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Version vom 2009-02-14
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:08:12 +0200
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1 <?xml version="1.0" encoding="utf-8" ?>
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5 <html>
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7 <head><title></title>
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9 <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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10 <meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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12 <meta name="src" content="Einst_Zurth_de_1906_02.tex" />
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13 <meta name="date" content="2005-03-10 17:56:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Zurth_de_1906_02.css" />
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15 </head><body >
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17 <!--l. 13--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 14--><p class="noindent">
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21 </p><!--l. 15--><p class="noindent"><span
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22 class="cmr-12x-x-120">7. </span><span
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23 class="cmbxti-10x-x-144">Zur Theorie der Brownschen Bewegung; </span> <br/><span
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24 class="cmbxti-10x-x-144">von A.</span>
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25 <span
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26 class="cmbxti-10x-x-144">Einstein.</span></p></div>
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27 <div class="center" >
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29 <!--l. 19--><p class="noindent">
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30 </p><!--l. 20--><p class="noindent">--------</p></div>
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31 <!--l. 24--><p class="indent"> Kurz nach dem Erscheinen meiner Arbeit über die durch <br/>die Molekulartheorie der Wärme geforderte Bewegung von in <br/>Flüssigkeiten suspendierten Teilchen<sup ><span
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32 class="cmr-8">1</span></sup>) teilte mir Hr. Sieden-<br/>topf (Jena) mit, daß er und andere Physiker -- zuerst wohl
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33 <br/>Hr. Prof. Gouy (Lyon) -- durch direkte Beobachtung zu der <br/>Überzeugung gelangt seien, daß die sogenannte Brownsche <br/>Bewegung durch die ungeordnete Wärmebewegung der Flüssig-<br/>keitsmoleküle verursacht sei.<sup ><span
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34 class="cmr-8">2</span></sup>) Nicht nur die qualitativen Eigen-<br/>schaften der Brownschen Bewegung, sondern auch die Größen-<br/>ordnung der von den Teilchen zurückgelegten Wege entspricht <br/>durchaus den Resultaten der Theorie. Ich will hier nicht <br/>eine Vergleichung des mir zur Verfügung stehenden dürftigen <br/>Erfahrungsmaterials mit den Resultaten der Theorie anstellen, <br/>sondern diese Vergleichung denjenigen überlassen, welche das
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35 <br/>Thema experimentell behandeln.
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36 </p><!--l. 43--><p class="indent"> Die nachfolgende Arbeit soll meine oben genannte Arbeit <br/>in einigen Punkten ergänzen. Wir leiten hier nicht nur die <br/>fortschreitende, sondern auch die Rotationsbewegung suspen-<br/>dierter Teilchen ab für den einfachsten Spezialfall, daß die <br/>Teilchen Kugelgestalt besitzen. Wir zeigen ferner bis zu wie
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37 <br/>kurzen Beobachtungszeiten das in jener Abhandlung gegebene <br/>Resultat gilt.
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38 </p><!--l. 52--><p class="indent"> Für die herleitung wollen wir uns hier einer allgemeineren <br/>Methode bedienen, teils um zu zeigen, wie die Brownsche <br/>Bewegung mit den Grundlagen der molekularen Theorie der <br/>Wärme zusammenhängt, teils um die Formeln für die fort-<br/>schreitende und für die rotierende Bewegung durch eine ein-<br/>heitliche Untersuchung entwickeln zu können. Es sei näm-<br/>lich <span
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39 class="cmmi-12"><img
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40 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>ein beobachtbarer Parameter eines im Temperatur-<br/>----------
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41 </p><!--l. 63--><p class="indent"> 1) A. Einstein, Ann. d. Phys. <span
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42 class="cmbx-12">17. </span>p. 549. 1905.
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43 </p><!--l. 65--><p class="indent"> 2) M. Gouy, Journ. de Phys. (2) <span
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44 class="cmbx-12">7. </span>p. 561. 1888. <pb/>
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45 </p><!--l. 69--><p class="indent">
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47 </p><!--l. 70--><p class="noindent">gleichgewicht befindlichen physikalischen Systems und es sei <br/>angenommen, daß das System bei jedem (möglichen) Wert <br/>von <span
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48 class="cmmi-12"><img
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49 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>im sogenannten indifferenten Gleichgewicht sich be-<br/>finde. Nach der klassischen Thermodynamik, die zwischen
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50 <br/>Wärme und anderen Energiearten <span
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51 class="cmti-12">prinzipiell </span>unterscheidet, <br/>finden spontane Änderungen von <span
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52 class="cmmi-12"><img
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53 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>nicht statt, wohl aber <br/>nach der molekularen Theorie der Wärme. Wir wollen im <br/>nachfolgenden untersuchen, nach welchen Gesetzen jene Ände-<br/>rungen gemäß der letzteren Theorie stattfinden müssen. Wir <br/>haben dann jene Gesetze auf folgende spezialfälle anzuwenden:
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54 </p><!--l. 83--><p class="indent"> 1. <span
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55 class="cmmi-12"><img
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56 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>ist die <span
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57 class="cmmi-12">x</span>-Koordinate des Schwerpunktes eines in einer <br/>(der Schwerkraft nicht unterworfenen) homogenen Flüssigkeit <br/>suspendierten Teilchens von Kugelgestalt.
