Einstein, Albert. 'Kinetische Theorie des Waermegleichgewichtes und des zweiten Hauptsatzes der Thermodynamik'. Annalen der Physik, 9 (1902)

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        <p class="noindent">
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        <p class="indent"/>
        <div class="center">
          <p class="noindent"/>
          <p class="noindent">
            <span class="cmsy-10">§ </span>
          4. Beweis dafür, dass die Grösse
            <span class="cmmi-10">h </span>
          positiv ist.</p>
        </div>
        <p class="indent"> Sei
          <span class="cmmi-10">
            <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle"/>
          </span>
        (
          <span class="cmmi-10">x</span>
        ) eine homogene, quadratische Function der Variabeln
          <br/>
          <span class="cmmi-10">x</span>
          <sub>
            <span class="cmr-7">1</span>
          </sub>
          <span class="cmmi-10">...</span>
          <span class="cmmi-10">x</span>
          <sub>
            <span class="cmmi-7">n</span>
          </sub>
          <span class="cmmi-10">. </span>
        Wir betrachten die Grösse
          <span class="cmmi-10">z </span>
        =
          <span class="cmsy-10">
            <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmsy10-73.png" alt=" integral " class="10x-x-73"/>
          </span>
          <span class="cmmi-10">dx</span>
          <sub>
            <span class="cmr-7">1</span>
          </sub>
          <span class="cmmi-10">...</span>
          <span class="cmmi-10">x</span>
          <sub>
            <span class="cmmi-7">n</span>
          </sub>
          <span class="cmmi-10">, </span>
        wobei
          <br/>
        die Integrationsgrenzen dadurch bestimmt sein mögen, dass
          <br/>
          <span class="cmmi-10">
            <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle"/>
          </span>
        (
          <span class="cmmi-10">x</span>
        ) zwischen einem gewissen Wert
          <span class="cmmi-10">y </span>
        und
          <span class="cmmi-10">y </span>
        +
          <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1"/>
        liege, wobei
          <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1"/>
          <br/>
        eine Constante sei. Wir behaupten, dass
          <span class="cmmi-10">z</span>
        , welches allein
          <br/>
        von
          <span class="cmmi-10">y </span>
        Function ist, stets mit wachsendem
          <span class="cmmi-10">y </span>
        zunimmt, wenn
          <br/>
          <span class="cmmi-10">n > </span>
        </p>
        <p class="indent"> Führen wir die neuen Variabeln ein
          <span class="cmmi-10">x</span>
          <sub>
            <span class="cmr-7">1</span>
          </sub>
        =
          <span class="cmmi-10">
            <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b"/>
          x</span>
          <sub>
            <span class="cmr-7">1</span>
          </sub>
          <sup>
            <span class="cmsy-7">'</span>
          </sup>
          <span class="cmmi-10">...</span>
          <span class="cmmi-10">x</span>
          <sub>
            <span class="cmmi-7">n</span>
          </sub>
        =
          <span class="cmmi-10">
            <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b"/>
          x</span>
          <sub>
            <span class="cmmi-7">n</span>
          </sub>
          <sup>
            <span class="cmsy-7">'</span>
          </sup>
          <span class="cmmi-10">, </span>
          <br/>
        wobei
          <span class="cmmi-10">
            <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b"/>
          </span>
        = const., dann </p>
        <center class="par-math-display">
          <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190221x.png" alt=" integral z = an dx1'... d xn'. " class="par-math-display"/>
        </center>
        <p class="nopar"/>
        <p class="noindent">Ferner erhalten wir
          <span class="cmmi-10">
            <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle"/>
          </span>
        (
          <span class="cmmi-10">x</span>
        ) =
          <span class="cmmi-10">
            <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b"/>
          </span>
          <sup>
            <span class="cmr-7">2</span>
          </sup>
          <span class="cmmi-10">
            <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle"/>
          </span>
        (
          <span class="cmmi-10">x</span>
          <span class="cmsy-10">'</span>
        </p>
        <p class="indent"> Die Integrationsgrenzen des gewonnenen Integrals lauten
          <br/>
        also für
          <span class="cmmi-10">
            <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle"/>
          </span>
        (
          <span class="cmmi-10">x</span>
          <span class="cmsy-10">'</span>
        </p>
        <center class="par-math-display">
          <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190222x.png" alt=" y y D a2- und a2-+ a2-. " class="par-math-display"/>
        </center>
        <p class="nopar"/>
        <p class="noindent">Ist ferner
          <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1"/>
        unendlich klein, was wir annehmen, so erhalten </p>
        <center class="par-math-display">
          <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190223x.png" alt=" integral z = an-2 dx1' ... dxn'. " class="par-math-display"/>
        </center>
        <p class="nopar"/>
        <p class="noindent">Hierbei ist
          <span class="cmmi-10">y</span>
          <span class="cmsy-10">' </span>
        zwischen den </p>
        <center class="par-math-display">
          <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190224x.png" alt="y-- und y--+ D. a2 a2 " class="par-math-display"/>
        </center>
        <p class="nopar"/>
        <p class="noindent">Obige Gleichung lässt sich auch </p>
        <center class="par-math-display">
          <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190225x.png" alt=" ( ) z(y) = an- 2z-y- . a2 " class="par-math-display"/>
        </center>
        <p class="nopar"/>
        <p class="noindent">Wählt man
          <span class="cmmi-10">
            <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b"/>
          </span>
        positiv und
          <span class="cmmi-10">n > </span>
        2
          <span class="cmmi-10">, </span>
        so ist also </p>
        <center class="par-math-display">
          <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190226x.png" alt=" z(y) -(-y-)> 1, z a2 " class="par-math-display"/>
        </center>
        <p class="nopar"/>
        <p class="noindent">was zu beweisen </p>
        <p class="indent"> Dieses Resultat benutzen wir, um zu beweisen, dass
          <span class="cmmi-10">h </span>
          <br/>
        positiv </p>
        <p class="indent"> Wir </p>
        <center class="par-math-display">
          <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190227x.png" alt=" w'(E) h = 12------, w (E) " class="par-math-display"/>
        </center>
        <p class="nopar"/>
        <p class="noindent"/>
        <center class="par-math-display">
          <img src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190228x.png" alt=" integral w (E) = dp1 ... d qn, " class="par-math-display"/>
        </center>
        <p class="nopar"> </p>
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