06-.36 1.617-.36L6.632 6H7l7.932-2.895A2.363 2.363 0 0 1 18 5.36v9.28a2.36 2.36 0 0 1-3.069 2.25l.084.03L7 14.997H6v1.057Zm9.637-11.057a.415.415 0 0 0-.083.008L8 7.638v5.536l7.424 1.786.104.02c.035.01.072.02.109.02.2 0 .363-.16.363-.36V5.36c0-.2-.163-.363-.363-.363Zm-9.638 3h-.874a1.82 1.82 0 0 0-.625.111l-.15.063a2.128 2.128 0 0 0-.689.517c-.42.47-.661 1.123-.661 1.81 0 .34.06.678.176.992.114.308.28.585.485.816.4.447.925.691 1.464.691h.874v-5Z" clip-rule="evenodd"/><path d="M20 8.997h2a1 1 0 1 1 0 2h-2a1 1 0 1 1 0-2ZM20.707 14.293l1.414 1.414a1 1 0 0 1-1.414 1.414l-1.414-1.414a1 1 0 0 1 1.414-1.414Z"/></g></symbol><symbol id="icon-eds-i-card-medium" viewBox="0 0 24 24"><path d="M19.615 2c.315 0 .716.067 1.14.279.76.38 1.245 1.107 1.245 2.106v15.23c0 .315-.067.716-.279 1.14-.38.76-1.107 1.245-2.106 1.245H4.385a2.56 2.56 0 0 1-1.14-.279C2.485 21.341 2 20.614 2 19.615V4.385c0-.315.067-.716.279-1.14C2.659 2.485 3.386 2 4.385 2h15.23Zm0 2H4.385c-.213 0-.265.034-.317.14A.71.71 0 0 0 4 4.385v15.23c0 .213.034.265.14.317a.71.71 0 0 0 .245.068h15.23c.213 0 .265-.034.317-.14a.71.71 0 0 0 .068-.245V4.385c0-.213-.034-.265-.14-.317A.71.71 0 0 0 19.615 4ZM17 16a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h10Zm0-3a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h10Zm-.5-7A1.5 1.5 0 0 1 18 7.5v3a1.5 1.5 0 0 1-1.5 1.5h-9A1.5 1.5 0 0 1 6 10.5v-3A1.5 1.5 0 0 1 7.5 6h9ZM16 8H8v2h8V8Z"/></symbol><symbol id="icon-eds-i-cart-medium" viewBox="0 0 24 24"><path d="M5.76 1a1 1 0 0 1 .994.902L7.155 6h13.34c.18 0 .358.02.532.057l.174.045a2.5 2.5 0 0 1 1.693 3.103l-2.069 7.03c-.36 1.099-1.398 1.823-2.49 1.763H8.65c-1.272.015-2.352-.927-2.546-2.244L4.852 3H2a1 1 0 0 1-.993-.883L1 2a1 1 0 0 1 1-1h3.76Zm2.328 14.51a.555.555 0 0 0 .55.488l9.751.001a.533.533 0 0 0 .527-.357l2.059-7a.5.5 0 0 0-.48-.642H7.351l.737 7.51ZM18 19a2 2 0 1 1 0 4 2 2 0 0 1 0-4ZM8 19a2 2 0 1 1 0 4 2 2 0 0 1 0-4Z"/></symbol><symbol id="icon-eds-i-check-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm5.125 4.72a1 1 0 0 1 .156 1.405l-6 7.5a1 1 0 0 1-1.421.143l-3-2.5a1 1 0 0 1 1.28-1.536l2.217 1.846 5.362-6.703a1 1 0 0 1 1.406-.156Z"/></symbol><symbol id="icon-eds-i-check-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm5.125 6.72a1 1 0 0 0-1.406.155l-5.362 6.703-2.217-1.846a1 1 0 1 0-1.28 1.536l3 2.5a1 1 0 0 0 1.42-.143l6-7.5a1 1 0 0 0-.155-1.406Z"/></symbol><symbol id="icon-eds-i-chevron-down-medium" viewBox="0 0 24 24"><path d="M3.305 8.28a1 1 0 0 0-.024 1.415l7.495 7.762c.314.345.757.543 1.224.543.467 0 .91-.198 1.204-.522l7.515-7.783a1 1 0 1 0-1.438-1.39L12 15.845l-7.28-7.54A1 1 0 0 0 3.4 8.2l-.096.082Z"/></symbol><symbol id="icon-eds-i-chevron-down-small" viewBox="0 0 16 16"><path d="M13.692 5.278a1 1 0 0 1 .03 1.414L9.103 11.51a1.491 1.491 0 0 1-2.188.019L2.278 6.692a1 1 0 0 1 1.444-1.384L8 9.771l4.278-4.463a1 1 0 0 1 1.318-.111l.096.081Z"/></symbol><symbol id="icon-eds-i-chevron-left-medium" viewBox="0 0 24 24"><path d="M15.72 3.305a1 1 0 0 0-1.415-.024l-7.762 7.495A1.655 1.655 0 0 0 6 12c0 .467.198.91.522 1.204l7.783 7.515a1 1 0 1 0 1.39-1.438L8.155 12l7.54-7.28A1 1 0 0 0 15.8 3.4l-.082-.096Z"/></symbol><symbol id="icon-eds-i-chevron-left-small" viewBox="0 0 16 16"><path d="M10.722 2.308a1 1 0 0 0-1.414-.03L4.49 6.897a1.491 1.491 0 0 0-.019 2.188l4.838 4.637a1 1 0 1 0 1.384-1.444L6.229 8l4.463-4.278a1 1 0 0 0 .111-1.318l-.081-.096Z"/></symbol><symbol id="icon-eds-i-chevron-right-medium" viewBox="0 0 24 24"><path d="M8.28 3.305a1 1 0 0 1 1.415-.024l7.762 7.495c.345.314.543.757.543 1.224 0 .467-.198.91-.522 1.204l-7.783 7.515a1 1 0 1 1-1.39-1.438L15.845 12l-7.54-7.28A1 1 0 0 1 8.2 3.4l.082-.096Z"/></symbol><symbol id="icon-eds-i-chevron-right-small" viewBox="0 0 16 16"><path d="M5.278 2.308a1 1 0 0 1 1.414-.03l4.819 4.619a1.491 1.491 0 0 1 .019 2.188l-4.838 4.637a1 1 0 1 1-1.384-1.444L9.771 8 5.308 3.722a1 1 0 0 1-.111-1.318l.081-.096Z"/></symbol><symbol id="icon-eds-i-chevron-up-medium" viewBox="0 0 24 24"><path d="M20.695 15.72a1 1 0 0 0 .024-1.415l-7.495-7.762A1.655 1.655 0 0 0 12 6c-.467 0-.91.198-1.204.522l-7.515 7.783a1 1 0 1 0 1.438 1.39L12 8.155l7.28 7.54a1 1 0 0 0 1.319.106l.096-.082Z"/></symbol><symbol id="icon-eds-i-chevron-up-small" viewBox="0 0 16 16"><path d="M13.692 10.722a1 1 0 0 0 .03-1.414L9.103 4.49a1.491 1.491 0 0 0-2.188-.019L2.278 9.308a1 1 0 0 0 1.444 1.384L8 6.229l4.278 4.463a1 1 0 0 0 1.318.111l.096-.081Z"/></symbol><symbol id="icon-eds-i-citations-medium" viewBox="0 0 24 24"><path d="M15.59 1a1 1 0 0 1 .706.291l5.41 5.385a1 1 0 0 1 .294.709v13.077c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742h-5.843a1 1 0 1 1 0-2h5.843a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.8L15.178 3H5.545a.543.543 0 0 0-.538.451L5 3.538v8.607a1 1 0 0 1-2 0V3.538A2.542 2.542 0 0 1 5.545 1h10.046ZM5.483 14.35c.197.26.17.62-.049.848l-.095.083-.016.011c-.36.24-.628.45-.804.634-.393.409-.59.93-.59 1.562.077-.019.192-.028.345-.028.442 0 .84.158 1.195.474.355.316.532.716.532 1.2 0 .501-.173.9-.518 1.198-.345.298-.767.446-1.266.446-.672 0-1.209-.195-1.612-.585-.403-.39-.604-.976-.604-1.757 0-.744.11-1.39.33-1.938.222-.549.49-1.009.807-1.38a4.28 4.28 0 0 1 .992-.88c.07-.043.148-.087.232-.133a.881.881 0 0 1 1.121.245Zm5 0c.197.26.17.62-.049.848l-.095.083-.016.011c-.36.24-.628.45-.804.634-.393.409-.59.93-.59 1.562.077-.019.192-.028.345-.028.442 0 .84.158 1.195.474.355.316.532.716.532 1.2 0 .501-.173.9-.518 1.198-.345.298-.767.446-1.266.446-.672 0-1.209-.195-1.612-.585-.403-.39-.604-.976-.604-1.757 0-.744.11-1.39.33-1.938.222-.549.49-1.009.807-1.38a4.28 4.28 0 0 1 .992-.88c.07-.043.148-.087.232-.133a.881.881 0 0 1 1.121.245Z"/></symbol><symbol id="icon-eds-i-clipboard-check-medium" viewBox="0 0 24 24"><path d="M14.4 1c1.238 0 2.274.865 2.536 2.024L18.5 3C19.886 3 21 4.14 21 5.535v14.93C21 21.86 19.886 23 18.5 23h-13C4.114 23 3 21.86 3 20.465V5.535C3 4.14 4.114 3 5.5 3h1.57c.27-1.147 1.3-2 2.53-2h4.8Zm4.115 4-1.59.024A2.601 2.601 0 0 1 14.4 7H9.6c-1.23 0-2.26-.853-2.53-2H5.5c-.27 0-.5.234-.5.535v14.93c0 .3.23.535.5.535h13c.27 0 .5-.234.5-.535V5.535c0-.3-.23-.535-.485-.535Zm-1.909 4.205a1 1 0 0 1 .19 1.401l-5.334 7a1 1 0 0 1-1.344.23l-2.667-1.75a1 1 0 1 1 1.098-1.672l1.887 1.238 4.769-6.258a1 1 0 0 1 1.401-.19ZM14.4 3H9.6a.6.6 0 0 0-.6.6v.8a.6.6 0 0 0 .6.6h4.8a.6.6 0 0 0 .6-.6v-.8a.6.6 0 0 0-.6-.6Z"/></symbol><symbol id="icon-eds-i-clipboard-report-medium" viewBox="0 0 24 24"><path d="M14.4 1c1.238 0 2.274.865 2.536 2.024L18.5 3C19.886 3 21 4.14 21 5.535v14.93C21 21.86 19.886 23 18.5 23h-13C4.114 23 3 21.86 3 20.465V5.535C3 4.14 4.114 3 5.5 3h1.57c.27-1.147 1.3-2 2.53-2h4.8Zm4.115 4-1.59.024A2.601 2.601 0 0 1 14.4 7H9.6c-1.23 0-2.26-.853-2.53-2H5.5c-.27 0-.5.234-.5.535v14.93c0 .3.23.535.5.535h13c.27 0 .5-.234.5-.535V5.535c0-.3-.23-.535-.485-.535Zm-2.658 10.929a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h7.857Zm0-3.929a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h7.857ZM14.4 3H9.6a.6.6 0 0 0-.6.6v.8a.6.6 0 0 0 .6.6h4.8a.6.6 0 0 0 .6-.6v-.8a.6.6 0 0 0-.6-.6Z"/></symbol><symbol id="icon-eds-i-close-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18ZM8.707 7.293 12 10.585l3.293-3.292a1 1 0 0 1 1.414 1.414L13.415 12l3.292 3.293a1 1 0 0 1-1.414 1.414L12 13.415l-3.293 3.292a1 1 0 1 1-1.414-1.414L10.585 12 7.293 8.707a1 1 0 0 1 1.414-1.414Z"/></symbol><symbol id="icon-eds-i-cloud-upload-medium" viewBox="0 0 24 24"><path d="m12.852 10.011.028-.004L13 10l.075.003.126.017.086.022.136.052.098.052.104.074.082.073 3 3a1 1 0 0 1 0 1.414l-.094.083a1 1 0 0 1-1.32-.083L14 13.416V20a1 1 0 0 1-2 0v-6.586l-1.293 1.293a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l3-3 .112-.097.11-.071.114-.054.105-.035.118-.025Zm.587-7.962c3.065.362 5.497 2.662 5.992 5.562l.013.085.207.073c2.117.782 3.496 2.845 3.337 5.097l-.022.226c-.297 2.561-2.503 4.491-5.124 4.502a1 1 0 1 1-.009-2c1.619-.007 2.967-1.186 3.147-2.733.179-1.542-.86-2.979-2.487-3.353-.512-.149-.894-.579-.981-1.165-.21-2.237-2-4.035-4.308-4.308-2.31-.273-4.497 1.06-5.25 3.19l-.049.113c-.234.468-.718.756-1.176.743-1.418.057-2.689.857-3.32 2.084a3.668 3.668 0 0 0 .262 3.798c.796 1.136 2.169 1.764 3.583 1.635a1 1 0 1 1 .182 1.992c-2.125.194-4.193-.753-5.403-2.48a5.668 5.668 0 0 1-.403-5.86c.85-1.652 2.449-2.79 4.323-3.092l.287-.039.013-.028c1.207-2.741 4.125-4.404 7.186-4.042Z"/></symbol><symbol id="icon-eds-i-collection-medium" viewBox="0 0 24 24"><path d="M21 7a1 1 0 0 1 1 1v12.5a2.5 2.5 0 0 1-2.5 2.5H8a1 1 0 0 1 0-2h11.5a.5.5 0 0 0 .5-.5V8a1 1 0 0 1 1-1Zm-5.5-5A2.5 2.5 0 0 1 18 4.5v12a2.5 2.5 0 0 1-2.5 2.5h-11A2.5 2.5 0 0 1 2 16.5v-12A2.5 2.5 0 0 1 4.5 2h11Zm0 2h-11a.5.5 0 0 0-.5.5v12a.5.5 0 0 0 .5.5h11a.5.5 0 0 0 .5-.5v-12a.5.5 0 0 0-.5-.5ZM13 13a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h6Zm0-3.5a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h6ZM13 6a1 1 0 0 1 0 2H7a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-conference-series-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M4.5 2A2.5 2.5 0 0 0 2 4.5v11A2.5 2.5 0 0 0 4.5 18h2.37l-2.534 2.253a1 1 0 0 0 1.328 1.494L9.88 18H11v3a1 1 0 1 0 2 0v-3h1.12l4.216 3.747a1 1 0 0 0 1.328-1.494L17.13 18h2.37a2.5 2.5 0 0 0 2.5-2.5v-11A2.5 2.5 0 0 0 19.5 2h-15ZM20 6V4.5a.5.5 0 0 0-.5-.5h-15a.5.5 0 0 0-.5.5V6h16ZM4 8v7.5a.5.5 0 0 0 .5.5h15a.5.5 0 0 0 .5-.5V8H4Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-delivery-medium" viewBox="0 0 24 24"><path d="M8.51 20.598a3.037 3.037 0 0 1-3.02 0A2.968 2.968 0 0 1 4.161 19L3.5 19A2.5 2.5 0 0 1 1 16.5v-11A2.5 2.5 0 0 1 3.5 3h10a2.5 2.5 0 0 1 2.45 2.004L16 5h2.527c.976 0 1.855.585 2.27 1.49l2.112 4.62a1 1 0 0 1 .091.416v4.856C23 17.814 21.889 19 20.484 19h-.523a1.01 1.01 0 0 1-.121-.007 2.96 2.96 0 0 1-1.33 1.605 3.037 3.037 0 0 1-3.02 0A2.968 2.968 0 0 1 14.161 19H9.838a2.968 2.968 0 0 1-1.327 1.597Zm-2.024-3.462a.955.955 0 0 0-.481.73L5.999 18l.001.022a.944.944 0 0 0 .388.777l.098.065c.316.181.712.181 1.028 0A.97.97 0 0 0 8 17.978a.95.95 0 0 0-.486-.842 1.037 1.037 0 0 0-1.028 0Zm10 0a.955.955 0 0 0-.481.73l-.005.156a.944.944 0 0 0 .388.777l.098.065c.316.181.712.181 1.028 0a.97.97 0 0 0 .486-.886.95.95 0 0 0-.486-.842 1.037 1.037 0 0 0-1.028 0ZM21 12h-5v3.17a3.038 3.038 0 0 1 2.51.232 2.993 2.993 0 0 1 1.277 1.45l.058.155.058-.005.581-.002c.27 0 .516-.263.516-.618V12Zm-7.5-7h-10a.5.5 0 0 0-.5.5v11a.5.5 0 0 0 .5.5h.662a2.964 2.964 0 0 1 1.155-1.491l.172-.107a3.037 3.037 0 0 1 3.022 0A2.987 2.987 0 0 1 9.843 17H13.5a.5.5 0 0 0 .5-.5v-11a.5.5 0 0 0-.5-.5Zm5.027 2H16v3h4.203l-1.224-2.677a.532.532 0 0 0-.375-.316L18.527 7Z"/></symbol><symbol id="icon-eds-i-download-medium" viewBox="0 0 24 24"><path d="M22 18.5a3.5 3.5 0 0 1-3.5 3.5h-13A3.5 3.5 0 0 1 2 18.5V18a1 1 0 0 1 2 0v.5A1.5 1.5 0 0 0 5.5 20h13a1.5 1.5 0 0 0 1.5-1.5V18a1 1 0 0 1 2 0v.5Zm-3.293-7.793-6 6-.063.059-.093.069-.081.048-.105.049-.104.034-.056.013-.118.017L12 17l-.076-.003-.122-.017-.113-.03-.085-.032-.063-.03-.098-.058-.06-.043-.05-.043-6.04-6.037a1 1 0 0 1 1.414-1.414l4.294 4.29L11 3a1 1 0 0 1 2 0l.001 10.585 4.292-4.292a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414Z"/></symbol><symbol id="icon-eds-i-edit-medium" viewBox="0 0 24 24"><path d="M17.149 2a2.38 2.38 0 0 1 1.699.711l2.446 2.46a2.384 2.384 0 0 1 .005 3.38L10.01 19.906a1 1 0 0 1-.434.257l-6.3 1.8a1 1 0 0 1-1.237-1.237l1.8-6.3a1 1 0 0 1 .257-.434L15.443 2.718A2.385 2.385 0 0 1 17.15 2Zm-3.874 5.689-7.586 7.536-1.234 4.319 4.318-1.234 7.54-7.582-3.038-3.039ZM17.149 4a.395.395 0 0 0-.286.126L14.695 6.28l3.029 3.029 2.162-2.173a.384.384 0 0 0 .106-.197L20 6.864c0-.103-.04-.2-.119-.278l-2.457-2.47A.385.385 0 0 0 17.149 4Z"/></symbol><symbol id="icon-eds-i-education-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M12.41 2.088a1 1 0 0 0-.82 0l-10 4.5a1 1 0 0 0 0 1.824L3 9.047v7.124A3.001 3.001 0 0 0 4 22a3 3 0 0 0 1-5.83V9.948l1 .45V14.5a1 1 0 0 0 .087.408L7 14.5c-.913.408-.912.41-.912.41l.001.003.003.006.007.015a1.988 1.988 0 0 0 .083.16c.054.097.131.225.236.373.21.297.53.68.993 1.057C8.351 17.292 9.824 18 12 18c2.176 0 3.65-.707 4.589-1.476.463-.378.783-.76.993-1.057a4.162 4.162 0 0 0 .319-.533l.007-.015.003-.006v-.003h.002s0-.002-.913-.41l.913.408A1 1 0 0 0 18 14.5v-4.103l4.41-1.985a1 1 0 0 0 0-1.824l-10-4.5ZM16 11.297l-3.59 1.615a1 1 0 0 1-.82 0L8 11.297v2.94a3.388 3.388 0 0 0 .677.739C9.267 15.457 10.294 16 12 16s2.734-.543 3.323-1.024a3.388 3.388 0 0 0 .677-.739v-2.94ZM4.437 7.5 12 4.097 19.563 7.5 12 10.903 4.437 7.5ZM3 19a1 1 0 1 1 2 0 1 1 0 0 1-2 0Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-error-diamond-medium" viewBox="0 0 24 24"><path d="M12.002 1c.702 0 1.375.279 1.871.775l8.35 8.353a2.646 2.646 0 0 1 .001 3.744l-8.353 8.353a2.646 2.646 0 0 1-3.742 0l-8.353-8.353a2.646 2.646 0 0 1 0-3.744l8.353-8.353.156-.142c.424-.362.952-.58 1.507-.625l.21-.008Zm0 2a.646.646 0 0 0-.38.123l-.093.08-8.34 8.34a.646.646 0 0 0-.18.355L3 12c0 .171.068.336.19.457l8.353 8.354a.646.646 0 0 0 .914 0l8.354-8.354a.646.646 0 0 0-.001-.914l-8.351-8.354A.646.646 0 0 0 12.002 3ZM12 14.5a1.5 1.5 0 0 1 .144 2.993L12 17.5a1.5 1.5 0 0 1 0-3ZM12 6a1 1 0 0 1 1 1v5a1 1 0 0 1-2 0V7a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-error-filled-medium" viewBox="0 0 24 24"><path d="M12.002 1c.702 0 1.375.279 1.871.775l8.35 8.353a2.646 2.646 0 0 1 .001 3.744l-8.353 8.353a2.646 2.646 0 0 1-3.742 0l-8.353-8.353a2.646 2.646 0 0 1 0-3.744l8.353-8.353.156-.142c.424-.362.952-.58 1.507-.625l.21-.008ZM12 14.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 14.5ZM12 6a1 1 0 0 0-1 1v5a1 1 0 0 0 2 0V7a1 1 0 0 0-1-1Z"/></symbol><symbol id="icon-eds-i-external-link-medium" viewBox="0 0 24 24"><path d="M9 2a1 1 0 1 1 0 2H4.6c-.371 0-.6.209-.6.5v15c0 .291.229.5.6.5h14.8c.371 0 .6-.209.6-.5V15a1 1 0 0 1 2 0v4.5c0 1.438-1.162 2.5-2.6 2.5H4.6C3.162 22 2 20.938 2 19.5v-15C2 3.062 3.162 2 4.6 2H9Zm6 0h6l.075.003.126.