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href="#Correlation_between_statistical_negentropy_and_Gibbs'_free_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Correlation between statistical negentropy and Gibbs' free energy</span> </div> </a> <ul id="toc-Correlation_between_statistical_negentropy_and_Gibbs'_free_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Brillouin's_negentropy_principle_of_information" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Brillouin's_negentropy_principle_of_information"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Brillouin's negentropy principle of information</span> </div> </a> <ul id="toc-Brillouin's_negentropy_principle_of_information-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div 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Available in 17 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-17" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">17 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Negentropie" title="Negentropie – Bavarian" lang="bar" hreflang="bar" data-title="Negentropie" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Neguentropia" title="Neguentropia – Catalan" lang="ca" hreflang="ca" data-title="Neguentropia" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Negentropie" title="Negentropie – German" lang="de" hreflang="de" data-title="Negentropie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Neguentrop%C3%ADa" title="Neguentropía – Spanish" lang="es" hreflang="es" data-title="Neguentropía" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/N%C3%A9guentropie" title="Néguentropie – French" lang="fr" hreflang="fr" data-title="Néguentropie" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Neghentropia" title="Neghentropia – Italian" lang="it" hreflang="it" data-title="Neghentropia" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B3%D0%B5%D0%BD%D1%82%D1%80%D0%BE%D0%BF%D0%B8%D1%8F" title="Негентропия – Kyrgyz" lang="ky" hreflang="ky" data-title="Негентропия" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Negentropija" title="Negentropija – Lithuanian" lang="lt" hreflang="lt" data-title="Negentropija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Negentropie" title="Negentropie – Dutch" lang="nl" hreflang="nl" data-title="Negentropie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%8D%E3%82%B2%E3%83%B3%E3%83%88%E3%83%AD%E3%83%94%E3%83%BC" title="ネゲントロピー – Japanese" lang="ja" hreflang="ja" data-title="ネゲントロピー" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Negentropi" title="Negentropi – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Negentropi" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Negentropi" title="Negentropi – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Negentropi" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Negentropia" title="Negentropia – Polish" lang="pl" hreflang="pl" data-title="Negentropia" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sintropia" title="Sintropia – Portuguese" lang="pt" hreflang="pt" data-title="Sintropia" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B3%D1%8D%D0%BD%D1%82%D1%80%D0%BE%D0%BF%D0%B8%D1%8F" title="Негэнтропия – Russian" lang="ru" hreflang="ru" data-title="Негэнтропия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Negentropy" title="Negentropy – Simple English" lang="en-simple" hreflang="en-simple" data-title="Negentropy" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B3%D0%B5%D0%BD%D1%82%D1%80%D0%BE%D0%BF%D1%96%D1%8F" title="Негентропія – Ukrainian" lang="uk" hreflang="uk" data-title="Негентропія" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q252552#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu 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.hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Entropy_and_life#Negative_entropy" title="Entropy and life">Negative entropy</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Syntropy" redirects here. For other uses, see <a href="/wiki/Syntropy_(software)" title="Syntropy (software)">Syntropy (software)</a>.</div> <p>In <a href="/wiki/Information_theory" title="Information theory">information theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, <b>negentropy</b> is used as a measure of distance to normality. The concept and phrase "<b>negative entropy</b>" was introduced by <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> in his 1944 popular-science book <i><a href="/wiki/What_is_Life%3F_(Schr%C3%B6dinger)" class="mw-redirect" title="What is Life? (Schrödinger)">What is Life?</a></i><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Later, <a href="/wiki/French_people" title="French people">French</a> <a href="/wiki/Physicist" title="Physicist">physicist</a> <a href="/wiki/L%C3%A9on_Brillouin" title="Léon Brillouin">Léon Brillouin</a> shortened the phrase to <i>néguentropie</i> (negentropy).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In 1974, <a href="/wiki/Albert_Szent-Gy%C3%B6rgyi" title="Albert Szent-Györgyi">Albert Szent-Györgyi</a> proposed replacing the term <i>negentropy</i> with <i><b>syntropy</b></i>. That term may have originated in the 1940s with the Italian mathematician <a href="/wiki/Luigi_Fantappi%C3%A8" title="Luigi Fantappiè">Luigi Fantappiè</a>, who tried to construct a unified theory of <a href="/wiki/Biology" title="Biology">biology</a> and <a href="/wiki/Physics" title="Physics">physics</a>. <a href="/wiki/Buckminster_Fuller" title="Buckminster Fuller">Buckminster Fuller</a> tried to popularize this usage, but <i>negentropy</i> remains common. </p><p>In a note to <i><a href="/wiki/What_is_Life%3F" class="mw-redirect" title="What is Life?">What is Life?</a></i> Schrödinger explained his use of this phrase. </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>... if I had been catering for them [physicists] alone I should have let the discussion turn on <i><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">free energy</a></i> instead. It is the more familiar notion in this context. But this highly technical term seemed linguistically too near to <i><a href="/wiki/Energy" title="Energy">energy</a></i> for making the average reader alive to the contrast between the two things.