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58 </p><!--l. 88--><p class="indent"> 2. <span
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59 class="cmmi-12"><img
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60 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>ist der Drehwinkel, welcher die Lage eines in einer <br/>Flüssigkeit suspendierten, um einen Durchmesser drehbaren <br/>Teilchens von Kugelgestalt bestimmt.
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61 </p>
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62 <div class="center" >
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63
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64 <!--l. 94--><p class="noindent">
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65 </p><!--l. 95--><p class="noindent"><span
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66 class="cmsy-10x-x-120">§ </span>1. Über einen Fall thermodynamischen Gleichgewichtes.</p></div>
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67 <!--l. 100--><p class="indent"> In einer Umgebung von der absoluten Temperatur <span
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68 class="cmmi-12">T </span>be-<br/>finde sich ein physikalisches System, das mit dieser Umgebung <br/>in thermischer Wechselwirkung stehe und im Zustand des Tem-<br/>peraturgleichgewichtes sei. Dies System, das also ebenfalls die <br/>absolute Temperatur <span
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69 class="cmmi-12">T </span>besitzt, sei im Sinne der molekularen <br/>Theorie der Wärme vollständig bestimmt<sup ><span
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70 class="cmr-8">1</span></sup>) durch die Zustands-<br/>variabeln <span
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71 class="cmmi-12">p</span><sub >
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72 <span
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73 class="cmr-8">1</span></sub><span
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74 class="cmmi-12">...</span><span
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75 class="cmmi-12">p</span><sub ><span
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76 class="cmmi-8">n</span></sub> Als Zustandsvariable <span
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77 class="cmmi-12">p</span><sub ><span
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78 class="cmr-8">1</span></sub><span
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79 class="cmmi-12">...</span><span
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80 class="cmmi-12">p</span><sub ><span
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81 class="cmmi-8">n</span></sub> können in <br/>den zu behandelnden Spezialfallen die Koordinaten und Ge-<br/>schwindigkeitskomponenten aller das betrachtete System bilden-<br/>der Atome gewählt werden.
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82 </p><!--l. 113--><p class="indent"> Es gilt für die Wahrscheinlichkeit dafür, daß in einem zufällig <br/>herausgegriffenen Zeitpunkt die Zustandsvariabeln <span
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83 class="cmmi-12">p</span><sub ><span
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84 class="cmr-8">1</span></sub><span
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85 class="cmmi-12">...</span><span
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86 class="cmmi-12">p</span><sub ><span
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87 class="cmmi-8">n</span></sub> in <br/>dem <span
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88 class="cmmi-12">n</span>-fach unendlich kleinen Gebiete (<span
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89 class="cmmi-12">dp</span><sub ><span
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90 class="cmr-8">1</span></sub><span
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91 class="cmmi-12">...</span><span
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92 class="cmmi-12">dp</span><sub ><span
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93 class="cmmi-8">n</span></sub>) liegen, die <br/>Gleichung<sup ><span
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94 class="cmr-8">2</span></sup>):
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95 </p>
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96 <table width="100%"
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97 class="equation"><tr><td><a
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98 id="x1-2r1"></a>
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99 <center class="math-display" >
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100 <img
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101 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_020x.png" alt="dw = C e- NRT Ed p ...dp , 1 n " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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102
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103 <!--l. 123--><p class="nopar">
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104 </p><!--l. 127--><p class="noindent">wobei <span
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105 class="cmmi-12">C </span>eine Konstante, <span
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106 class="cmmi-12">R </span>die universelle Konstante der Gas-<br/>gleichung, <span
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107 class="cmmi-12">N </span>die Anzahl der wirklichen Moleküle in einem <br/>Grammmolekül und <span
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108 class="cmmi-12">E </span>die Energie bedeutet.
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109 </p><!--l. 132--><p class="noindent">----------
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110 </p><!--l. 135--><p class="indent"> 1) Vgl. Ann. d. Phys. <span
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111 class="cmbx-12">17. </span>p. 549. 1905.