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.054.114.035.105.03.148L22 3v6a1 1 0 0 1-2 0V5.414l-6.693 6.693a1 1 0 0 1-1.414-1.414L18.584 4H15a1 1 0 0 1-.993-.883L14 3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-external-link-small" viewBox="0 0 16 16"><path d="M5 1a1 1 0 1 1 0 2l-2-.001V13L13 13v-2a1 1 0 0 1 2 0v2c0 1.15-.93 2-2.067 2H3.067C1.93 15 1 14.15 1 13V3c0-1.15.93-2 2.067-2H5Zm4 0h5l.075.003.126.017.111.03.111.044.098.052.096.067.09.08.044.047.073.093.051.083.054.113.035.105.03.148L15 2v5a1 1 0 0 1-2 0V4.414L9.107 8.307a1 1 0 0 1-1.414-1.414L11.584 3H9a1 1 0 0 1-.993-.883L8 2a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-file-download-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3ZM12 7a1 1 0 0 1 1 1v6.585l2.293-2.292a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414l-4 4a1.008 1.008 0 0 1-.112.097l-.11.071-.114.054-.105.035-.149.03L12 18l-.075-.003-.126-.017-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08-4-4a1 1 0 0 1 1.414-1.414L11 14.585V8a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-file-report-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742H5.545c-.674 0-1.32-.267-1.798-.742A2.535 2.535 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .142.057.278.158.379.102.102.242.159.387.159h12.91a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.915L14.085 3ZM16 17a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm0-3a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm-4.793-6.207L13 9.585l1.793-1.792a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414l-2.5 2.5a1 1 0 0 1-1.414 0L10.5 9.915l-1.793 1.792a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l2.5-2.5a1 1 0 0 1 1.414 0Z"/></symbol><symbol id="icon-eds-i-file-text-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3ZM16 15a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm0-4a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm-5-4a1 1 0 0 1 0 2H8a1 1 0 1 1 0-2h3Z"/></symbol><symbol id="icon-eds-i-file-upload-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3Zm-2.233 4.011.058-.007L12 7l.075.003.126.017.111.03.111.044.098.052.104.074.082.073 4 4a1 1 0 0 1 0 1.414l-.094.083a1 1 0 0 1-1.32-.083L13 10.415V17a1 1 0 0 1-2 0v-6.585l-2.293 2.292a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l4-4 .112-.097.11-.071.114-.054.105-.035.118-.025Z"/></symbol><symbol id="icon-eds-i-filter-medium" viewBox="0 0 24 24"><path d="M21 2a1 1 0 0 1 .82 1.573L15 13.314V18a1 1 0 0 1-.31.724l-.09.076-4 3A1 1 0 0 1 9 21v-7.684L2.18 3.573a1 1 0 0 1 .707-1.567L3 2h18Zm-1.921 2H4.92l5.9 8.427a1 1 0 0 1 .172.45L11 13v6l2-1.5V13a1 1 0 0 1 .117-.469l.064-.104L19.079 4Z"/></symbol><symbol id="icon-eds-i-funding-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M23 8A7 7 0 1 0 9 8a7 7 0 0 0 14 0ZM9.006 12.225A4.07 4.07 0 0 0 6.12 11.02H2a.979.979 0 1 0 0 1.958h4.12c.558 0 1.094.222 1.489.617l2.207 2.288c.27.27.27.687.012.944a.656.656 0 0 1-.928 0L7.744 15.67a.98.98 0 0 0-1.386 1.384l1.157 1.158c.535.536 1.244.791 1.946.765l.041.002h6.922c.874 0 1.597.748 1.597 1.688 0 .203-.146.354-.309.354H7.755c-.487 0-.96-.178-1.339-.504L2.64 17.259a.979.979 0 0 0-1.28 1.482L5.137 22c.733.631 1.66.979 2.618.979h9.957c1.26 0 2.267-1.043 2.267-2.312 0-2.006-1.584-3.646-3.555-3.646h-4.529a2.617 2.617 0 0 0-.681-2.509l-2.208-2.287ZM16 3a5 5 0 1 0 0 10 5 5 0 0 0 0-10Zm.979 3.5a.979.979 0 1 0-1.958 0v3a.979.979 0 1 0 1.958 0v-3Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-hashtag-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18ZM9.52 18.189a1 1 0 1 1-1.964-.378l.437-2.274H6a1 1 0 1 1 0-2h2.378l.592-3.076H6a1 1 0 0 1 0-2h3.354l.51-2.65a1 1 0 1 1 1.964.378l-.437 2.272h3.04l.51-2.65a1 1 0 1 1 1.964.378l-.438 2.272H18a1 1 0 0 1 0 2h-1.917l-.592 3.076H18a1 1 0 0 1 0 2h-2.893l-.51 2.652a1 1 0 1 1-1.964-.378l.437-2.274h-3.04l-.51 2.652Zm.895-4.652h3.04l.591-3.076h-3.04l-.591 3.076Z"/></symbol><symbol id="icon-eds-i-home-medium" viewBox="0 0 24 24"><path d="M5 22a1 1 0 0 1-1-1v-8.586l-1.293 1.293a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l10-10a1 1 0 0 1 1.414 0l10 10a1 1 0 0 1-1.414 1.414L20 12.415V21a1 1 0 0 1-1 1H5Zm7-17.585-6 5.999V20h5v-4a1 1 0 0 1 2 0v4h5v-9.585l-6-6Z"/></symbol><symbol id="icon-eds-i-image-medium" viewBox="0 0 24 24"><path d="M19.615 2A2.385 2.385 0 0 1 22 4.385v15.23A2.385 2.385 0 0 1 19.615 22H4.385A2.385 2.385 0 0 1 2 19.615V4.385A2.385 2.385 0 0 1 4.385 2h15.23Zm0 2H4.385A.385.385 0 0 0 4 4.385v15.23c0 .213.172.385.385.385h1.244l10.228-8.76a1 1 0 0 1 1.254-.037L20 13.392V4.385A.385.385 0 0 0 19.615 4Zm-3.07 9.283L8.703 20h10.912a.385.385 0 0 0 .385-.385v-3.713l-3.455-2.619ZM9.5 6a3.5 3.5 0 1 1 0 7 3.5 3.5 0 0 1 0-7Zm0 2a1.5 1.5 0 1 0 0 3 1.5 1.5 0 0 0 0-3Z"/></symbol><symbol id="icon-eds-i-impact-factor-medium" viewBox="0 0 24 24"><path d="M16.49 2.672c.74.694.986 1.765.632 2.712l-.04.1-1.549 3.54h1.477a2.496 2.496 0 0 1 2.485 2.34l.005.163c0 .618-.23 1.21-.642 1.675l-7.147 7.961a2.48 2.48 0 0 1-3.554.165 2.512 2.512 0 0 1-.633-2.712l.042-.103L9.108 15H7.46c-1.393 0-2.379-1.11-2.455-2.369L5 12.473c0-.593.142-1.145.628-1.692l7.307-7.944a2.48 2.48 0 0 1 3.555-.165ZM14.43 4.164l-7.33 7.97c-.083.093-.101.214-.101.34 0 .277.19.526.46.526h4.163l.097-.009c.015 0 .03.003.046.009.181.078.264.32.186.5l-2.554 5.817a.512.512 0 0 0 .127.552.48.48 0 0 0 .69-.033l7.155-7.97a.513.513 0 0 0 .13-.34.497.497 0 0 0-.49-.502h-3.988a.355.355 0 0 1-.328-.497l2.555-5.844a.512.512 0 0 0-.127-.552.48.48 0 0 0-.69.033Z"/></symbol><symbol id="icon-eds-i-info-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm0 7a1 1 0 0 1 1 1v5h1.5a1 1 0 0 1 0 2h-5a1 1 0 0 1 0-2H11v-4h-.5a1 1 0 0 1-.993-.883L9.5 11a1 1 0 0 1 1-1H12Zm0-4.5a1.5 1.5 0 0 1 .144 2.993L12 8.5a1.5 1.5 0 0 1 0-3Z"/></symbol><symbol id="icon-eds-i-info-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 9h-1.5a1 1 0 0 0-1 1l.007.117A1 1 0 0 0 10.5 12h.5v4H9.5a1 1 0 0 0 0 2h5a1 1 0 0 0 0-2H13v-5a1 1 0 0 0-1-1Zm0-4.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 5.5Z"/></symbol><symbol id="icon-eds-i-journal-medium" viewBox="0 0 24 24"><path d="M18.5 1A2.5 2.5 0 0 1 21 3.5v14a2.5 2.5 0 0 1-2.5 2.5h-13a.5.5 0 1 0 0 1H20a1 1 0 0 1 0 2H5.5A2.5 2.5 0 0 1 3 20.5v-17A2.5 2.5 0 0 1 5.5 1h13ZM7 3H5.5a.5.5 0 0 0-.5.5v14.549l.016-.002c.104-.02.211-.035.32-.042L5.5 18H7V3Zm11.5 0H9v15h9.5a.5.5 0 0 0 .5-.5v-14a.5.5 0 0 0-.5-.5ZM16 5a1 1 0 0 1 1 1v4a1 1 0 0 1-1 1h-5a1 1 0 0 1-1-1V6a1 1 0 0 1 1-1h5Zm-1 2h-3v2h3V7Z"/></symbol><symbol id="icon-eds-i-mail-medium" viewBox="0 0 24 24"><path d="M20.462 3C21.875 3 23 4.184 23 5.619v12.762C23 19.816 21.875 21 20.462 21H3.538C2.125 21 1 19.816 1 18.381V5.619C1 4.184 2.125 3 3.538 3h16.924ZM21 8.158l-7.378 6.258a2.549 2.549 0 0 1-3.253-.008L3 8.16v10.222c0 .353.253.619.538.619h16.924c.285 0 .538-.266.538-.619V8.158ZM20.462 5H3.538c-.264 0-.5.228-.534.542l8.65 7.334c.2.165.492.165.684.007l8.656-7.342-.001-.025c-.044-.3-.274-.516-.531-.516Z"/></symbol><symbol id="icon-eds-i-mail-send-medium" viewBox="0 0 24 24"><path d="M20.444 5a2.562 2.562 0 0 1 2.548 2.37l.007.078.001.123v7.858A2.564 2.564 0 0 1 20.444 18H9.556A2.564 2.564 0 0 1 7 15.429l.001-7.977.007-.082A2.561 2.561 0 0 1 9.556 5h10.888ZM21 9.331l-5.46 3.51a1 1 0 0 1-1.08 0L9 9.332v6.097c0 .317.251.571.556.571h10.888a.564.564 0 0 0 .556-.571V9.33ZM20.444 7H9.556a.543.543 0 0 0-.32.105l5.763 3.706 5.766-3.706a.543.543 0 0 0-.32-.105ZM4.308 5a1 1 0 1 1 0 2H2a1 1 0 1 1 0-2h2.308Zm0 5.5a1 1 0 0 1 0 2H2a1 1 0 0 1 0-2h2.308Zm0 5.5a1 1 0 0 1 0 2H2a1 1 0 0 1 0-2h2.308Z"/></symbol><symbol id="icon-eds-i-mentions-medium" viewBox="0 0 24 24"><path d="m9.452 1.293 5.92 5.92 2.92-2.92a1 1 0 0 1 1.415 1.414l-2.92 2.92 5.92 5.92a1 1 0 0 1 0 1.415 10.371 10.371 0 0 1-10.378 2.584l.652 3.258A1 1 0 0 1 12 23H2a1 1 0 0 1-.874-1.486l4.789-8.62C4.194 9.074 4.9 4.43 8.038 1.292a1 1 0 0 1 1.414 0Zm-2.355 13.59L3.699 21h7.081l-.689-3.442a10.392 10.392 0 0 1-2.775-2.396l-.22-.28Zm1.69-11.427-.07.09a8.374 8.374 0 0 0 11.737 11.737l.089-.071L8.787 3.456Z"/></symbol><symbol id="icon-eds-i-menu-medium" viewBox="0 0 24 24"><path d="M21 4a1 1 0 0 1 0 2H3a1 1 0 1 1 0-2h18Zm-4 7a1 1 0 0 1 0 2H3a1 1 0 0 1 0-2h14Zm4 7a1 1 0 0 1 0 2H3a1 1 0 0 1 0-2h18Z"/></symbol><symbol id="icon-eds-i-metrics-medium" viewBox="0 0 24 24"><path d="M3 22a1 1 0 0 1-1-1V3a1 1 0 0 1 1-1h6a1 1 0 0 1 1 1v7h4V8a1 1 0 0 1 1-1h6a1 1 0 0 1 1 1v13a1 1 0 0 1-.883.993L21 22H3Zm17-2V9h-4v11h4Zm-6-8h-4v8h4v-8ZM8 4H4v16h4V4Z"/></symbol><symbol id="icon-eds-i-news-medium" viewBox="0 0 24 24"><path d="M17.384 3c.975 0 1.77.787 1.77 1.762v13.333c0 .462.354.846.815.899l.107.006.109-.006a.915.915 0 0 0 .809-.794l.006-.105V8.19a1 1 0 0 1 2 0v9.905A2.914 2.914 0 0 1 20.077 21H3.538a2.547 2.547 0 0 1-1.644-.601l-.147-.135A2.516 2.516 0 0 1 1 18.476V4.762C1 3.787 1.794 3 2.77 3h14.614Zm-.231 2H3v13.476c0 .11.035.216.1.304l.054.063c.101.1.24.157.384.157l13.761-.001-.026-.078a2.88 2.88 0 0 1-.115-.655l-.004-.17L17.153 5ZM14 15.021a.979.979 0 1 1 0 1.958H6a.979.979 0 1 1 0-1.958h8Zm0-8c.54 0 .979.438.979.979v4c0 .54-.438.979-.979.979H6A.979.979 0 0 1 5.021 12V8c0-.54.438-.979.979-.979h8Zm-.98 1.958H6.979v2.041h6.041V8.979Z"/></symbol><symbol id="icon-eds-i-newsletter-medium" viewBox="0 0 24 24"><path d="M21 10a1 1 0 0 1 1 1v9.5a2.5 2.5 0 0 1-2.5 2.5h-15A2.5 2.5 0 0 1 2 20.5V11a1 1 0 0 1 2 0v.439l8 4.888 8-4.889V11a1 1 0 0 1 1-1Zm-1 3.783-7.479 4.57a1 1 0 0 1-1.042 0l-7.48-4.57V20.5a.5.5 0 0 0 .501.5h15a.5.5 0 0 0 .5-.5v-6.717ZM15 9a1 1 0 0 1 0 2H9a1 1 0 0 1 0-2h6Zm2.5-8A2.5 2.5 0 0 1 20 3.5V9a1 1 0 0 1-2 0V3.5a.5.5 0 0 0-.5-.5h-11a.5.5 0 0 0-.5.5V9a1 1 0 1 1-2 0V3.5A2.5 2.5 0 0 1 6.5 1h11ZM15 5a1 1 0 0 1 0 2H9a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-notifcation-medium" viewBox="0 0 24 24"><path d="M14 20a1 1 0 0 1 0 2h-4a1 1 0 0 1 0-2h4ZM3 18l-.133-.007c-1.156-.124-1.156-1.862 0-1.986l.3-.012C4.32 15.923 5 15.107 5 14V9.5C5 5.368 8.014 2 12 2s7 3.368 7 7.5V14c0 1.107.68 1.923 1.832 1.995l.301.012c1.156.124 1.156 1.862 0 1.986L21 18H3Zm9-14C9.17 4 7 6.426 7 9.5V14c0 .671-.146 1.303-.416 1.858L6.51 16h10.979l-.073-.142a4.192 4.192 0 0 1-.412-1.658L17 14V9.5C17 6.426 14.83 4 12 4Z"/></symbol><symbol id="icon-eds-i-publish-medium" viewBox="0 0 24 24"><g><path d="M16.296 1.291A1 1 0 0 0 15.591 1H5.545A2.542 2.542 0 0 0 3 3.538V13a1 1 0 1 0 2 0V3.538l.007-.087A.543.543 0 0 1 5.545 3h9.633L20 7.8v12.662a.534.534 0 0 1-.158.379.548.548 0 0 1-.387.159H11a1 1 0 1 0 0 2h8.455c.674 0 1.32-.267 1.798-.742A2.534 2.534 0 0 0 22 20.462V7.385a1 1 0 0 0-.294-.709l-5.41-5.385Z"/><path d="M10.762 16.647a1 1 0 0 0-1.525-1.294l-4.472 5.271-2.153-1.665a1 1 0 1 0-1.224 1.582l2.91 2.25a1 1 0 0 0 1.374-.144l5.09-6ZM16 10a1 1 0 1 1 0 2H8a1 1 0 1 1 0-2h8ZM12 7a1 1 0 0 0-1-1H8a1 1 0 1 0 0 2h3a1 1 0 0 0 1-1Z"/></g></symbol><symbol id="icon-eds-i-refresh-medium" viewBox="0 0 24 24"><g><path d="M7.831 5.636H6.032A8.76 8.76 0 0 1 9 3.631 8.549 8.549 0 0 1 12.232 3c.603 0 1.192.063 1.76.182C17.979 4.017 21 7.632 21 12a1 1 0 1 0 2 0c0-5.296-3.674-9.746-8.591-10.776A10.61 10.61 0 0 0 5 3.851V2.805a1 1 0 0 0-.987-1H4a1 1 0 0 0-1 1v3.831a1 1 0 0 0 1 1h3.831a1 1 0 0 0 .013-2h-.013ZM17.968 18.364c-1.59 1.632-3.784 2.636-6.2 2.636C6.948 21 3 16.993 3 12a1 1 0 1 0-2 0c0 6.053 4.799 11 10.768 11 2.788 0 5.324-1.082 7.232-2.85v1.045a1 1 0 1 0 2 0v-3.831a1 1 0 0 0-1-1h-3.831a1 1 0 0 0 0 2h1.799Z"/></g></symbol><symbol id="icon-eds-i-search-medium" viewBox="0 0 24 24"><path d="M11 1c5.523 0 10 4.477 10 10 0 2.4-.846 4.604-2.256 6.328l3.963 3.965a1 1 0 0 1-1.414 1.414l-3.965-3.963A9.959 9.959 0 0 1 11 21C5.477 21 1 16.523 1 11S5.477 1 11 1Zm0 2a8 8 0 1 0 0 16 8 8 0 0 0 0-16Z"/></symbol><symbol id="icon-eds-i-settings-medium" viewBox="0 0 24 24"><path d="M11.382 1h1.24a2.508 2.508 0 0 1 2.334 1.63l.523 1.378 1.59.933 1.444-.224c.954-.132 1.89.3 2.422 1.101l.095.155.598 1.066a2.56 2.56 0 0 1-.195 2.848l-.894 1.161v1.896l.92 1.163c.6.768.707 1.812.295 2.674l-.09.17-.606 1.08a2.504 2.504 0 0 1-2.531 1.25l-1.428-.223-1.589.932-.523 1.378a2.512 2.512 0 0 1-2.155 1.625L12.65 23h-1.27a2.508 2.508 0 0 1-2.334-1.63l-.524-1.379-1.59-.933-1.443.225c-.954.132-1.89-.3-2.422-1.101l-.095-.155-.598-1.066a2.56 2.56 0 0 1 .195-2.847l.891-1.161v-1.898l-.919-1.162a2.562 2.562 0 0 1-.295-2.674l.09-.17.606-1.08a2.504 2.504 0 0 1 2.531-1.25l1.43.223 1.618-.938.524-1.375.07-.167A2.507 2.507 0 0 1 11.382 1Zm.003 2a.509.509 0 0 0-.47.338l-.65 1.71a1 1 0 0 1-.434.51L7.6 6.85a1 1 0 0 1-.655.123l-1.762-.275a.497.497 0 0 0-.498.252l-.61 1.088a.562.562 0 0 0 .04.619l1.13 1.43a1 1 0 0 1 .216.62v2.585a1 1 0 0 1-.207.61L4.15 15.339a.568.568 0 0 0-.036.634l.601 1.072a.494.494 0 0 0 .484.26l1.78-.278a1 1 0 0 1 .66.126l2.2 1.292a1 1 0 0 1 .43.507l.648 1.71a.508.508 0 0 0 .467.338h1.263a.51.51 0 0 0 .47-.34l.65-1.708a1 1 0 0 1 .428-.507l2.201-1.292a1 1 0 0 1 .66-.126l1.763.275a.497.497 0 0 0 .498-.252l.61-1.088a.562.562 0 0 0-.04-.619l-1.13-1.43a1 1 0 0 1-.216-.62v-2.585a1 1 0 0 1 .207-.61l1.105-1.437a.568.568 0 0 0 .037-.634l-.601-1.072a.494.494 0 0 0-.484-.26l-1.78.278a1 1 0 0 1-.66-.126l-2.2-1.292a1 1 0 0 1-.43-.507l-.649-1.71A.508.508 0 0 0 12.62 3h-1.234ZM12 8a4 4 0 1 1 0 8 4 4 0 0 1 0-8Zm0 2a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"/></symbol><symbol id="icon-eds-i-shipping-medium" viewBox="0 0 24 24"><path d="M16.515 2c1.406 0 2.706.728 3.352 1.902l2.02 3.635.02.042.036.089.031.105.012.058.01.073.004.075v11.577c0 .64-.244 1.255-.683 1.713a2.356 2.356 0 0 1-1.701.731H4.386a2.356 2.356 0 0 1-1.702-.731 2.476 2.476 0 0 1-.683-1.713V7.948c.01-.217.083-.43.22-.6L4.2 3.905C4.833 2.755 6.089 2.032 7.486 2h9.029ZM20 9H4v10.556a.49.49 0 0 0 .075.26l.053.07a.356.356 0 0 0 .257.114h15.23c.094 0 .186-.04.258-.115a.477.477 0 0 0 .127-.33V9Zm-2 7.5a1 1 0 0 1 0 2h-4a1 1 0 0 1 0-2h4ZM16.514 4H13v3h6.3l-1.183-2.13c-.288-.522-.908-.87-1.603-.87ZM11 3.999H7.51c-.679.017-1.277.36-1.566.887L4.728 7H11V3.999Z"/></symbol><symbol id="icon-eds-i-step-guide-medium" viewBox="0 0 24 24"><path d="M11.394 9.447a1 1 0 1 0-1.788-.894l-.88 1.759-.019-.02a1 1 0 1 0-1.414 1.415l1 1a1 1 0 0 0 1.601-.26l1.5-3ZM12 11a1 1 0 0 1 1-1h3a1 1 0 1 1 0 2h-3a1 1 0 0 1-1-1ZM12 17a1 1 0 0 1 1-1h3a1 1 0 1 1 0 2h-3a1 1 0 0 1-1-1ZM10.947 14.105a1 1 0 0 1 .447 1.342l-1.5 3a1 1 0 0 1-1.601.26l-1-1a1 1 0 1 1 1.414-1.414l.02.019.879-1.76a1 1 0 0 1 1.341-.447Z"/><path d="M5.545 1A2.542 2.542 0 0 0 3 3.538v16.924A2.542 2.542 0 0 0 5.545 23h12.91A2.542 2.542 0 0 0 21 20.462V7.5a1 1 0 0 0-.293-.707l-5.5-5.5A1 1 0 0 0 14.5 1H5.545ZM5 3.538C5 3.245 5.24 3 5.545 3h8.54L19 7.914v12.547c0 .294-.24.539-.546.539H5.545A.542.542 0 0 1 5 20.462V3.538Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-submission-medium" viewBox="0 0 24 24"><g><path d="M5 3.538C5 3.245 5.24 3 5.545 3h9.633L20 7.8v12.662a.535.535 0 0 1-.158.379.549.549 0 0 1-.387.159H6a1 1 0 0 1-1-1v-2.5a1 1 0 1 0-2 0V20a3 3 0 0 0 3 3h13.455c.673 0 1.32-.266 1.798-.742A2.535 2.535 0 0 0 22 20.462V7.385a1 1 0 0 0-.294-.709l-5.41-5.385A1 1 0 0 0 15.591 1H5.545A2.542 2.542 0 0 0 3 3.538V7a1 1 0 0 0 2 0V3.538Z"/><path d="m13.707 13.707-4 4a1 1 0 0 1-1.414 0l-.083-.094a1 1 0 0 1 .083-1.32L10.585 14 2 14a1 1 0 1 1 0-2l8.583.001-2.29-2.294a1 1 0 0 1 1.414-1.414l4.037 4.04.043.05.043.06.059.098.03.063.031.085.03.113.017.122L14 13l-.004.087-.017.118-.013.056-.034.104-.049.105-.048.081-.07.093-.058.063Z"/></g></symbol><symbol id="icon-eds-i-table-1-medium" viewBox="0 0 24 24"><path d="M4.385 22a2.56 2.56 0 0 1-1.14-.279C2.485 21.341 2 20.614 2 19.615V4.385c0-.315.067-.716.279-1.14C2.659 2.485 3.386 2 4.385 2h15.23c.315 0 .716.067 1.14.279.76.38 1.245 1.107 1.245 2.106v15.23c0 .315-.067.716-.279 1.14-.38.76-1.107 1.245-2.106 1.245H4.385ZM4 19.615c0 .213.034.265.14.317a.71.71 0 0 0 .245.068H8v-4H4v3.615ZM20 16H10v4h9.615c.213 0 .265-.034.317-.14a.71.71 0 0 0 .068-.245V16Zm0-2v-4H10v4h10ZM4 14h4v-4H4v4ZM19.615 4H10v4h10V4.385c0-.213-.034-.265-.14-.317A.71.71 0 0 0 19.615 4ZM8 4H4.385l-.082.002c-.146.01-.19.047-.235.138A.71.71 0 0 0 4 4.385V8h4V4Z"/></symbol><symbol id="icon-eds-i-table-2-medium" viewBox="0 0 24 24"><path d="M4.384 22A2.384 2.384 0 0 1 2 19.616V4.384A2.384 2.384 0 0 1 4.384 2h15.232A2.384 2.384 0 0 1 22 4.384v15.232A2.384 2.384 0 0 1 19.616 22H4.384ZM10 15H4v4.616c0 .212.172.384.384.384H10v-5Zm5 0h-3v5h3v-5Zm5 0h-3v5h2.616a.384.384 0 0 0 .384-.384V15ZM10 9H4v4h6V9Zm5 0h-3v4h3V9Zm5 0h-3v4h3V9Zm-.384-5H4.384A.384.384 0 0 0 4 4.384V7h16V4.384A.384.384 0 0 0 19.616 4Z"/></symbol><symbol id="icon-eds-i-tag-medium" viewBox="0 0 24 24"><path d="m12.621 1.998.127.004L20.496 2a1.5 1.5 0 0 1 1.497 1.355L22 3.5l-.005 7.669c.038.456-.133.905-.447 1.206l-9.02 9.018a2.075 2.075 0 0 1-2.932 0l-6.99-6.99a2.075 2.075 0 0 1 .001-2.933L11.61 2.47c.246-.258.573-.418.881-.46l.131-.011Zm.286 2-8.885 8.886a.075.075 0 0 0 0 .106l6.987 6.988c.03.03.077.03.106 0l8.883-8.883L19.999 4l-7.092-.002ZM16 6.5a1.5 1.5 0 0 1 .144 2.993L16 9.5a1.5 1.5 0 0 1 0-3Z"/></symbol><symbol id="icon-eds-i-trash-medium" viewBox="0 0 24 24"><path d="M12 1c2.717 0 4.913 2.232 4.997 5H21a1 1 0 0 1 0 2h-1v12.5c0 1.389-1.152 2.5-2.556 2.5H6.556C5.152 23 4 21.889 4 20.5V8H3a1 1 0 1 1 0-2h4.003l.001-.051C7.114 3.205 9.3 1 12 1Zm6 7H6v12.5c0 .238.19.448.454.492l.102.008h10.888c.315 0 .556-.232.556-.5V8Zm-4 3a1 1 0 0 1 1 1v6.005a1 1 0 0 1-2 0V12a1 1 0 0 1 1-1Zm-4 0a1 1 0 0 1 1 1v6a1 1 0 0 1-2 0v-6a1 1 0 0 1 1-1Zm2-8c-1.595 0-2.914 1.32-2.996 3h5.991v-.02C14.903 4.31 13.589 3 12 3Z"/></symbol><symbol id="icon-eds-i-user-account-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 16c-1.806 0-3.52.994-4.664 2.698A8.947 8.947 0 0 0 12 21a8.958 8.958 0 0 0 4.664-1.301C15.52 17.994 13.806 17 12 17Zm0-14a9 9 0 0 0-6.25 15.476C7.253 16.304 9.54 15 12 15s4.747 1.304 6.25 3.475A9 9 0 0 0 12 3Zm0 3a4 4 0 1 1 0 8 4 4 0 0 1 0-8Zm0 2a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"/></symbol><symbol id="icon-eds-i-user-add-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm9 10a1 1 0 0 1 1 1v3h3a1 1 0 0 1 0 2h-3v3a1 1 0 0 1-2 0v-3h-3a1 1 0 0 1 0-2h3v-3a1 1 0 0 1 1-1Zm-5.545-.