</p></blockquote> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Information_theory">Information theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Negentropy&action=edit&section=1" title="Edit section: Information theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Missing_information plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/44px-Wiki_letter_w.svg.png" decoding="async" width="44" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/66px-Wiki_letter_w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/88px-Wiki_letter_w.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>is missing information</b> about the mathematical treatment of negentropy in information theory.<span class="hide-when-compact"> Please expand the section to include this information. Further details may exist on the <a href="/wiki/Talk:Negentropy#Section_on_information_theory_is_deficient" title="Talk:Negentropy">talk page</a>.</span> <span class="date-container"><i>(<span class="date">November 2024</span>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Information_theory" title="Information theory">information theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a>, negentropy is used as a measure of distance to normality.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Out of all <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a> with a given mean and variance, the normal or <a href="/wiki/Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian distribution</a> is the one with the highest <a href="/wiki/Entropy" title="Entropy">entropy</a>. Negentropy measures the difference in entropy between a given distribution and the Gaussian distribution with the same mean and variance. Thus, negentropy is always nonnegative, is invariant by any linear invertible change of coordinates, and vanishes <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the signal is Gaussian. </p><p>Negentropy is defined as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(p_{x})=S(\varphi _{x})-S(p_{x})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>S</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J(p_{x})=S(\varphi _{x})-S(p_{x})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24c2408865833917de154c1e12f9da9ef5102c5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.6ex; height:2.843ex;" alt="{\displaystyle J(p_{x})=S(\varphi _{x})-S(p_{x})\,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\varphi _{x})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\varphi _{x})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b3c5195a23a6d803a57a8668aef7c82f8708a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.001ex; height:2.843ex;" alt="{\displaystyle S(\varphi _{x})}"></span> is the <a href="/wiki/Differential_entropy" title="Differential entropy">differential entropy</a> of the Gaussian density with the same <a href="/wiki/Mean" title="Mean">mean</a> and <a href="/wiki/Variance" title="Variance">variance</a> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5055ed65713825b48aa6ee05118c072e6f026a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.431ex; height:2.009ex;" alt="{\displaystyle p_{x}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(p_{x})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(p_{x})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b7d85b69adbc556c01c834f99f298a7cdcb347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.65ex; height:2.843ex;" alt="{\displaystyle S(p_{x})}"></span> is the differential entropy of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5055ed65713825b48aa6ee05118c072e6f026a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.431ex; height:2.009ex;" alt="{\displaystyle p_{x}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(p_{x})=-\int p_{x}(u)\log p_{x}(u)\,du}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo>∫</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(p_{x})=-\int p_{x}(u)\log p_{x}(u)\,du}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef1d3a2e1909fbecaffd37d909baffe28e0733d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.165ex; height:5.676ex;" alt="{\displaystyle S(p_{x})=-\int p_{x}(u)\log p_{x}(u)\,du}"></span></dd></dl> <p>Negentropy is used in <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>. It is related to network <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">entropy</a>, which is used in <a href="/wiki/Independent_component_analysis" title="Independent component analysis">independent component analysis</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The negentropy of a distribution is equal to the <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a> between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5055ed65713825b48aa6ee05118c072e6f026a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.431ex; height:2.009ex;" alt="{\displaystyle p_{x}}"></span> and a Gaussian distribution with the same mean and variance as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5055ed65713825b48aa6ee05118c072e6f026a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.431ex; height:2.009ex;" alt="{\displaystyle p_{x}}"></span> (see <i><a href="/wiki/Differential_entropy#Maximization_in_the_normal_distribution" title="Differential entropy">Differential entropy § Maximization in the normal distribution</a></i> for a proof). In particular, it is always nonnegative. </p> <div class="mw-heading mw-heading2"><h2 id="Correlation_between_statistical_negentropy_and_Gibbs'_free_energy"><span id="Correlation_between_statistical_negentropy_and_Gibbs.27_free_energy"></span>Correlation between statistical negentropy and Gibbs' free energy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Negentropy&action=edit&section=2" title="Edit section: Correlation between statistical negentropy and Gibbs' free energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wykres_Gibbsa.