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378Z"/></symbol><symbol id="icon-eds-i-user-assign-medium" viewBox="0 0 24 24"><path d="M16.226 13.298a1 1 0 0 1 1.414-.01l.084.093a1 1 0 0 1-.073 1.32L15.39 17H22a1 1 0 0 1 0 2h-6.611l2.262 2.298a1 1 0 0 1-1.425 1.404l-3.939-4a1 1 0 0 1 0-1.404l3.94-4Zm-3.771-.449a1 1 0 1 1-.91 1.781 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 10.5 20a1 1 0 0 1 .993.883L11.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Z"/></symbol><symbol id="icon-eds-i-user-block-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm9 10a5 5 0 1 1 0 10 5 5 0 0 1 0-10Zm-5.545-.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM15 18a3 3 0 0 0 4.294 2.707l-4.001-4c-.188.391-.293.83-.293 1.293Zm3-3c-.463 0-.902.105-1.294.293l4.001 4A3 3 0 0 0 18 15Z"/></symbol><symbol id="icon-eds-i-user-check-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm13.647 12.237a1 1 0 0 1 .116 1.41l-5.091 6a1 1 0 0 1-1.375.144l-2.909-2.25a1 1 0 1 1 1.224-1.582l2.153 1.665 4.472-5.271a1 1 0 0 1 1.41-.116Zm-8.139-.977c.22.214.428.44.622.678a1 1 0 1 1-1.548 1.266 6.025 6.025 0 0 0-1.795-1.49.86.86 0 0 1-.163-.048l-.079-.036a5.721 5.721 0 0 0-2.62-.63l-.194.006c-2.76.134-5.022 2.177-5.592 4.864l-.035.175-.035.213c-.03.201-.05.405-.06.61L3.003 20 10 20a1 1 0 0 1 .993.883L11 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876l.005-.223.02-.356.02-.222.03-.248.022-.15c.02-.133.044-.265.071-.397.44-2.178 1.725-4.105 3.595-5.301a7.75 7.75 0 0 1 3.755-1.215l.12-.004a7.908 7.908 0 0 1 5.87 2.252Z"/></symbol><symbol id="icon-eds-i-user-delete-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6ZM4.763 13.227a7.713 7.713 0 0 1 7.692-.378 1 1 0 1 1-.91 1.781 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20H11.5a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897Zm11.421 1.543 2.554 2.553 2.555-2.553a1 1 0 0 1 1.414 1.414l-2.554 2.554 2.554 2.555a1 1 0 0 1-1.414 1.414l-2.555-2.554-2.554 2.554a1 1 0 0 1-1.414-1.414l2.553-2.555-2.553-2.554a1 1 0 0 1 1.414-1.414Z"/></symbol><symbol id="icon-eds-i-user-edit-medium" viewBox="0 0 24 24"><path d="m19.876 10.77 2.831 2.83a1 1 0 0 1 0 1.415l-7.246 7.246a1 1 0 0 1-.572.284l-3.277.446a1 1 0 0 1-1.125-1.13l.461-3.277a1 1 0 0 1 .283-.567l7.23-7.246a1 1 0 0 1 1.415-.001Zm-7.421 2.08a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 7.5 20a1 1 0 0 1 .993.883L8.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378Zm6.715.042-6.29 6.3-.23 1.639 1.633-.222 6.302-6.302-1.415-1.415ZM9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Z"/></symbol><symbol id="icon-eds-i-user-linked-medium" viewBox="0 0 24 24"><path d="M15.65 6c.31 0 .706.066 1.122.274C17.522 6.65 18 7.366 18 8.35v12.3c0 .31-.066.706-.274 1.122-.375.75-1.092 1.228-2.076 1.228H3.35a2.52 2.52 0 0 1-1.122-.274C1.478 22.35 1 21.634 1 20.65V8.35c0-.31.066-.706.274-1.122C1.65 6.478 2.366 6 3.35 6h12.3Zm0 2-12.376.002c-.134.007-.17.04-.21.12A.672.672 0 0 0 3 8.35v12.3c0 .198.028.24.122.287.09.044.2.063.228.063h.887c.788-2.269 2.814-3.5 5.263-3.5 2.45 0 4.475 1.231 5.263 3.5h.887c.198 0 .24-.028.287-.122.044-.09.063-.2.063-.228V8.35c0-.198-.028-.24-.122-.287A.672.672 0 0 0 15.65 8ZM9.5 19.5c-1.36 0-2.447.51-3.06 1.5h6.12c-.613-.99-1.7-1.5-3.06-1.5ZM20.65 1A2.35 2.35 0 0 1 23 3.348V15.65A2.35 2.35 0 0 1 20.65 18H20a1 1 0 0 1 0-2h.65a.35.35 0 0 0 .35-.35V3.348A.35.35 0 0 0 20.65 3H8.35a.35.35 0 0 0-.35.348V4a1 1 0 1 1-2 0v-.652A2.35 2.35 0 0 1 8.35 1h12.3ZM9.5 10a3.5 3.5 0 1 1 0 7 3.5 3.5 0 0 1 0-7Zm0 2a1.5 1.5 0 1 0 0 3 1.5 1.5 0 0 0 0-3Z"/></symbol><symbol id="icon-eds-i-user-multiple-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm6 0a5 5 0 0 1 0 10 1 1 0 0 1-.117-1.993L15 9a3 3 0 0 0 0-6 1 1 0 0 1 0-2ZM9 3a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm8.857 9.545a7.99 7.99 0 0 1 2.651 1.715A8.31 8.31 0 0 1 23 20.134V21a1 1 0 0 1-1 1h-3a1 1 0 0 1 0-2h1.995l-.005-.153a6.307 6.307 0 0 0-1.673-3.945l-.204-.209a5.99 5.99 0 0 0-1.988-1.287 1 1 0 1 1 .732-1.861Zm-3.349 1.715A8.31 8.31 0 0 1 17 20.134V21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.877c.044-4.343 3.387-7.908 7.638-8.115a7.908 7.908 0 0 1 5.87 2.252ZM9.016 14l-.285.006c-3.104.15-5.58 2.718-5.725 5.9L3.004 20h11.991l-.005-.153a6.307 6.307 0 0 0-1.673-3.945l-.204-.209A5.924 5.924 0 0 0 9.3 14.008L9.016 14Z"/></symbol><symbol id="icon-eds-i-user-notify-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm10 18v1a1 1 0 0 1-2 0v-1h-3a1 1 0 0 1 0-2v-2.818C14 13.885 15.777 12 18 12s4 1.885 4 4.182V19a1 1 0 0 1 0 2h-3Zm-6.545-8.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM18 14c-1.091 0-2 .964-2 2.182V19h4v-2.818c0-1.165-.832-2.098-1.859-2.177L18 14Z"/></symbol><symbol id="icon-eds-i-user-remove-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm3.455 9.85a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM22 17a1 1 0 0 1 0 2h-8a1 1 0 0 1 0-2h8Z"/></symbol><symbol id="icon-eds-i-user-single-medium" viewBox="0 0 24 24"><path d="M12 1a5 5 0 1 1 0 10 5 5 0 0 1 0-10Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm-.406 9.008a8.965 8.965 0 0 1 6.596 2.494A9.161 9.161 0 0 1 21 21.025V22a1 1 0 0 1-1 1H4a1 1 0 0 1-1-1v-.985c.05-4.825 3.815-8.777 8.594-9.007Zm.39 1.992-.299.006c-3.63.175-6.518 3.127-6.678 6.775L5 21h13.998l-.009-.268a7.157 7.157 0 0 0-1.97-4.573l-.214-.213A6.967 6.967 0 0 0 11.984 14Z"/></symbol><symbol id="icon-eds-i-warning-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm0 11.5a1.5 1.5 0 0 1 .144 2.993L12 17.5a1.5 1.5 0 0 1 0-3ZM12 6a1 1 0 0 1 1 1v5a1 1 0 0 1-2 0V7a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-warning-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 13.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 14.5ZM12 6a1 1 0 0 0-1 1v5a1 1 0 0 0 2 0V7a1 1 0 0 0-1-1Z"/></symbol><symbol id="icon-chevron-left-medium" viewBox="0 0 24 24"><path d="M15.7194 3.3054C15.3358 2.90809 14.7027 2.89699 14.3054 3.28061L6.54342 10.7757C6.19804 11.09 6 11.5335 6 12C6 12.4665 6.19804 12.91 6.5218 13.204L14.3054 20.7194C14.7027 21.103 15.3358 21.0919 15.7194 20.6946C16.103 20.2973 16.0919 19.6642 15.6946 19.2806L8.155 12L15.6946 4.71939C16.0614 4.36528 16.099 3.79863 15.8009 3.40105L15.7194 3.3054Z"/></symbol><symbol id="icon-chevron-right-medium" viewBox="0 0 24 24"><path d="M8.28061 3.3054C8.66423 2.90809 9.29729 2.89699 9.6946 3.28061L17.4566 10.7757C17.802 11.09 18 11.5335 18 12C18 12.4665 17.802 12.91 17.4782 13.204L9.6946 20.7194C9.29729 21.103 8.66423 21.0919 8.28061 20.6946C7.89699 20.2973 7.90809 19.6642 8.3054 19.2806L15.845 12L8.3054 4.71939C7.93865 4.36528 7.90098 3.79863 8.19908 3.40105L8.28061 3.3054Z"/></symbol><symbol id="icon-eds-alerts" viewBox="0 0 32 32"><path d="M28 12.667c.736 0 1.333.597 1.333 1.333v13.333A3.333 3.333 0 0 1 26 30.667H6a3.333 3.333 0 0 1-3.333-3.334V14a1.333 1.333 0 1 1 2.666 0v1.252L16 21.769l10.667-6.518V14c0-.736.597-1.333 1.333-1.333Zm-1.333 5.71-9.972 6.094c-.427.26-.963.26-1.39 0l-9.972-6.094v8.956c0 .368.299.667.667.667h20a.667.667 0 0 0 .667-.667v-8.956ZM19.333 12a1.333 1.333 0 1 1 0 2.667h-6.666a1.333 1.333 0 1 1 0-2.667h6.666Zm4-10.667a3.333 3.333 0 0 1 3.334 3.334v6.666a1.333 1.333 0 1 1-2.667 0V4.667A.667.667 0 0 0 23.333 4H8.667A.667.667 0 0 0 8 4.667v6.666a1.333 1.333 0 1 1-2.667 0V4.667a3.333 3.333 0 0 1 3.334-3.334h14.666Zm-4 5.334a1.333 1.333 0 0 1 0 2.666h-6.666a1.333 1.333 0 1 1 0-2.666h6.666Z"/></symbol><symbol id="icon-eds-arrow-up" viewBox="0 0 24 24"><path fill-rule="evenodd" d="m13.002 7.408 4.88 4.88a.99.99 0 0 0 1.32.08l.09-.08c.39-.39.39-1.03 0-1.42l-6.58-6.58a1.01 1.01 0 0 0-1.42 0l-6.58 6.58a1 1 0 0 0-.09 1.32l.08.1a1 1 0 0 0 1.42-.01l4.88-4.87v11.59a.99.99 0 0 0 .88.99l.12.01c.55 0 1-.45 1-1V7.408z" class="layer"/></symbol><symbol id="icon-eds-checklist" viewBox="0 0 32 32"><path d="M19.2 1.333a3.468 3.468 0 0 1 3.381 2.699L24.667 4C26.515 4 28 5.52 28 7.38v19.906c0 1.86-1.485 3.38-3.333 3.38H7.333c-1.848 0-3.333-1.52-3.333-3.38V7.38C4 5.52 5.485 4 7.333 4h2.093A3.468 3.468 0 0 1 12.8 1.333h6.4ZM9.426 6.667H7.333c-.36 0-.666.312-.666.713v19.906c0 .401.305.714.666.714h17.334c.36 0 .666-.313.666-.714V7.38c0-.4-.305-.713-.646-.714l-2.121.033A3.468 3.468 0 0 1 19.2 9.333h-6.4a3.468 3.468 0 0 1-3.374-2.666Zm12.715 5.606c.586.446.7 1.283.253 1.868l-7.111 9.334a1.333 1.333 0 0 1-1.792.306l-3.556-2.333a1.333 1.333 0 1 1 1.463-2.23l2.517 1.651 6.358-8.344a1.333 1.333 0 0 1 1.868-.252ZM19.2 4h-6.4a.8.8 0 0 0-.8.8v1.067a.8.8 0 0 0 .8.8h6.4a.8.8 0 0 0 .8-.8V4.8a.8.8 0 0 0-.8-.8Z"/></symbol><symbol id="icon-eds-citation" viewBox="0 0 36 36"><path d="M23.25 1.5a1.5 1.5 0 0 1 1.06.44l8.25 8.25a1.5 1.5 0 0 1 .44 1.06v19.5c0 2.105-1.645 3.75-3.75 3.75H18a1.5 1.5 0 0 1 0-3h11.25c.448 0 .75-.302.75-.75V11.873L22.628 4.5H8.31a.811.811 0 0 0-.8.68l-.011.13V16.5a1.5 1.5 0 0 1-3 0V5.31A3.81 3.81 0 0 1 8.31 1.5h14.94ZM8.223 20.358a.984.984 0 0 1-.192 1.378l-.048.034c-.54.36-.942.676-1.206.951-.59.614-.885 1.395-.885 2.343.115-.028.288-.042.518-.042.662 0 1.26.237 1.791.711.533.474.799 1.074.799 1.799 0 .753-.259 1.352-.777 1.799-.518.446-1.151.669-1.9.669-1.006 0-1.812-.293-2.417-.878C3.302 28.536 3 27.657 3 26.486c0-1.115.165-2.085.496-2.907.331-.823.734-1.513 1.209-2.071.475-.558.971-.997 1.49-1.318a6.01 6.01 0 0 1 .347-.2 1.321 1.321 0 0 1 1.681.368Zm7.5 0a.984.984 0 0 1-.192 1.378l-.048.034c-.54.36-.942.676-1.206.951-.59.614-.885 1.395-.885 2.343.115-.028.288-.042.518-.042.662 0 1.26.237 1.791.711.533.474.799 1.074.799 1.799 0 .753-.259 1.352-.777 1.799-.518.446-1.151.669-1.9.669-1.006 0-1.812-.293-2.417-.878-.604-.586-.906-1.465-.906-2.636 0-1.115.165-2.085.496-2.907.331-.823.734-1.513 1.209-2.071.475-.558.971-.997 1.49-1.318a6.01 6.01 0 0 1 .347-.2 1.321 1.321 0 0 1 1.681.368Z"/></symbol><symbol id="icon-eds-i-access-indicator" viewBox="0 0 16 16"><circle cx="4.5" cy="11.5" r="3.5" style="fill:currentColor"/><path fill-rule="evenodd" d="M4 3v3a1 1 0 0 1-2 0V2.923C2 1.875 2.84 1 3.909 1h5.909a1 1 0 0 1 .713.298l3.181 3.231a1 1 0 0 1 .288.702v7.846c0 .505-.197.993-.554 1.354a1.902 1.902 0 0 1-1.355.569H10a1 1 0 1 1 0-2h2V5.64L9.4 3H4Z" clip-rule="evenodd" style="fill:#222"/></symbol><symbol id="icon-eds-i-github-medium" viewBox="0 0 24 24"><path d="M 11.964844 0 C 5.347656 0 0 5.269531 0 11.792969 C 0 17.003906 3.425781 21.417969 8.179688 22.976562 C 8.773438 23.09375 8.992188 22.722656 8.992188 22.410156 C 8.992188 22.136719 8.972656 21.203125 8.972656 20.226562 C 5.644531 20.929688 4.953125 18.820312 4.953125 18.820312 C 4.417969 17.453125 3.625 17.101562 3.625 17.101562 C 2.535156 16.378906 3.703125 16.378906 3.703125 16.378906 C 4.914062 16.457031 5.546875 17.589844 5.546875 17.589844 C 6.617188 19.386719 8.339844 18.878906 9.03125 18.566406 C 9.132812 17.804688 9.449219 17.277344 9.785156 16.984375 C 7.132812 16.710938 4.339844 15.695312 4.339844 11.167969 C 4.339844 9.878906 4.8125 8.824219 5.566406 8.003906 C 5.445312 7.710938 5.03125 6.5 5.683594 4.878906 C 5.683594 4.878906 6.695312 4.566406 8.972656 6.089844 C 9.949219 5.832031 10.953125 5.703125 11.964844 5.699219 C 12.972656 5.699219 14.003906 5.835938 14.957031 6.089844 C 17.234375 4.566406 18.242188 4.878906 18.242188 4.878906 C 18.898438 6.5 18.480469 7.710938 18.363281 8.003906 C 19.136719 8.824219 19.589844 9.878906 19.589844 11.167969 C 19.589844 15.695312 16.796875 16.691406 14.125 16.984375 C 14.558594 17.355469 14.933594 18.058594 14.933594 19.171875 C 14.933594 20.753906 14.914062 22.019531 14.914062 22.410156 C 14.914062 22.722656 15.132812 23.09375 15.726562 22.976562 C 20.480469 21.414062 23.910156 17.003906 23.910156 11.792969 C 23.929688 5.269531 18.558594 0 11.964844 0 Z M 11.964844 0 "/></symbol><symbol id="icon-eds-i-limited-access" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 3v3a1 1 0 0 1-2 0V2.923C2 1.875 2.84 1 3.909 1h5.909a1 1 0 0 1 .713.298l3.181 3.231a1 1 0 0 1 .288.702V6a1 1 0 1 1-2 0v-.36L9.4 3H4ZM3 8a1 1 0 0 1 1 1v1a1 1 0 1 1-2 0V9a1 1 0 0 1 1-1Zm10 0a1 1 0 0 1 1 1v1a1 1 0 1 1-2 0V9a1 1 0 0 1 1-1Zm-3.5 6a1 1 0 0 1-1 1h-1a1 1 0 1 1 0-2h1a1 1 0 0 1 1 1Zm2.441-1a1 1 0 0 1 2 0c0 .73-.246 1.306-.706 1.664a1.61 1.61 0 0 1-.876.334l-.032.002H11.5a1 1 0 1 1 0-2h.441ZM4 13a1 1 0 0 0-2 0c0 .73.247 1.306.706 1.664a1.609 1.609 0 0 0 .876.334l.032.002H4.5a1 1 0 1 0 0-2H4Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-subjects-medium" viewBox="0 0 24 24"><g id="icon-subjects-copy" stroke="none" stroke-width="1" fill-rule="evenodd"><path d="M13.3846154,2 C14.7015971,2 15.7692308,3.06762994 15.7692308,4.38461538 L15.7692308,7.15384615 C15.7692308,8.47082629 14.7015955,9.53846154 13.3846154,9.53846154 L13.1038388,9.53925278 C13.2061091,9.85347965 13.3815528,10.1423885 13.6195822,10.3804178 C13.9722182,10.7330539 14.436524,10.9483278 14.9293854,10.9918129 L15.1153846,11 C16.2068332,11 17.2535347,11.433562 18.0254647,12.2054189 C18.6411944,12.8212361 19.0416785,13.6120766 19.1784166,14.4609738 L19.6153846,14.4615385 C20.932386,14.4615385 22,15.5291672 22,16.8461538 L22,19.6153846 C22,20.9323924 20.9323924,22 19.6153846,22 L16.8461538,22 C15.5291672,22 14.4615385,20.932386 14.4615385,19.6153846 L14.4615385,16.8461538 C14.4615385,15.5291737 15.5291737,14.4615385 16.8461538,14.4615385 L17.126925,14.460779 C17.0246537,14.1465537 16.8492179,13.857633 16.6112344,13.6196157 C16.2144418,13.2228606 15.6764136,13 15.1153846,13 C14.0239122,13 12.9771569,12.5664197 12.2053686,11.7946314 C12.1335167,11.7227795 12.0645962,11.6485444 11.9986839,11.5721119 C11.9354038,11.6485444 11.8664833,11.7227795 11.7946314,11.7946314 C11.0228431,12.5664197 9.97608778,13 8.88461538,13 C8.323576,13 7.78552852,13.2228666 7.38881294,13.6195822 C7.15078359,13.8576115 6.97533988,14.1465203 6.8730696,14.4607472 L7.15384615,14.4615385 C8.47082629,14.4615385 9.53846154,15.5291737 9.53846154,16.8461538 L9.53846154,19.6153846 C9.53846154,20.932386 8.47083276,22 7.15384615,22 L4.38461538,22 C3.06762347,22 2,20.9323876 2,19.6153846 L2,16.8461538 C2,15.5291721 3.06762994,14.4615385 4.38461538,14.4615385 L4.8215823,14.4609378 C4.95831893,13.6120029 5.3588057,12.8211623 5.97459937,12.2053686 C6.69125996,11.488708 7.64500941,11.0636656 8.6514968,11.0066017 L8.88461538,11 C9.44565477,11 9.98370225,10.7771334 10.3804178,10.3804178 C10.6184472,10.1423885 10.7938909,9.85347965 10.8961612,9.53925278 L10.6153846,9.53846154 C9.29840448,9.53846154 8.23076923,8.47082629 8.23076923,7.15384615 L8.23076923,4.38461538 C8.23076923,3.06762994 9.29840286,2 10.6153846,2 L13.3846154,2 Z M7.15384615,16.4615385 L4.38461538,16.4615385 C4.17220099,16.4615385 4,16.63374 4,16.8461538 L4,19.6153846 C4,19.8278134 4.17218833,20 4.38461538,20 L7.15384615,20 C7.36626945,20 7.53846154,19.8278103 7.53846154,19.6153846 L7.53846154,16.8461538 C7.53846154,16.6337432 7.36625679,16.4615385 7.15384615,16.4615385 Z M19.6153846,16.4615385 L16.8461538,16.4615385 C16.6337432,16.4615385 16.4615385,16.6337432 16.4615385,16.8461538 L16.4615385,19.6153846 C16.4615385,19.8278103 16.6337306,20 16.8461538,20 L19.6153846,20 C19.8278229,20 20,19.8278229 20,19.6153846 L20,16.8461538 C20,16.6337306 19.8278103,16.4615385 19.6153846,16.4615385 Z M13.3846154,4 L10.6153846,4 C10.4029708,4 10.2307692,4.17220099 10.2307692,4.38461538 L10.2307692,7.15384615 C10.2307692,7.36625679 10.402974,7.53846154 10.6153846,7.53846154 L13.3846154,7.53846154 C13.597026,7.53846154 13.7692308,7.36625679 13.7692308,7.15384615 L13.7692308,4.38461538 C13.7692308,4.17220099 13.5970292,4 13.3846154,4 Z" id="Shape" fill-rule="nonzero"/></g></symbol><symbol id="icon-eds-small-arrow-left" viewBox="0 0 16 17"><path stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M14 8.092H2m0 0L8 2M2 8.092l6 6.035"/></symbol><symbol id="icon-eds-small-arrow-right" viewBox="0 0 16 16"><g fill-rule="evenodd" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2"><path d="M2 8.092h12M8 2l6 6.092M8 14.127l6-6.035"/></g></symbol><symbol id="icon-orcid-logo" viewBox="0 0 40 40"><path fill-rule="evenodd" d="M12.281 10.453c.875 0 1.578-.719 1.578-1.578 0-.86-.703-1.578-1.578-1.578-.875 0-1.578.703-1.578 1.578 0 .86.703 1.578 1.578 1.578Zm-1.203 18.641h2.406V12.359h-2.406v16.735Z"/><path fill-rule="evenodd" d="M17.016 12.36h6.5c6.187 0 8.906 4.421 8.906 8.374 0 4.297-3.36 8.375-8.875 8.375h-6.531V12.36Zm6.234 14.578h-3.828V14.53h3.703c4.688 0 6.828 2.844 6.828 6.203 0 2.063-1.25 6.203-6.703 6.203Z" clip-rule="evenodd"/></symbol></svg> </div> <a class="c-skip-link" href="#main">Skip to main content</a> <header class="eds-c-header" data-eds-c-header> <div class="eds-c-header__container" data-eds-c-header-expander-anchor> <div class="eds-c-header__brand"> <a href="https://link.springer.com" data-test=springerlink-logo data-track="click_imprint_logo" data-track-context="unified header" data-track-action="click logo link" data-track-category="unified header" data-track-label="link" > <img src="/oscar-static/images/darwin/header/img/logo-springer-nature-link-3149409f62.svg" alt="Springer Nature Link"> </a> </div> <a class="c-header__link eds-c-header__link" id="identity-account-widget" href='https://idp.springer.com/auth/personal/springernature?redirect_uri=https://link.springer.com/article/10.1007/s10773-023-05458-x?'><span class="eds-c-header__widget-fragment-title">Log in</span></a> </div> <nav class="eds-c-header__nav" aria-label="header navigation"> <div class="eds-c-header__nav-container"> <div class="eds-c-header__item eds-c-header__item--menu"> <a href="#eds-c-header-nav" class="eds-c-header__link" data-eds-c-header-expander> <svg class="eds-c-header__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-menu-medium"></use> </svg><span>Menu</span> </a> </div> <div class="eds-c-header__item eds-c-header__item--inline-links"> <a class="eds-c-header__link" href="https://link.springer.com/journals/" data-track="nav_find_a_journal" data-track-context="unified header" data-track-action="click find a journal" data-track-category="unified header" data-track-label="link" > Find a journal </a> <a class="eds-c-header__link" href="https://www.springernature.com/gp/authors" data-track="nav_how_to_publish" data-track-context="unified header" data-track-action="click publish with us link" data-track-category="unified header" data-track-label="link" > Publish with us </a> <a class="eds-c-header__link" href="https://link.springernature.com/home/" data-track="nav_track_your_research" data-track-context="unified header" data-track-action="click track your research" data-track-category="unified header" data-track-label="link" > Track your research </a> </div> <div class="eds-c-header__link-container"> <div class="eds-c-header__item eds-c-header__item--divider"> <a href="#eds-c-header-popup-search" class="eds-c-header__link" data-eds-c-header-expander data-eds-c-header-test-search-btn> <svg class="eds-c-header__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-search-medium"></use> </svg><span>Search</span> </a> </div> <div id="ecommerce-header-cart-icon-link" class="eds-c-header__item ecommerce-cart" style="display:inline-block"> <a class="eds-c-header__link" href="https://order.springer.com/public/cart" style="appearance:none;border:none;background:none;color:inherit;position:relative"> <svg id="eds-i-cart" class="eds-c-header__icon" xmlns="http://www.w3.org/2000/svg" height="24" width="24" viewBox="0 0 24 24" aria-hidden="true" focusable="false"> <path fill="currentColor" fill-rule="nonzero" d="M2 1a1 1 0 0 0 0 2l1.659.001 2.257 12.808a2.599 2.599 0 0 0 2.435 2.185l.167.004 9.976-.001a2.613 2.613 0 0 0 2.61-1.748l.03-.106 1.755-7.82.032-.107a2.546 2.546 0 0 0-.311-1.986l-.108-.157a2.604 2.604 0 0 0-2.197-1.076L6.042 5l-.56-3.17a1 1 0 0 0-.864-.82l-.12-.007L2.001 1ZM20.35 6.996a.63.63 0 0 1 .54.26.55.55 0 0 1 .082.505l-.028.1L19.2 15.63l-.022.05c-.094.177-.282.299-.526.317l-10.145.002a.61.61 0 0 1-.618-.515L6.394 6.999l13.955-.003ZM18 19a2 2 0 1 0 0 4 2 2 0 0 0 0-4ZM8 19a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"></path> </svg><span>Cart</span><span class="cart-info" style="display:none;position:absolute;top:10px;right:45px;background-color:#C65301;color:#fff;width:18px;height:18px;font-size:11px;border-radius:50%;line-height:17.5px;text-align:center"></span></a> <script>(function () { var exports = {}; if (window.fetch) { "use strict"; Object.defineProperty(exports, "__esModule", { value: true }); exports.headerWidgetClientInit = void 0; var headerWidgetClientInit = function (getCartInfo) { document.body.addEventListener("updatedCart", function () { updateCartIcon(); }, false); return updateCartIcon(); function updateCartIcon() { return getCartInfo() .then(function (res) { return res.json(); }) .then(refreshCartState) .catch(function (_) { }); } function refreshCartState(json) { var indicator = document.querySelector("#ecommerce-header-cart-icon-link .cart-info"); /* istanbul ignore else */ if (indicator && json.itemCount) { indicator.style.display = 'block'; indicator.textContent = json.itemCount > 9 ? '9+' : json.itemCount.toString(); var moreThanOneItem = json.itemCount > 1; indicator.setAttribute('title', "there ".concat(moreThanOneItem ? "are" : "is", " ").concat(json.itemCount, " item").concat(moreThanOneItem ? "s" : "", " in your cart")); } return json; } }; exports.headerWidgetClientInit = headerWidgetClientInit; headerWidgetClientInit( function () { return window.fetch("https://cart.springer.com/cart-info", { credentials: "include", headers: { Accept: "application/json" } }) } ) }})()</script> </div> </div> </div> </nav> </header> <article lang="en" id="main" class="app-masthead__colour-20"> <section class="app-masthead " aria-label="article masthead"> <div class="app-masthead__container"> <div class="app-article-masthead u-sans-serif js-context-bar-sticky-point-masthead" data-track-component="article" data-test="masthead-component"> <div class="app-article-masthead__info"> <nav aria-label="breadcrumbs" data-test="breadcrumbs"> <ol class="c-breadcrumbs c-breadcrumbs--contrast" itemscope itemtype="https://schema.org/BreadcrumbList"> <li class="c-breadcrumbs__item" id="breadcrumb0" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <a href="/" class="c-breadcrumbs__link" itemprop="item" data-track="click_breadcrumb" data-track-context="article page" data-track-category="article" data-track-action="breadcrumbs" data-track-label="breadcrumb1"><span itemprop="name">Home</span></a><meta itemprop="position" content="1"> <svg class="c-breadcrumbs__chevron" role="img" aria-hidden="true" focusable="false" width="10" height="10" viewBox="0 0 10 10"> <path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/> </svg> </li> <li class="c-breadcrumbs__item" id="breadcrumb1" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <a href="/journal/10773" class="c-breadcrumbs__link" itemprop="item" data-track="click_breadcrumb" data-track-context="article page" data-track-category="article" data-track-action="breadcrumbs" data-track-label="breadcrumb2"><span itemprop="name">International Journal of Theoretical Physics</span></a><meta itemprop="position" content="2"> <svg class="c-breadcrumbs__chevron" role="img" aria-hidden="true" focusable="false" width="10" height="10" viewBox="0 0 10 10"> <path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/> </svg> </li> <li class="c-breadcrumbs__item" id="breadcrumb2" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <span itemprop="name">Article</span><meta itemprop="position" content="3"> </li> </ol> </nav> <h1 class="c-article-title" data-test="article-title" data-article-title="">The Reversibility Paradox: Role of the Velocity Reversal Step</h1> <ul class="c-article-identifiers"> <li class="c-article-identifiers__item" data-test="article-category">Research</li> <li class="c-article-identifiers__item"> <a href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="click" data-track-action="open access" data-track-label="link" class="u-color-open-access" data-test="open-access">Open access</a> </li> <li class="c-article-identifiers__item"> Published: <time datetime="2023-09-02">02 September 2023</time> </li> </ul> <ul class="c-article-identifiers c-article-identifiers--cite-list"> <li class="c-article-identifiers__item"> <span data-test="journal-volume">Volume 62</span>, article number <span data-test="article-number">200</span>, (<span data-test="article-publication-year">2023</span>) </li> <li class="c-article-identifiers__item c-article-identifiers__item--cite"> <a href="#citeas" data-track="click" data-track-action="cite this article" data-track-category="article body" data-track-label="link">Cite this article</a> </li> </ul> <div class="app-article-masthead__buttons" data-test="download-article-link-wrapper" data-track-context="masthead"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1007/s10773-023-05458-x.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="button" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-eds-i-download-medium"/></svg> </a> </div> </div> <p class="app-article-masthead__access"> <svg width="16" height="16" focusable="false" role="img" aria-hidden="true"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-check-filled-medium"></use></svg> You have full access to this <a href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="click" data-track-action="open access" data-track-label="link">open access</a> article</p> </div> </div> <div class="app-article-masthead__brand"> <a href="/journal/10773" class="app-article-masthead__journal-link" data-track="click_journal_home" data-track-action="journal homepage" data-track-context="article page" data-track-label="link"> <picture> <source type="image/webp" media="(min-width: 768px)" width="120" height="159" srcset="https://media.springernature.com/w120/springer-static/cover-hires/journal/10773?as=webp, https://media.springernature.com/w316/springer-static/cover-hires/journal/10773?as=webp 2x"> <img width="72" height="95" src="https://media.springernature.com/w72/springer-static/cover-hires/journal/10773?as=webp" srcset="https://media.springernature.com/w144/springer-static/cover-hires/journal/10773?as=webp 2x" alt=""> </picture> <span class="app-article-masthead__journal-title">International Journal of Theoretical Physics</span> </a> <a href="https://link.springer.com/journal/10773/aims-and-scope" class="app-article-masthead__submission-link" data-track="click_aims_and_scope" data-track-action="aims and scope" data-track-context="article page" data-track-label="link"> Aims and scope <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> <a href="https://submission.nature.com/new-submission/10773/3" class="app-article-masthead__submission-link" data-track="click_submit_manuscript" data-track-context="article masthead on springerlink article page" data-track-action="submit manuscript" data-track-label="link"> Submit manuscript <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> </div> </div> </div> </section> <div class="c-article-main u-container u-mt-24 u-mb-32 l-with-sidebar" id="main-content" data-component="article-container"> <main class="u-serif js-main-column" data-track-component="article body"> <div class="c-context-bar u-hide" data-test="context-bar" data-context-bar aria-hidden="true"> <div class="c-context-bar__container u-container"> <div class="c-context-bar__title"> The Reversibility Paradox: Role of the Velocity Reversal Step </div> <div data-test="inCoD" data-track-context="sticky banner"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1007/s10773-023-05458-x.