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Wykres_Gibbsa.svg/275px-Wykres_Gibbsa.svg.png" decoding="async" width="275" height="317" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Wykres_Gibbsa.svg/413px-Wykres_Gibbsa.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Wykres_Gibbsa.svg/550px-Wykres_Gibbsa.svg.png 2x" data-file-width="695" data-file-height="800" /></a><figcaption><a href="/wiki/Willard_Gibbs" class="mw-redirect" title="Willard Gibbs">Willard Gibbs</a>’ 1873 <b>available energy</b> (<a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">free energy</a>) graph, which shows a plane perpendicular to the axis of <i>v</i> (<a href="/wiki/Volume" title="Volume">volume</a>) and passing through point A, which represents the initial state of the body. MN is the section of the surface of <a href="/wiki/Dissipated_energy" class="mw-redirect" title="Dissipated energy">dissipated energy</a>. Qε and Qη are sections of the planes <i>η</i> = 0 and <i>ε</i> = 0, and therefore parallel to the axes of ε (<a href="/wiki/Internal_energy" title="Internal energy">internal energy</a>) and η (<a href="/wiki/Entropy" title="Entropy">entropy</a>) respectively. AD and AE are the energy and entropy of the body in its initial state, AB and AC its <i>available energy</i> (<a href="/wiki/Gibbs_energy" class="mw-redirect" title="Gibbs energy">Gibbs energy</a>) and its <i>capacity for entropy</i> (the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume) respectively.</figcaption></figure> <p>There is a physical quantity closely linked to <a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">free energy</a> (<a href="/wiki/Free_enthalpy" class="mw-redirect" title="Free enthalpy">free enthalpy</a>), with a unit of entropy and isomorphic to negentropy known in statistics and information theory. In 1873, <a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Willard Gibbs</a> created a diagram illustrating the concept of free energy corresponding to <a href="/wiki/Free_enthalpy" class="mw-redirect" title="Free enthalpy">free enthalpy</a>. On the diagram one can see the quantity called <a href="/wiki/Capacity_for_entropy" class="mw-redirect" title="Capacity for entropy">capacity for entropy</a>. This quantity is the amount of entropy that may be increased without changing an internal energy or increasing its volume.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In other words, it is a difference between maximum possible, under assumed conditions, entropy and its actual entropy. It corresponds exactly to the definition of negentropy adopted in statistics and information theory. A similar physical quantity was introduced in 1869 by <a href="/wiki/Fran%C3%A7ois_Jacques_Dominique_Massieu" class="mw-redirect" title="François Jacques Dominique Massieu">Massieu</a> for the <a href="/wiki/Isothermal_process" title="Isothermal process">isothermal process</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> (both quantities differs just with a figure sign) and then <a href="/wiki/Max_Planck" title="Max Planck">Planck</a> for the <a href="/wiki/Isothermal_process" title="Isothermal process">isothermal</a>-<a href="/wiki/Isobaric_process" title="Isobaric process">isobaric</a> process.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> More recently, the Massieu–Planck <a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">thermodynamic potential</a>, known also as <i><a href="/wiki/Free_entropy" title="Free entropy">free entropy</a></i>, has been shown to play a great role in the so-called entropic formulation of <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> applied among the others in molecular biology<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> and thermodynamic non-equilibrium processes.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=S_{\max }-S=-\Phi =-k\ln Z\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mo>−</mo> <mi>S</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>ln</mi> <mo>⁡</mo> <mi>Z</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=S_{\max }-S=-\Phi =-k\ln Z\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b41ebd3529c9c8c0f9571ef1e04ee4c6f17e284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.109ex; height:2.509ex;" alt="{\displaystyle J=S_{\max }-S=-\Phi =-k\ln Z\,}"></span></dd></dl></dd></dl> <dl><dd><dl><dd>where:</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is <a href="/wiki/Entropy" title="Entropy">entropy</a></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> is negentropy (Gibbs "capacity for entropy")</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span> is the <a href="/wiki/Free_entropy" title="Free entropy">Massieu potential</a></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> is the <a href="/wiki/Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition function</a></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a></dd></dl></dd></dl> <p>In particular, mathematically the negentropy (the negative entropy function, in physics interpreted as free entropy) is the <a href="/wiki/Convex_conjugate" title="Convex conjugate">convex conjugate</a> of <a href="/wiki/LogSumExp" title="LogSumExp">LogSumExp</a> (in physics interpreted as the free energy). </p> <div class="mw-heading mw-heading2"><h2 id="Brillouin's_negentropy_principle_of_information"><span id="Brillouin.27s_negentropy_principle_of_information"></span>Brillouin's negentropy principle of information</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Negentropy&action=edit&section=3" title="Edit section: Brillouin's negentropy principle of information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1953, <a href="/wiki/L%C3%A9on_Brillouin" title="Léon Brillouin">Léon Brillouin</a> derived a general equation<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> stating that the changing of an information bit value requires at least <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kT\ln 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>T</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kT\ln 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27e47d20dbbf4eea36787b171322e10e19e85814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.724ex; height:2.176ex;" alt="{\displaystyle kT\ln 2}"></span> energy. This is the same energy as the work <a href="/wiki/Le%C3%B3_Szil%C3%A1rd" class="mw-redirect" title="Leó Szilárd">Leó Szilárd</a>'s engine produces in the idealistic case. In his book,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> he further explored this problem concluding that any cause of this bit value change (measurement, decision about a yes/no question, erasure, display, etc.) will require the same amount of energy. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Negentropy&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Exergy" title="Exergy">Exergy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li> <li><a href="/wiki/Entropy_in_thermodynamics_and_information_theory" title="Entropy in thermodynamics and information theory">Entropy in thermodynamics and information theory</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Negentropy&action=edit&section=5" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 20em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Schrödinger, Erwin, <i>What is Life – the Physical Aspect of the Living Cell</i>, Cambridge University Press, 1944</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Brillouin, Leon: (1953) "Negentropy Principle of Information", <i>J. of Applied Physics</i>, v. <b>24(9)</b>, pp. 1152–1163</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Léon Brillouin, <i>La science et la théorie de l'information</i>, Masson, 1959</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Aapo Hyvärinen, <a rel="nofollow" class="external text" href="http://www.cis.hut.fi/aapo/papers/NCS99web/node32.html">Survey on Independent Component Analysis, node32: Negentropy</a>, Heli University of Technology Laboratory of Computer and Information Science</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Aapo Hyvärinen and Erkki Oja, <a rel="nofollow" class="external text" href="http://www.cis.hut.fi/aapo/papers/IJCNN99_tutorialweb/node14.html">Independent Component Analysis: A Tutorial, node14: Negentropy</a>, Helsinki University of Technology Laboratory of Computer and Information Science</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Ruye Wang, <a rel="nofollow" class="external text" href="http://fourier.eng.hmc.edu/e161/lectures/ica/node4.html">Independent Component Analysis, node4: Measures of Non-Gaussianity</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">P. Comon, Independent Component Analysis – a new concept?, <i>Signal Processing</i>, <b>36</b> 287–314, 1994.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Didier G. Leibovici and Christian Beckmann, <a rel="nofollow" class="external text" href="http://www.fmrib.ox.ac.uk/analysis/techrep/tr01dl1/tr01dl1/tr01dl1.html">An introduction to Multiway Methods for Multi-Subject fMRI experiment</a>, FMRIB Technical Report 2001, Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital, Headley Way, Headington, Oxford, UK.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Willard Gibbs, <a rel="nofollow" class="external text" href="http://www.ufn.ru/ufn39/ufn39_4/Russian/r394b.pdf">A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces</a>, <i>Transactions of the Connecticut Academy</i>, 382–404 (1873)</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Massieu, M. F. (1869a). Sur les fonctions caractéristiques des divers fluides. <i>C. R. Acad. Sci.</i> LXIX:858–862.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Massieu, M. F. (1869b). Addition au precedent memoire sur les fonctions caractéristiques. <i>C. R. Acad. Sci.</i> LXIX:1057–1061.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Massieu, M. F. (1869), <i>Compt. Rend.</i> <b>69</b> (858): 1057.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Planck, M. (1945). <i>Treatise on Thermodynamics</i>. Dover, New York.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Antoni Planes, Eduard Vives, <a rel="nofollow" class="external text" href="http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html">Entropic Formulation of Statistical Mechanics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081011011717/http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html">Archived</a> 2008-10-11 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Entropic variables and Massieu–Planck functions 2000-10-24 Universitat de Barcelona</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">John A. Scheilman, <a rel="nofollow" class="external text" href="http://www.biophysj.org/cgi/reprint/73/6/2960.pdf">Temperature, Stability, and the Hydrophobic Interaction</a>, <i>Biophysical Journal</i> <b>73</b> (December 1997), 2960–2964, Institute of Molecular Biology, University of Oregon, Eugene, Oregon 97403 USA</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Z. Hens and X. de Hemptinne, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/chao-dyn/9604008">Non-equilibrium Thermodynamics approach to Transport Processes in Gas Mixtures</a>, Department of Chemistry, Catholic University of Leuven, Celestijnenlaan 200 F, B-3001 Heverlee, Belgium</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Leon Brillouin, The negentropy principle of information, <i>J. Applied Physics</i> <b>24</b>, 1152–1163 1953</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Leon Brillouin, <i>Science and Information theory</i>, Dover, 1956</span> </li> </ol></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/negentropy" class="extiw" title="wiktionary:Special:Search/negentropy">negentropy</a></b></i> in Wiktionary, the free dictionary.</div></div> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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