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="button" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-eds-i-download-medium"/></svg> </a> </div> </div> </div> </div> </div> <div class="c-article-header"> <header> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-P_M_-Binder-Aff1" data-author-popup="auth-P_M_-Binder-Aff1" data-author-search="Binder, P.M." data-corresp-id="c1">P.M. Binder<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0001-5242-6319"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0001-5242-6319</a></span><sup class="u-js-hide"><a href="#Aff1">1</a></sup> </li></ul> <div data-test="article-metrics"> <ul class="app-article-metrics-bar u-list-reset"> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>1722 <span class="app-article-metrics-bar__label">Accesses</span></p> </li> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-citations-medium"></use> </svg>2 <span class="app-article-metrics-bar__label">Citations</span></p> </li> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-altmetric-medium"></use> </svg>5 <span class="app-article-metrics-bar__label">Altmetric</span></p> </li> <li class="app-article-metrics-bar__item app-article-metrics-bar__item--metrics"> <p class="app-article-metrics-bar__details"><a href="/article/10.1007/s10773-023-05458-x/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>The reversibility, or Loschmidt paradox, is a thought experiment in which microscopic reversibility is exploited to generate an apparently spontaneous trajectory in which entropy decreases, thus violating the Second Law of Thermodynamics (SLT). To achieve this, the system is allowed to evolve from a low-entropy to a high-entropy configuration, at which point all velocities of its molecules are reversed, thus producing the desired effect. In this paper we focus on the velocity reversal step. Implementing this step requires the measurement of velocities and positions, which must be stored in a memory. Even in the case of a reversible measurement, the erasure of the stored information, necessary to reset the measuring device to its original state, has an entropy cost that offsets the entropy decrease during the reverse evolution of the system. This cost is sufficient to explain why the described procedure does not violate the SLT.</p></div></div></section> <div data-test="cobranding-download"> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-nature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1038/s42005-020-00396-0?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1038/s42005-020-00396-0">Time-reversal of an unknown quantum state </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">21 July 2020</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1007%2Fs10701-015-9871-z/MediaObjects/10701_2015_9871_Fig1_HTML.gif" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s10701-015-9871-z?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1007/s10701-015-9871-z">Lanford’s Theorem and the Emergence of Irreversibility </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">19 February 2015</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1007%2Fs10701-022-00557-x/MediaObjects/10701_2022_557_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s10701-022-00557-x?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1007/s10701-022-00557-x">The Time-Identity Tradeoff </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">10 March 2022</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1732305800, embedded_user: 'null' } }); </script> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=10773" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>The purpose of this paper is to re-examine the reversibility paradox (RP), related to the Second Law of Thermodynamics (SLT), using modern developments in the physics of measurement and information. From this perspective we produce an account of the paradox that is sufficient to explain it: a step in the procedure of the paradox requires a measurement with an eventual entropy cost that offsets the entropy decrease one can achieve in a different step, thus avoiding a violation of the SLT.</p><p>The RP was formulated in the 1860s in the context of the infancy of kinetic theory. A complete, well-documented early history is presented in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Brush, S.G.: The Kind of Motion We Call Heat. North-Holland, Amsterdam (1976)" href="/article/10.1007/s10773-023-05458-x#ref-CR1" id="ref-link-section-d40198907e336">1</a>]. The initial formulation was made during discussions between Maxwell, Tait and Thompson (Lord Kelvin). It states that, if a system starts in a highly ordered state and one allows it to approach equilibrium, one can then simply at some point reverse the velocities of all its particles and thus produce a trajectory for which the entropy decreases, and therefore violate the SLT. The argument was independently formulated by Loschmidt in the 1870s and became known to Boltzmann. Brush [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Brush, S.G.: The Kind of Motion We Call Heat. North-Holland, Amsterdam (1976)" href="/article/10.1007/s10773-023-05458-x#ref-CR1" id="ref-link-section-d40198907e339">1</a>] credits Loschmidt with motivating Boltzmann into formulating his H-theorem in terms of probabilities. It is precisely the velocity reversal step that this paper concentrates on.</p><p>In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10773-023-05458-x#Sec2">2</a> we review some of the best-known resolutions of the RP. In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10773-023-05458-x#Sec3">3</a> we look in more detail at the velocity reversal step, introduce elements from the modern theory of information and measurement, and recast the RP in such terms. All this results in a new understanding of the paradox. We discuss the findings in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10773-023-05458-x#Sec4">4</a>.</p></div></div></section><section data-title="Previous Explanations of the Paradox"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Previous Explanations of the Paradox</h2><div class="c-article-section__content" id="Sec2-content"><p>The point of the paradox is that one can construct processes that violate the SLT using an apparently simple reversal of velocities under the existence of reversible dynamics. This section describes previous explanations or resolutions of the paradox.</p><p>In the late 1800s it was argued that, since most available states in equilibrium have high entropy, encountering a configuration that evolved to a low-entropy state is statistically improbable [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Brush, S.G.: The Kind of Motion We Call Heat. North-Holland, Amsterdam (1976)" href="/article/10.1007/s10773-023-05458-x#ref-CR1" id="ref-link-section-d40198907e365">1</a>]. More recent improbability argument address either the relatively small number of low-entropy initial conditions [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Sklar, L.: Physics and Chance. Cambridge University Press, Cambridge (1993)" href="/article/10.1007/s10773-023-05458-x#ref-CR2" id="ref-link-section-d40198907e368">2</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Holian, B.L., Hoover, W.G., Posch, H.A.: Resolution of the Loschmidt paradox: The origin of irreversible behavior in reversible atomistic dynamics. Phys. Rev. Lett. 59, 10–13 (1987)" href="/article/10.1007/s10773-023-05458-x#ref-CR3" id="ref-link-section-d40198907e371">3</a>] or the rarity of trajectories in which entropy decreases [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Evans, D.J., Searles, D.J.: The fluctuation theorem. Adv. Phys. 51, 1529–1585 (2002)" href="/article/10.1007/s10773-023-05458-x#ref-CR4" id="ref-link-section-d40198907e374">4</a>]. Note that such arguments do not explain the RP – they just make its implementation rather improbable. Similarly, we see sensitive dependence on initial conditions arguments as moot, since one can at least conceptually reverse velocities exactly. Even if not, a reversed trajectory in which entropy decreases should be feasible at least for short times, thus still violating the SLT.</p><p>A different current of thought is that microscopic reversibility and macroscopic irreversibility are not incompatible. Swendsen [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Swendsen, R.H.: Explaining irreversibility. Am. J. Phys. 76, 643–648 (2008)" href="/article/10.1007/s10773-023-05458-x#ref-CR5" id="ref-link-section-d40198907e380">5</a>] has studied the case of free expansion of noninteracting particles, initially confined in a small volume, and with velocities following a Maxwell-Boltzmann distribution, into a large container with perfectly reflecting walls. He developed an exact solution for the 1D case, and showed that, while the dynamics is reversible, it mimics the irreversible approach to equilibrium one expects as the particles spread into the larger box. Lebowitz [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Lebowitz, J.L.: Microscopic origins of macroscopic irreversible behavior. Phys. A. 263, 516–527 (1999)" href="/article/10.1007/s10773-023-05458-x#ref-CR6" id="ref-link-section-d40198907e383">6</a>] has also stated that the paradox can be explained with three ingredients: a large separation of scales, the role of initial conditions, and probabilistic reasoning (not every state of a system evolves according to the SLT – only most of them).</p><p>Now, consider two scenarios. (1) A highly ordered macrostate is prepared (e.g., by non-adiabatic compression) and allowed to relax to a higher entropy state. Then the velocities are reversed, and the system comes back (close) to the initial microstate – not just macrostate. The velocity reversal step requires gaining a lot of knowledge about the system that we initially did not have. For the initial evolution we can at best offer a coarse-grained description, within which transitions from state to state are probabilistic. For the entire reversed evolution, we know “everything” about the system – not just its coarse-grained macroscopic configuration. (2) A highly ordered microstate is prepared, allowed to evolve to a state of higher entropy (as specified above); the velocities are reversed, and the system traces back its earlier evolution (close) to the initial microstate. This is allowed due to the reversibility of the system; in this case there is no paradox whatsoever. In scenario (1), however, a potential confusion arises due to the switch between the macro and a micro descriptions, which can be at the root of a misunderstanding of the paradox, as was pointed out almost 50 years ago [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Rothstein, J.: Loschmidt’s and Zermelo’s paradoxes do not exist. Found. Phys. 4, 83–89 (1974)" href="/article/10.1007/s10773-023-05458-x#ref-CR7" id="ref-link-section-d40198907e389">7</a>]. In both cases I have included the specification “close to,” in order to accommodate the earlier point about sensitive dependence.</p></div></div></section><section data-title="Measurement, Erasure, Information, and Entropy"><div class="c-article-section" id="Sec3-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec3"><span class="c-article-section__title-number">3 </span>Measurement, Erasure, Information, and Entropy</h2><div class="c-article-section__content" id="Sec3-content"><p>The SLT is formulated for closed systems; it is time to look at the only step in the RP where the system is open to its environment: the velocity reversal step.</p><p>In order to determine the velocities and positions of the particle components of the system, measurements have to been made by some agent so that, for example, mirror-like barriers can be placed at the appropriate positions and angles so that the particles can indeed reverse their velocities. This has already been noticed by Zurek [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Zurek, W.H.: Reversibility and stability of information processing systems. Phys. Rev. Lett. 53, 391–394 (1984)" href="/article/10.1007/s10773-023-05458-x#ref-CR8" id="ref-link-section-d40198907e403">8</a>]. We will call this agent generically a “Laplace demon.” As part of the measurement process, a record is written in its memory.</p><p>The all-knowing demon (“une vaste intelligence”) proposed by Laplace in a different context [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="De Laplace, P.S.: Philosophical Essay on Probabilities. Dover, New York (1951). Translated from the 1814 French original by Truscott, F. W., Emory, F. L" href="/article/10.1007/s10773-023-05458-x#ref-CR9" id="ref-link-section-d40198907e409">9</a>] shares some features with the better-known Maxwell’s demon [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Leff, H.S., Rex, A.F. (eds.): Maxwell’s Demon. Princeton University Press, Princeton, USA (2014)" href="/article/10.1007/s10773-023-05458-x#ref-CR10" id="ref-link-section-d40198907e412">10</a>]. They both collect information about a system, store it in their memory, take subsequent actions that can presumably violate the SLT, and will either affect their environment during the measurement process, or need to have their memory reset at some point.</p><p>The Maxwell’s demon operates a trap between two containers so that it concentrates the faster particles in one of the containers. This creates a lower-entropy state that violates the SLT at no apparent cost. After 150 years of discussion in the literature, Bennett [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Bennett, C.H.: The thermodynamics of computation – a review. Int. J. Theor. Phys. 21, 905–940 (1982)" href="/article/10.1007/s10773-023-05458-x#ref-CR11" id="ref-link-section-d40198907e418">11</a>] proposed a solution for why the SLT is not violated after all. It invokes Landauer’s principle [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Landauer, R.: Reversibility and heat generation in the computing process. IBM J. Res. Develop. 5, 183–191 (1961)" href="/article/10.1007/s10773-023-05458-x#ref-CR12" id="ref-link-section-d40198907e421">12</a>], which states that during irreversible operations such as memory erasure the entropy in non-information bearing degrees of freedom in the demon or its environment must increase. Landauer’s principle has been recently verified experimentally [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Bérut, A., Arakelyan, A., Petrosyan, A., Ciliberto, S., Dillenschneider, R., Lutz, E.: Experimental verification of Landauer’s principle linking information and thermodynamics. Nature. 483, 187–189 (2012)" href="#ref-CR13" id="ref-link-section-d40198907e424">13</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Jun, Y., Gavrilov, M., Bechhoeffer, J.: High-precision test of Landauer’s principle in a feedback trap. Phys. Rev. Lett. 113, 190601 (2014)" href="#ref-CR14" id="ref-link-section-d40198907e424_1">14</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Hong, J., Lambson, B., Dhuey, S., Bokor, J.: Experimental test of Landauer’s principle in single-bit operations on nanomagnetic memory bits. Sci. Adv. 2, e1501492 (2016)" href="/article/10.1007/s10773-023-05458-x#ref-CR15" id="ref-link-section-d40198907e427">15</a>] in a variety of physical systems. This principle is an extension of an earlier discovery by Szilard [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Szilard, L.: Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. Zeit Phys. 53, 840–856 (1929)" href="/article/10.1007/s10773-023-05458-x#ref-CR16" id="ref-link-section-d40198907e430">16</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Toyabe, S., Sagawa, T., Ueda, M., Muneyuki, E., Sano, M.: Information heat engine: Converting information to energy by feedback control. Nat. Phys. 6, 988–992 (2010)" href="/article/10.1007/s10773-023-05458-x#ref-CR17" id="ref-link-section-d40198907e434">17</a>] that erasing a measurement has a thermodynamic cost, but which now has been extended to apply to irreversible measurements.</p><p>To elaborate on the above: following [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Bennett, C.H.: The thermodynamics of computation – a review. Int. J. Theor. Phys. 21, 905–940 (1982)" href="/article/10.1007/s10773-023-05458-x#ref-CR11" id="ref-link-section-d40198907e441">11</a>], a Maxwell’s demon can potentially perform either irreversible or reversible measurements. In the first case, the measurement of each bit of information will require the generation of at least <i>k</i> ln <i>2</i> of entropy, thus avoiding a violation of the SLT. A reversible measurement requires a standard state for the bits where the measurement will be recorded, so that no previous information is overwritten. Then, the irreversible step is not the measurement, but instead the erasure, or resetting of the memory to the standard state. This is where Landauer’s principle comes in.</p><p>Thus, in order to restore the demon (Maxwell’s or Laplace’s) to its original state after a reversible measurement, there is an entropy cost, which offsets the decrease in entropy in the system as it follows its reverse path. In the RP scenario, the SLT therefore, still holds. As an example, for <i>N</i> monatomic particles and 20 bits per coordinate, the required entropy increase is <i>120Nk</i> ln <i>2.</i> The decrease of entropy in the reverse trajectory is bounded from above by this number, and will usually be less, depending on the level of coarse-graining of the system used to calculate the entropy.</p></div></div></section><section data-title="Discussion and Conclusions"><div class="c-article-section" id="Sec4-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec4"><span class="c-article-section__title-number">4 </span>Discussion and Conclusions</h2><div class="c-article-section__content" id="Sec4-content"><p>The SLT is valid for closed systems. In this paper we have shown that in the standard formulation of the Reversibility Paradox, the system in question must interact, even if for a brief time, with a measuring device. There is an ensuing transfer of information that, either is irreversible, or in order to satisfy the constraints of the SLT of no changes in the environment, needs to be reset. The last step has an entropy cost that offsets the decrease of entropy resulting from the reversed trajectory. Thus, the SLT is not violated. The resolution of the paradox presented here closely mirrors that of the Maxwell’s demon offered by Bennett. While this explanation does not challenge the vast effort in going from microscopic to macroscopic descriptions of physical systems, it simply suggests that this machinery is not the way to explain the reversibility paradox.</p><p>To reinforce the argument, consider any ideal gas process in which the entropy of the gas decreases – for example, an isothermal compression. This decrease is legitimate, as long as there is an equal, or greater increase of entropy somewhere else (e.g., one, or multiple thermal reservoirs) so that the overall entropy of the closed system (gas + reservoir) does not decrease. The set-up of the RP is no different, although it contains an illusion. Our sight is directed to the forward and backwards “closed” evolution, while misdirecting us from the crucial interaction with the environment that protects the integrity of the SLT through a sequence of measurement, information storage, and erasure with an entropy increase through Landauer’s principle, if needed. None of this was known in the 1800s, but as was done with Maxwell’s demon 40 years ago, the complete story can now be told. The other two paradoxes from that time, Gibbs’ and the recurrence paradox, never involve opening the system to an external agent and hence their resolutions are totally different.</p><p>The final remark comes in the context that I have written this paper thinking about classical gases, which were familiar to scientists in the 1800s. By now, a real-life Maxwell’s demon has been implemented in a laser cooling experiment [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Raizen, M.G.: Comprehensive control of atomic motion. Science. 324, 1403–1406 (2009)" href="/article/10.1007/s10773-023-05458-x#ref-CR18" id="ref-link-section-d40198907e476">18</a>], in which ultracold atoms confined by magnetic traps are isolated with the use of carefully tuned laser beams. In this set-up, scattered photons, one per atom, carry information about the turning point and energy of each atom, thus constituting the memory. As these photons are absorbed far away from the set-up with possible re-emission of other photons uncorrelated with the original process, this information is lost and the entropy of the demon - contained in the radiation field - increases [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Binder, P.-M.: Reflections on a wall of light. Science. 322, 1334–1335 (2008)" href="/article/10.1007/s10773-023-05458-x#ref-CR19" id="ref-link-section-d40198907e479">19</a>]. The demon here is an inert, physical device – but the point is that it is real – not just an idle object of speculation.</p></div></div></section> </div> <div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">Brush, S.G.: The Kind of Motion We Call Heat. North-Holland, Amsterdam (1976)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 1" href="http://scholar.google.com/scholar_lookup?&title=The%20Kind%20of%20Motion%20We%20Call%20Heat&publication_year=1976&author=Brush%2CSG"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="2."><p class="c-article-references__text" id="ref-CR2">Sklar, L.: Physics and Chance. Cambridge University Press, Cambridge (1993)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1017/CBO9780511624933" data-track-item_id="10.1017/CBO9780511624933" data-track-value="book reference" data-track-action="book reference" href="https://doi.org/10.1017%2FCBO9780511624933" aria-label="Book reference 2" data-doi="10.1017/CBO9780511624933">Book</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 2" href="http://scholar.google.com/scholar_lookup?&title=Physics%20and%20Chance&doi=10.1017%2FCBO9780511624933&publication_year=1993&author=Sklar%2CL"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="3."><p class="c-article-references__text" id="ref-CR3">Holian, B.L., Hoover, W.G., Posch, H.A.: Resolution of the Loschmidt paradox: The origin of irreversible behavior in reversible atomistic dynamics. Phys. Rev. Lett. <b>59</b>, 10–13 (1987)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.59.10" data-track-item_id="10.1103/PhysRevLett.59.10" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.59.10" aria-label="Article reference 3" data-doi="10.1103/PhysRevLett.59.10">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=1987PhRvL..59...10H" aria-label="ADS reference 3">ADS</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=894674" aria-label="MathSciNet reference 3">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 3" href="http://scholar.google.com/scholar_lookup?&title=Resolution%20of%20the%20Loschmidt%20paradox%3A%20The%20origin%20of%20irreversible%20behavior%20in%20reversible%20atomistic%20dynamics&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.59.10&volume=59&pages=10-13&publication_year=1987&author=Holian%2CBL&author=Hoover%2CWG&author=Posch%2CHA"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="4."><p class="c-article-references__text" id="ref-CR4">Evans, D.J., Searles, D.J.: The fluctuation theorem. Adv. Phys. <b>51</b>, 1529–1585 (2002)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1080/00018730210155133" data-track-item_id="10.1080/00018730210155133" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1080%2F00018730210155133" aria-label="Article reference 4" data-doi="10.1080/00018730210155133">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2002AdPhy..51.1529E" aria-label="ADS reference 4">ADS</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 4" href="http://scholar.google.com/scholar_lookup?&title=The%20fluctuation%20theorem&journal=Adv.%20Phys.&doi=10.1080%2F00018730210155133&volume=51&pages=1529-1585&publication_year=2002&author=Evans%2CDJ&author=Searles%2CDJ"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="5."><p class="c-article-references__text" id="ref-CR5">Swendsen, R.H.: Explaining irreversibility. Am. J. Phys. <b>76</b>, 643–648 (2008)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1119/1.2894523" data-track-item_id="10.1119/1.2894523" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1119%2F1.2894523" aria-label="Article reference 5" data-doi="10.1119/1.2894523">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2008AmJPh..76..643S" aria-label="ADS reference 5">ADS</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 5" href="http://scholar.google.com/scholar_lookup?&title=Explaining%20irreversibility&journal=Am.%20J.%20Phys.&doi=10.1119%2F1.2894523&volume=76&pages=643-648&publication_year=2008&author=Swendsen%2CRH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="6."><p class="c-article-references__text" id="ref-CR6">Lebowitz, J.L.: Microscopic origins of macroscopic irreversible behavior. Phys. A. <b>263</b>, 516–527 (1999)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/S0378-4371(98)00514-7" data-track-item_id="10.1016/S0378-4371(98)00514-7" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2FS0378-4371%2898%2900514-7" aria-label="Article reference 6" data-doi="10.1016/S0378-4371(98)00514-7">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1723369" aria-label="MathSciNet reference 6">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 6" href="http://scholar.google.com/scholar_lookup?&title=Microscopic%20origins%20of%20macroscopic%20irreversible%20behavior&journal=Phys.%20A&doi=10.1016%2FS0378-4371%2898%2900514-7&volume=263&pages=516-527&publication_year=1999&author=Lebowitz%2CJL"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="7."><p class="c-article-references__text" id="ref-CR7">Rothstein, J.: Loschmidt’s and Zermelo’s paradoxes do not exist. Found. Phys. <b>4</b>, 83–89 (1974)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/BF00708557" data-track-item_id="10.1007/BF00708557" data-track-value="article reference" data-track-action="article reference" href="https://link.springer.com/doi/10.1007/BF00708557" aria-label="Article reference 7" data-doi="10.1007/BF00708557">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=1974FoPh....4...83R" aria-label="ADS reference 7">ADS</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 7" href="http://scholar.google.com/scholar_lookup?&title=Loschmidt%E2%80%99s%20and%20Zermelo%E2%80%99s%20paradoxes%20do%20not%20exist&journal=Found.%20Phys.&doi=10.1007%2FBF00708557&volume=4&pages=83-89&publication_year=1974&author=Rothstein%2CJ"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="8."><p class="c-article-references__text" id="ref-CR8">Zurek, W.H.: Reversibility and stability of information processing systems. Phys. Rev. Lett. <b>53</b>, 391–394 (1984)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.53.391" data-track-item_id="10.1103/PhysRevLett.53.391" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.53.391" aria-label="Article reference 8" data-doi="10.1103/PhysRevLett.53.391">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=1984PhRvL..53..391Z" aria-label="ADS reference 8">ADS</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 8" href="http://scholar.google.com/scholar_lookup?&title=Reversibility%20and%20stability%20of%20information%20processing%20systems&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.53.391&volume=53&pages=391-394&publication_year=1984&author=Zurek%2CWH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="9."><p class="c-article-references__text" id="ref-CR9">De Laplace, P.S.: Philosophical Essay on Probabilities. Dover, New York (1951). Translated from the 1814 French original by Truscott, F. W., Emory, F. L</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0045.06905" aria-label="MATH reference 9">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 9" href="http://scholar.google.com/scholar_lookup?&title=Philosophical%20Essay%20on%20Probabilities&publication_year=1951&author=De%20Laplace%2CPS"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="10."><p class="c-article-references__text" id="ref-CR10">Leff, H.S., Rex, A.F. (eds.): Maxwell’s Demon. Princeton University Press, Princeton, USA (2014)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 10" href="http://scholar.google.com/scholar_lookup?&title=Maxwell%E2%80%99s%20Demon&publication_year=2014"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="11."><p class="c-article-references__text" id="ref-CR11">Bennett, C.H.: The thermodynamics of computation – a review. Int. J. Theor. Phys. <b>21</b>, 905–940 (1982)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/BF02084158" data-track-item_id="10.1007/BF02084158" data-track-value="article reference" data-track-action="article reference" href="https://link.springer.com/doi/10.1007/BF02084158" aria-label="Article reference 11" data-doi="10.1007/BF02084158">Article</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 11" href="http://scholar.google.com/scholar_lookup?&title=The%20thermodynamics%20of%20computation%20%E2%80%93%20a%20review&journal=Int.%20J.%20Theor.%20Phys.&doi=10.1007%2FBF02084158&volume=21&pages=905-940&publication_year=1982&author=Bennett%2CCH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="12."><p class="c-article-references__text" id="ref-CR12">Landauer, R.: Reversibility and heat generation in the computing process. IBM J. Res. Develop. <b>5</b>, 183–191 (1961)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1147/rd.53.0183" data-track-item_id="10.1147/rd.53.0183" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1147%2Frd.53.0183" aria-label="Article reference 12" data-doi="10.1147/rd.53.0183">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=134833" aria-label="MathSciNet reference 12">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1160.68305" aria-label="MATH reference 12">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 12" href="http://scholar.google.com/scholar_lookup?&title=Reversibility%20and%20heat%20generation%20in%20the%20computing%20process&journal=IBM%20J.%20Res.%20Develop&doi=10.1147%2Frd.53.0183&volume=5&pages=183-191&publication_year=1961&author=Landauer%2CR"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="13."><p class="c-article-references__text" id="ref-CR13">Bérut, A., Arakelyan, A., Petrosyan, A., Ciliberto, S., Dillenschneider, R., Lutz, E.: Experimental verification of Landauer’s principle linking information and thermodynamics. Nature. <b>483</b>, 187–189 (2012)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/nature10872" data-track-item_id="10.1038/nature10872" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2Fnature10872" aria-label="Article reference 13" data-doi="10.1038/nature10872">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2012Natur.483..187B" aria-label="ADS reference 13">ADS</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 13" href="http://scholar.google.com/scholar_lookup?&title=Experimental%20verification%20of%20Landauer%E2%80%99s%20principle%20linking%20information%20and%20thermodynamics&journal=Nature&doi=10.1038%2Fnature10872&volume=483&pages=187-189&publication_year=2012&author=B%C3%A9rut%2CA&author=Arakelyan%2CA&author=Petrosyan%2CA&author=Ciliberto%2CS&author=Dillenschneider%2CR&author=Lutz%2CE"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="14."><p class="c-article-references__text" id="ref-CR14">Jun, Y., Gavrilov, M., Bechhoeffer, J.: High-precision test of Landauer’s principle in a feedback trap. Phys. Rev. Lett. <b>113</b>, 190601 (2014)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1103/PhysRevLett.113.190601" data-track-item_id="10.1103/PhysRevLett.113.190601" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1103%2FPhysRevLett.113.190601" aria-label="Article reference 14" data-doi="10.1103/PhysRevLett.113.190601">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2014PhRvL.113s0601J" aria-label="ADS reference 14">ADS</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 14" href="http://scholar.google.com/scholar_lookup?&title=High-precision%20test%20of%20Landauer%E2%80%99s%20principle%20in%20a%20feedback%20trap&journal=Phys.%20Rev.%20Lett.&doi=10.1103%2FPhysRevLett.113.190601&volume=113&publication_year=2014&author=Jun%2CY&author=Gavrilov%2CM&author=Bechhoeffer%2CJ"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="15."><p class="c-article-references__text" id="ref-CR15">Hong, J., Lambson, B., Dhuey, S., Bokor, J.: Experimental test of Landauer’s principle in single-bit operations on nanomagnetic memory bits. Sci. Adv. <b>2</b>, e1501492 (2016)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="16."><p class="c-article-references__text" id="ref-CR16">Szilard, L.: Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. Zeit Phys. <b>53</b>, 840–856 (1929)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/BF01341281" data-track-item_id="10.1007/BF01341281" data-track-value="article reference" data-track-action="article reference" href="https://link.springer.com/doi/10.1007/BF01341281" aria-label="Article reference 16" data-doi="10.1007/BF01341281">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=1929ZPhy...53..840S" aria-label="ADS reference 16">ADS</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?55.0488.06" aria-label="MATH reference 16">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 16" href="http://scholar.google.com/scholar_lookup?&title=%C3%9Cber%20die%20Entropieverminderung%20in%20einem%20thermodynamischen%20System%20bei%20Eingriffen%20intelligenter%20Wesen&journal=Zeit%20Phys.&doi=10.1007%2FBF01341281&volume=53&pages=840-856&publication_year=1929&author=Szilard%2CL"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="17."><p class="c-article-references__text" id="ref-CR17">Toyabe, S., Sagawa, T., Ueda, M., Muneyuki, E., Sano, M.: Information heat engine: Converting information to energy by feedback control. Nat. Phys. <b>6</b>, 988–992 (2010)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1038/nphys1821" data-track-item_id="10.1038/nphys1821" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1038%2Fnphys1821" aria-label="Article reference 17" data-doi="10.1038/nphys1821">Article</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 17" href="http://scholar.google.com/scholar_lookup?&title=Information%20heat%20engine%3A%20Converting%20information%20to%20energy%20by%20feedback%20control&journal=Nat.%20Phys.&doi=10.1038%2Fnphys1821&volume=6&pages=988-992&publication_year=2010&author=Toyabe%2CS&author=Sagawa%2CT&author=Ueda%2CM&author=Muneyuki%2CE&author=Sano%2CM"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="18."><p class="c-article-references__text" id="ref-CR18">Raizen, M.G.: Comprehensive control of atomic motion. Science. <b>324</b>, 1403–1406 (2009)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1126/science.1171506" data-track-item_id="10.1126/science.1171506" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1126%2Fscience.1171506" aria-label="Article reference 18" data-doi="10.1126/science.1171506">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="ads reference" data-track-action="ads reference" href="http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=2009Sci...324.1403R" aria-label="ADS reference 18">ADS</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 18" href="http://scholar.google.com/scholar_lookup?&title=Comprehensive%20control%20of%20atomic%20motion&journal=Science&doi=10.1126%2Fscience.1171506&volume=324&pages=1403-1406&publication_year=2009&author=Raizen%2CMG"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="19."><p class="c-article-references__text" id="ref-CR19">Binder, P.-M.: Reflections on a wall of light. Science. <b>322</b>, 1334–1335 (2008)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1126/science.1166681" data-track-item_id="10.1126/science.1166681" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1126%2Fscience.1166681" aria-label="Article reference 19" data-doi="10.1126/science.1166681">Article</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 19" href="http://scholar.google.com/scholar_lookup?&title=Reflections%20on%20a%20wall%20of%20light&journal=Science&doi=10.1126%2Fscience.1166681&volume=322&pages=1334-1335&publication_year=2008&author=Binder%2CP-M"> Google Scholar</a> </p></li></ol><p class="c-article-references__download u-hide-print"><a data-track="click" data-track-action="download citation references" data-track-label="link" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s10773-023-05458-x?format=refman&flavour=references">Download references<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p></div></div></div></section></div><section data-title="Acknowledgements"><div class="c-article-section" id="Ack1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Ack1">Acknowledgements</h2><div class="c-article-section__content" id="Ack1-content"><p>The author thanks Juan Manuel Pedraza, Samir Garzon, Ben Browning, Seneca Cox and Dallas Tada for conversations on this subject.</p></div></div></section><section data-title="Funding"><div class="c-article-section" id="Fun-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Fun">Funding</h2><div class="c-article-section__content" id="Fun-content"><p>Publication charges were paid by the Reid Hoffman Foundation.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">University of Hawaii, 200 W Kawili St, Hilo, HI, 96720-4091, USA</p><p class="c-article-author-affiliation__authors-list">P.M. Binder</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-P_M_-Binder-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">P.M. Binder</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?dc.creator=P.M.%20Binder" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=P.M.%20Binder" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22P.M.%20Binder%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="contributions">Contributions</h3><p>P.M.B., as the only author, did all the work related to this manuscript.</p><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:pbinder@hawaii.edu">P.M. Binder</a>.</p></div></div></section><section data-title="Ethics declarations"><div class="c-article-section" id="ethics-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="ethics">Ethics declarations</h2><div class="c-article-section__content" id="ethics-content"> <h3 class="c-article__sub-heading" id="FPar1">Competing Interests</h3> <p>The authors declare no competing interests.</p> </div></div></section><section data-title="Additional information"><div class="c-article-section" id="additional-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="additional-information">Additional information</h2><div class="c-article-section__content" id="additional-information-content"><h3 class="c-article__sub-heading">Publisher’s Note</h3><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"> <p><b>Open Access</b> This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit <a href="http://creativecommons.org/licenses/by/4.0/" rel="license">http://creativecommons.org/licenses/by/4.0/</a>.</p> <p class="c-article-rights"><a data-track="click" data-track-action="view rights and permissions" data-track-label="link" href="https://s100.copyright.com/AppDispatchServlet?title=The%20Reversibility%20Paradox%3A%20Role%20of%20the%20Velocity%20Reversal%20Step&author=P.M.%20Binder&contentID=10.1007%2Fs10773-023-05458-x&copyright=The%20Author%28s%29&publication=1572-9575&publicationDate=2023-09-02&publisherName=SpringerNature&orderBeanReset=true&oa=CC%20BY">Reprints and permissions</a></p></div></div></section><section aria-labelledby="article-info" data-title="About this article"><div class="c-article-section" id="article-info-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="article-info">About this article</h2><div class="c-article-section__content" id="article-info-content"><div class="c-bibliographic-information"><div class="u-hide-print c-bibliographic-information__column c-bibliographic-information__column--border"><a data-crossmark="10.1007/s10773-023-05458-x" target="_blank" rel="noopener" href="https://crossmark.crossref.org/dialog/?doi=10.1007/s10773-023-05458-x" data-track="click" data-track-action="Click Crossmark" data-track-label="link" data-test="crossmark"><img loading="lazy" width="57" height="81" alt="Check for updates. Verify currency and authenticity via CrossMark" src="data:image/svg+xml;base64,<svg height="81" width="57" xmlns="http://www.w3.org/2000/svg"><g fill="none" fill-rule="evenodd"><path d="m17.35 35.45 21.3-14.2v-17.03h-21.3" fill="#989898"/><path d="m38.65 35.45-21.3-14.2v-17.03h21.3" fill="#747474"/><path d="m28 .5c-12.98 0-23.5 10.52-23.5 23.5s10.52 23.5 23.5 23.5 23.5-10.52 23.5-23.5c0-6.23-2.48-12.21-6.88-16.62-4.41-4.4-10.39-6.88-16.62-6.88zm0 41.25c-9.8 0-17.75-7.95-17.75-17.75s7.95-17.75 17.75-17.75 17.75 7.95 17.75 17.75c0 4.71-1.87 9.22-5.2 12.55s-7.84 5.2-12.55 5.2z" fill="#535353"/><path d="m41 36c-5.81 6.23-15.23 7.45-22.43 2.9-7.21-4.55-10.16-13.57-7.03-21.5l-4.92-3.11c-4.95 10.7-1.19 23.42 8.78 29.71 9.97 6.3 23.07 4.22 30.6-4.86z" fill="#9c9c9c"/><path d="m.2 58.45c0-.75.11-1.42.33-2.01s.52-1.09.91-1.5c.38-.41.83-.73 1.34-.94.51-.22 1.06-.32 1.65-.32.56 0 1.06.11 1.51.35.44.23.81.5 1.1.81l-.91 1.01c-.24-.24-.49-.42-.75-.56-.27-.13-.58-.2-.93-.2-.39 0-.73.08-1.05.23-.31.16-.58.37-.81.66-.23.28-.41.63-.53 1.04-.13.41-.19.88-.19 1.39 0 1.04.23 1.86.68 2.46.45.59 1.06.88 1.84.88.41 0 .77-.07 1.07-.23s.59-.39.85-.68l.91 1c-.38.43-.8.76-1.28.99-.47.22-1 .34-1.58.34-.59 0-1.13-.1-1.64-.31-.5-.2-.94-.51-1.31-.91-.38-.4-.67-.9-.88-1.48-.22-.59-.33-1.26-.33-2.02zm8.4-5.33h1.61v2.54l-.05 1.33c.29-.27.61-.51.96-.72s.76-.31 1.24-.31c.73 0 1.27.23 1.61.71.33.47.5 1.14.5 2.02v4.31h-1.61v-4.1c0-.57-.08-.97-.25-1.21-.17-.23-.45-.35-.83-.35-.3 0-.56.08-.79.22-.23.15-.49.36-.78.64v4.8h-1.61zm7.37 6.45c0-.56.09-1.06.26-1.51.18-.45.42-.83.71-1.14.29-.3.63-.54 1.01-.71.39-.17.78-.25 1.18-.25.47 0 .88.08 1.23.24.36.16.65.38.89.67s.42.63.54 1.03c.12.41.18.84.18 1.32 0 .32-.02.57-.07.76h-4.36c.07.62.29 1.1.65 1.44.36.33.82.5 1.38.5.29 0 .57-.04.83-.13s.51-.21.76-.37l.55 1.01c-.33.21-.69.39-1.09.53-.41.14-.83.21-1.26.21-.48 0-.92-.08-1.34-.25-.41-.16-.76-.4-1.07-.7-.31-.31-.55-.69-.72-1.13-.18-.44-.26-.95-.26-1.52zm4.6-.62c0-.55-.11-.98-.34-1.28-.23-.31-.58-.47-1.06-.47-.41 0-.77.15-1.07.45-.31.29-.5.73-.58 1.3zm2.5.62c0-.57.09-1.08.28-1.53.18-.44.43-.82.75-1.13s.69-.54 1.1-.71c.42-.16.85-.24 1.31-.24.45 0 .84.08 1.17.23s.61.34.85.57l-.77 1.02c-.19-.16-.38-.28-.56-.37-.19-.09-.39-.14-.61-.14-.56 0-1.01.21-1.35.63-.35.41-.52.97-.52 1.67 0 .69.17 1.24.51 1.66.34.41.78.62 1.32.62.28 0 .54-.06.78-.17.24-.12.45-.26.64-.42l.67 1.03c-.33.29-.69.51-1.08.65-.39.15-.78.23-1.18.23-.46 0-.9-.08-1.31-.24-.4-.16-.75-.39-1.05-.7s-.53-.69-.7-1.13c-.17-.45-.25-.96-.25-1.53zm6.91-6.45h1.58v6.17h.05l2.54-3.16h1.77l-2.35 2.8 2.59 4.07h-1.75l-1.77-2.98-1.08 1.23v1.75h-1.58zm13.69 1.27c-.25-.11-.5-.17-.75-.17-.58 0-.87.39-.87 1.16v.75h1.34v1.27h-1.34v5.6h-1.61v-5.6h-.92v-1.2l.92-.07v-.72c0-.35.04-.68.13-.98.08-.31.21-.57.4-.79s.42-.39.71-.51c.28-.12.63-.18 1.04-.18.24 0 .48.02.69.07.22.05.41.1.57.17zm.48 5.18c0-.57.09-1.08.27-1.53.17-.44.41-.82.72-1.13.3-.31.65-.54 1.04-.71.39-.16.8-.24 1.23-.24s.84.08 1.24.24c.4.17.74.4 1.04.71s.54.69.72 1.13c.19.45.28.96.28 1.53s-.09 1.08-.28 1.53c-.18.44-.42.82-.72 1.13s-.64.54-1.04.7-.81.24-1.24.24-.84-.08-1.23-.24-.74-.39-1.04-.7c-.31-.31-.55-.69-.72-1.13-.18-.45-.27-.96-.27-1.53zm1.65 0c0 .69.14 1.24.43 1.66.28.41.68.62 1.18.62.51 0 .9-.21 1.19-.62.29-.42.44-.97.44-1.66 0-.7-.15-1.26-.44-1.67-.29-.42-.68-.63-1.19-.63-.5 0-.9.21-1.18.63-.29.41-.43.97-.43 1.67zm6.48-3.44h1.33l.12 1.21h.05c.24-.44.54-.79.88-1.02.35-.24.7-.36 1.07-.36.32 0 .59.05.78.14l-.28 1.4-.33-.09c-.11-.01-.23-.02-.38-.02-.27 0-.56.1-.86.31s-.55.58-.77 1.1v4.2h-1.61zm-47.87 15h1.61v4.1c0 .57.08.97.25 1.2.17.24.44.35.81.35.3 0 .57-.07.8-.22.22-.15.47-.39.73-.73v-4.7h1.61v6.87h-1.32l-.12-1.01h-.04c-.3.36-.63.64-.98.86-.35.21-.76.32-1.24.32-.73 0-1.27-.24-1.61-.71-.33-.47-.5-1.14-.5-2.02zm9.46 7.43v2.16h-1.61v-9.59h1.33l.12.72h.05c.29-.24.61-.45.97-.63.35-.17.72-.26 1.1-.26.43 0 .81.08 1.15.24.33.17.61.4.84.71.24.31.41.68.53 1.11.13.42.19.91.19 1.44 0 .59-.09 1.11-.25 1.57-.16.47-.38.85-.65 1.16-.27.32-.58.56-.94.73-.35.16-.72.25-1.1.25-.3 0-.6-.07-.9-.2s-.59-.31-.87-.56zm0-2.3c.26.22.5.37.73.45.24.09.46.13.66.13.46 0 .84-.2 1.15-.6.31-.39.46-.98.46-1.77 0-.69-.12-1.22-.35-1.61-.23-.38-.61-.57-1.13-.57-.49 0-.99.26-1.52.77zm5.87-1.69c0-.56.08-1.06.25-1.51.16-.45.37-.83.65-1.14.27-.3.58-.54.93-.71s.71-.25 1.08-.25c.39 0 .73.07 1 .2.27.14.54.32.81.55l-.06-1.1v-2.49h1.61v9.88h-1.33l-.11-.74h-.06c-.25.25-.54.46-.88.64-.33.18-.69.27-1.06.27-.87 0-1.56-.32-2.07-.95s-.76-1.51-.76-2.65zm1.67-.01c0 .74.13 1.31.4 1.7.26.38.65.58 1.15.58.51 0 .99-.26 1.44-.77v-3.21c-.24-.21-.48-.36-.7-.45-.23-.08-.46-.12-.7-.12-.45 0-.82.19-1.13.59-.31.39-.46.95-.46 1.68zm6.35 1.59c0-.73.32-1.3.97-1.71.64-.4 1.67-.68 3.08-.84 0-.17-.02-.34-.07-.51-.05-.16-.12-.3-.22-.43s-.22-.22-.38-.3c-.15-.06-.34-.1-.58-.1-.34 0-.68.07-1 .2s-.63.29-.93.47l-.59-1.08c.39-.24.81-.45 1.28-.63.47-.17.99-.26 1.54-.26.86 0 1.51.25 1.93.76s.63 1.25.63 2.21v4.07h-1.32l-.12-.76h-.05c-.3.27-.63.48-.98.66s-.73.27-1.14.27c-.61 0-1.1-.19-1.48-.56-.38-.36-.57-.85-.57-1.46zm1.57-.12c0 .3.09.53.27.67.19.14.42.21.71.21.28 0 .54-.07.77-.2s.48-.31.73-.56v-1.54c-.47.06-.86.13-1.18.23-.31.09-.57.19-.76.31s-.33.25-.41.4c-.09.15-.13.31-.13.48zm6.29-3.63h-.98v-1.2l1.06-.07.2-1.88h1.34v1.88h1.75v1.27h-1.75v3.28c0 .8.32 1.2.97 1.2.12 0 .24-.01.37-.04.12-.03.24-.07.34-.11l.28 1.19c-.19.06-.4.12-.64.17-.23.05-.49.08-.76.08-.4 0-.74-.06-1.02-.18-.27-.13-.49-.3-.67-.52-.17-.21-.3-.48-.37-.78-.08-.3-.12-.64-.12-1.01zm4.36 2.17c0-.56.09-1.06.27-1.51s.41-.83.71-1.14c.29-.3.63-.54 1.01-.71.39-.17.78-.25 1.18-.25.47 0 .88.08 1.23.24.36.16.65.38.89.67s.42.63.54 1.03c.12.41.18.84.18 1.32 0 .32-.02.57-.07.76h-4.37c.08.62.29 1.1.65 1.44.36.33.82.5 1.38.5.3 0 .58-.04.84-.13.25-.09.51-.21.76-.37l.54 1.01c-.32.21-.69.39-1.09.53s-.82.21-1.26.21c-.47 0-.92-.08-1.33-.25-.41-.16-.77-.4-1.08-.7-.3-.31-.54-.69-.72-1.13-.17-.44-.26-.95-.26-1.52zm4.61-.62c0-.55-.11-.98-.34-1.28-.23-.31-.58-.47-1.06-.47-.41 0-.77.15-1.08.45-.31.29-.5.73-.57 1.3zm3.01 2.23c.31.24.61.43.92.57.3.13.63.2.98.2.38 0 .65-.08.83-.23s.27-.35.27-.6c0-.14-.05-.26-.13-.37-.08-.1-.2-.2-.34-.28-.14-.09-.29-.16-.47-.23l-.53-.22c-.23-.09-.46-.18-.69-.3-.23-.11-.44-.24-.62-.4s-.33-.35-.45-.55c-.12-.21-.18-.46-.18-.75 0-.61.23-1.1.68-1.49.44-.38 1.06-.57 1.83-.57.48 0 .91.08 1.29.25s.71.36.99.57l-.74.98c-.24-.17-.49-.32-.73-.42-.25-.11-.51-.16-.78-.16-.35 0-.6.07-.76.21-.17.15-.25.33-.25.54 0 .14.04.26.12.36s.18.18.31.26c.14.07.29.14.46.21l.54.19c.23.09.47.18.7.29s.44.24.64.4c.19.16.34.35.46.58.11.23.17.5.17.82 0 .3-.06.58-.17.83-.12.26-.29.48-.51.68-.23.19-.51.34-.84.45-.34.11-.72.17-1.15.17-.48 0-.95-.09-1.41-.27-.46-.19-.86-.41-1.2-.68z" fill="#535353"/></g></svg>"></a></div><div class="c-bibliographic-information__column"><h3 class="c-article__sub-heading" id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Binder, P. The Reversibility Paradox: Role of the Velocity Reversal Step. <i>Int J Theor Phys</i> <b>62</b>, 200 (2023). https://doi.org/10.1007/s10773-023-05458-x</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s10773-023-05458-x?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2023-07-09">09 July 2023</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Accepted<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2023-08-24">24 August 2023</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Published<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2023-09-02">02 September 2023</time></span></p></li><li class="c-bibliographic-information__list-item c-bibliographic-information__list-item--full-width"><p><abbr title="Digital Object Identifier">DOI</abbr><span class="u-hide">: </span><span class="c-bibliographic-information__value">https://doi.org/10.1007/s10773-023-05458-x</span></p></li></ul><div data-component="share-box"><div class="c-article-share-box u-display-none" hidden=""><h3 class="c-article__sub-heading">Share this article</h3><p class="c-article-share-box__description">Anyone you share the following link with will be able to read this content:</p><button class="js-get-share-url c-article-share-box__button" type="button" id="get-share-url" data-track="click" data-track-label="button" data-track-external="" data-track-action="get shareable link">Get shareable link</button><div class="js-no-share-url-container u-display-none" hidden=""><p class="js-c-article-share-box__no-sharelink-info c-article-share-box__no-sharelink-info">Sorry, a shareable link is not currently available for this article.</p></div><div class="js-share-url-container u-display-none" hidden=""><p class="js-share-url c-article-share-box__only-read-input" id="share-url" data-track="click" data-track-label="button" data-track-action="select share url"></p><button class="js-copy-share-url c-article-share-box__button--link-like" type="button" id="copy-share-url" data-track="click" data-track-label="button" data-track-action="copy share url" data-track-external="">Copy to clipboard</button></div><p class="js-c-article-share-box__additional-info c-article-share-box__additional-info"> Provided by the Springer Nature SharedIt content-sharing initiative </p></div></div><h3 class="c-article__sub-heading">Keywords</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=Foundations%20of%20physics&facet-discipline="Physics"" data-track="click" data-track-action="view keyword" data-track-label="link">Foundations of physics</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=History%20of%20science&facet-discipline="Physics"" data-track="click" data-track-action="view keyword" data-track-label="link">History of science</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Loschmidt%20paradox&facet-discipline="Physics"" data-track="click" data-track-action="view keyword" data-track-label="link">Loschmidt paradox</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Landauer%E2%80%99s%20principle&facet-discipline="Physics"" data-track="click" data-track-action="view keyword" data-track-label="link">Landauer’s principle</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Maxwell%E2%80%99s%20demon&facet-discipline="Physics"" data-track="click" data-track-action="view keyword" data-track-label="link">Maxwell’s demon</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Second%20Law%20of%20Thermodynamics&facet-discipline="Physics"" data-track="click" data-track-action="view keyword" data-track-label="link">Second Law of Thermodynamics</a></span></li></ul><div data-component="article-info-list"></div></div></div></div></div></section> </div> </main> <div class="c-article-sidebar u-text-sm u-hide-print l-with-sidebar__sidebar" id="sidebar" data-container-type="reading-companion" data-track-component="reading companion"> <aside> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=10773" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div data-test="collections"> </div> <div data-test="editorial-summary"> </div> <div class="c-reading-companion"> <div class="c-reading-companion__sticky" data-component="reading-companion-sticky" data-test="reading-companion-sticky"> <div class="c-reading-companion__panel c-reading-companion__sections c-reading-companion__panel--active" id="tabpanel-sections"> <div class="u-lazy-ad-wrapper u-mt-16 u-hide" data-component-mpu><div class="c-ad c-ad--300x250"> <div class="c-ad__inner"> <p class="c-ad__label">Advertisement</p> <div id="div-gpt-ad-MPU1" class="div-gpt-ad grade-c-hide" data-pa11y-ignore data-gpt data-gpt-unitpath="/270604982/springerlink/10773/article" data-gpt-sizes="300x250" data-test="MPU1-ad" data-gpt-targeting="pos=MPU1;articleid=s10773-023-05458-x;"> </div> </div> </div> </div> </div> <div class="c-reading-companion__panel c-reading-companion__figures c-reading-companion__panel--full-width" id="tabpanel-figures"></div> <div class="c-reading-companion__panel c-reading-companion__references c-reading-companion__panel--full-width" id="tabpanel-references"></div> </div> </div> </aside> </div> </div> </article> <div class="app-elements"> <div class="eds-c-header__expander eds-c-header__expander--search" id="eds-c-header-popup-search"> <h2 class="eds-c-header__heading">Search</h2> <div class="u-container"> <search class="eds-c-header__search" role="search" aria-label="Search from the header"> <form method="GET" action="//link.springer.com/search" data-test="header-search" data-track="search" data-track-context="search from header" data-track-action="submit search form" data-track-category="unified header" data-track-label="form" > <label for="eds-c-header-search" class="eds-c-header__search-label">Search by keyword or author</label> <div class="eds-c-header__search-container"> <input id="eds-c-header-search" class="eds-c-header__search-input" autocomplete="off" name="query" type="search" value="" required> <button class="eds-c-header__search-button" type="submit"> <svg class="eds-c-header__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-search-medium"></use> </svg> <span class="u-visually-hidden">Search</span> </button> </div> </form> </search> </div> </div> <div class="eds-c-header__expander eds-c-header__expander--menu" id="eds-c-header-nav"> <h2 class="eds-c-header__heading">Navigation</h2> <ul class="eds-c-header__list"> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://link.springer.com/journals/" data-track="nav_find_a_journal" data-track-context="unified header" data-track-action="click find a journal" data-track-category="unified header" data-track-label="link" > Find a journal </a> </li> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://www.springernature.com/gp/authors" data-track="nav_how_to_publish" data-track-context="unified header" data-track-action="click publish with us link" data-track-category="unified header" data-track-label="link" > Publish with us </a> </li> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://link.springernature.com/home/" data-track="nav_track_your_research" data-track-context="unified header" data-track-action="click track your research" data-track-category="unified header" data-track-label="link" > Track your research </a> </li> </ul> </div> <footer > <div class="eds-c-footer" > <div class="eds-c-footer__container"> <div class="eds-c-footer__grid eds-c-footer__group--separator"> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Discover content</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/journals/a/1" data-track="nav_journals_a_z" data-track-action="journals a-z" data-track-context="unified footer" data-track-label="link">Journals A-Z</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/books/a/1" data-track="nav_books_a_z" data-track-action="books a-z" data-track-context="unified footer" data-track-label="link">Books A-Z</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Publish with us</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/journals" data-track="nav_journal_finder" data-track-action="journal finder" data-track-context="unified footer" data-track-label="link">Journal finder</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/authors" data-track="nav_publish_your_research" data-track-action="publish your research" data-track-context="unified footer" data-track-label="link">Publish your research</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="nav_open_access_publishing" data-track-action="open access publishing" data-track-context="unified footer" data-track-label="link">Open access publishing</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Products and services</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/products" data-track="nav_our_products" data-track-action="our products" data-track-context="unified footer" data-track-label="link">Our products</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/librarians" data-track="nav_librarians" data-track-action="librarians" data-track-context="unified footer" data-track-label="link">Librarians</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/societies" data-track="nav_societies" data-track-action="societies" data-track-context="unified footer" data-track-label="link">Societies</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/partners" data-track="nav_partners_and_advertisers" data-track-action="partners and advertisers" data-track-context="unified footer" data-track-label="link">Partners and advertisers</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Our imprints</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springer.com/" data-track="nav_imprint_Springer" data-track-action="Springer" data-track-context="unified footer" data-track-label="link">Springer</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.nature.com/" data-track="nav_imprint_Nature_Portfolio" data-track-action="Nature Portfolio" data-track-context="unified footer" data-track-label="link">Nature Portfolio</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.biomedcentral.com/" data-track="nav_imprint_BMC" data-track-action="BMC" data-track-context="unified footer" data-track-label="link">BMC</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.palgrave.com/" data-track="nav_imprint_Palgrave_Macmillan" data-track-action="Palgrave Macmillan" data-track-context="unified footer" data-track-label="link">Palgrave Macmillan</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.apress.com/" data-track="nav_imprint_Apress" data-track-action="Apress" data-track-context="unified footer" data-track-label="link">Apress</a></li> </ul> </div> </div> </div> <div class="eds-c-footer__container"> <nav aria-label="footer navigation"> <ul class="eds-c-footer__links"> <li class="eds-c-footer__item"> <button class="eds-c-footer__link" data-cc-action="preferences" data-track="dialog_manage_cookies" data-track-action="Manage cookies" data-track-context="unified footer" data-track-label="link"><span class="eds-c-footer__button-text">Your privacy choices/Manage cookies</span></button> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://www.springernature.com/gp/legal/ccpa" data-track="nav_california_privacy_statement" data-track-action="california privacy statement" data-track-context="unified footer" data-track-label="link">Your US state privacy rights</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://www.springernature.com/gp/info/accessibility" data-track="nav_accessibility_statement" data-track-action="accessibility statement" data-track-context="unified footer" data-track-label="link">Accessibility statement</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://link.springer.com/termsandconditions" data-track="nav_terms_and_conditions" data-track-action="terms and conditions" data-track-context="unified footer" data-track-label="link">Terms and conditions</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://link.springer.com/privacystatement" data-track="nav_privacy_policy" data-track-action="privacy policy" data-track-context="unified footer" data-track-label="link">Privacy policy</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://support.springernature.com/en/support/home" data-track="nav_help_and_support" data-track-action="help and support" data-track-context="unified footer" data-track-label="link">Help and support</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://support.springernature.com/en/support/solutions/articles/6000255911-subscription-cancellations" data-track-action="cancel contracts here">Cancel contracts here</a> </li> </ul> </nav> <div class="eds-c-footer__user"> <p class="eds-c-footer__user-info"> <span data-test="footer-user-ip">8.222.208.146</span> </p> <p class="eds-c-footer__user-info" data-test="footer-business-partners">Not affiliated</p> </div> <a href="https://www.springernature.com/" class="eds-c-footer__link"> <img src="/oscar-static/images/logo-springernature-white-19dd4ba190.svg" alt="Springer Nature" loading="lazy" width="200" height="20"/> </a> <p class="eds-c-footer__legal" data-test="copyright">© 2024 Springer Nature</p> </div> </div> </footer> </div> </body> </html>