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112 </p><!--l. 137--><p class="indent"> 2) l. c. <span
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113 class="cmsy-10x-x-120">§ </span>3 und 4. <pb/>
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114 </p><!--l. 141--><p class="indent">
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115
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116 </p><!--l. 142--><p class="indent"> Es sei <span
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117 class="cmmi-12"><img
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118 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>ein beobachtbarer Parameter des Systems und es <br/>entspreche jedem Wertsystem <span
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119 class="cmmi-12">p</span><sub ><span
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120 class="cmr-8">1</span></sub> <span
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121 class="cmmi-12">...p</span><sub ><span
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122 class="cmmi-8">n</span></sub> ein bestimmter Wert <span
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123 class="cmmi-12"><img
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124 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span><span
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125 class="cmti-12">. </span> <br/>Wir bezeichnen mit
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126 <span
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127 class="cmmi-12">Ad<img
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128 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>die Wahrscheinlichkeit dafür, daß in <br/>einem zufällig herausgegriffenen Zeitpunkt der Wert des Para-<br/>meters <span
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129 class="cmmi-12"><img
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130 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>zwischen <span
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131 class="cmmi-12"><img
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132 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>und <span
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133 class="cmmi-12"><img
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134 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>+ <span
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135 class="cmmi-12">d<img
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136 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>liege. Es ist dann
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137 </p>
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138 <table width="100%"
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139 class="equation"><tr><td><a
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140 id="x1-3r2"></a>
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141 <center class="math-display" >
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142 <img
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143 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_021x.png" alt=" integral
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144 -N--E A da = C e RT d p1...d pn, da " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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145 <!--l. 154--><p class="nopar">
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146 </p><!--l. 158--><p class="noindent">wenn das Integral der rechten Seite über alle Wertkombi-<br/>nationen der Zustandsvariabeln erstreckt wird, deren <span
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147 class="cmmi-12"><img
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148 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>-Wert <br/>zwischen <span
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149 class="cmmi-12"><img
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150 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>und <span
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151 class="cmmi-12"><img
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152 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>+ <span
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153 class="cmmi-12">d<img
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154 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>
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155 liegt.
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156 </p><!--l. 163--><p class="indent"> Wir beschränken uns auf den Fall, daß aus der Natur <br/>des Problems ohne weiteres klar ist, daß allen (möglichen) <br/>Werten von <span
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157 class="cmmi-12"><img
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158 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>dieselbe Wahrscheinlichkeit (Häufigkeit) zu-<br/>kommt, daß also die Größe <span
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159 class="cmmi-12">A </span>von <span
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160 class="cmmi-12"><img
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161 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>unabhängig ist.
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162 </p><!--l. 169--><p class="indent"> Es liege nun ein zweites physikalisches System vor, das <br/>sich von dem soeben betrachteten einzig darin unterscheide, <br/>daß auf das System eine nur von <span
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163 class="cmmi-12"><img
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164 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>
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165 abhängige Kraft vom <br/>Potential <img
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166 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> ( <span
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167 class="cmmi-12"><img
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168 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>) wirke. Ist <span
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169 class="cmmi-12">E </span>die Energie des vorhin betrachteten <br/>Systems, so ist <span
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170 class="cmmi-12">E </span>+ <img
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171 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> die Energie des jetzt betrachteten, so <br/>daß wir die der Gleichung (1) analoge Beziehung erhalten:
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172 </p>
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173 <center class="par-math-display" >
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174 <img
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175 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_022x.png" alt=" ' ' --N-(E+P) d w = C e R T d p1...d pn. " class="par-math-display" /></center>
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176 <!--l. 182--><p class="nopar">
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177 </p><!--l. 185--><p class="noindent">Hieraus folgt für die Wahrscheinlichkeit <span
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178 class="cmmi-12">dW </span>dafür, daß in <br/>einem beliebig herausgegriffenen Zeitpunkt der Wert von <span
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179 class="cmmi-12"><img
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180 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span> <br/>zwischen <span
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181 class="cmmi-12"><img
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182 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>und <span
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183 class="cmmi-12"><img
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184 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>+ <span
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185 class="cmmi-12">d<img
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186 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>liegt, die der Gleichung (2) analoge <br/>Beziehung:</p>
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187 <table width="100%"
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188 class="equation"><tr><td><a
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189 id="x1-4r3"></a>
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190
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191 <center class="math-display" >
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192 <img
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193 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_023x.png" alt=" integral ' ' -RNT (E+P) C-- - NRT P { d W = C e d p1...d pn = C e A da ' --N-P = A e RT d a, " class="math-display" /></center></td><td width="5%">(I)</td></tr></table>
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194 <!--l. 199--><p class="nopar">
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195 </p><!--l. 203--><p class="noindent">wobei <span
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196 class="cmmi-12">A</span><span
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197 class="cmsy-10x-x-120">' </span>von <span
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198 class="cmmi-12"><img
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199 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>unabhängig ist.
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200 </p><!--l. 205--><p class="indent"> Diese Beziehung, welche dem von Bolzmann in seinen <br/>gastheoretischen Untersuchungen vielfach benutzten Exponential-<br/>gesetz genau entspricht, ist für die molekulare Theorie der <br/>Wärme charakteristisch. Sie gibt Aufschluß darüber, wieviel <br/>sich ein einer konstanten äußeren Kraft unterworfener Parameter <br/>eines Systems infolge der ungeordneten Molekularbewegung
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201 <br/><pb/>
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202 </p><!--l. 215--><p class="indent">
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203
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204 </p><!--l. 216--><p class="noindent">von dem Werte entfernt, welcher dem stabilen Gleichgewicht <br/>entspricht.
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205 </p>
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206 <div class="center" >
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207
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208 <!--l. 220--><p class="noindent">
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209 </p><!--l. 221--><p class="noindent"><span
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210 class="cmsy-10x-x-120">§ </span>2. Anwendungsbeispiele für die in <span
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211 class="cmsy-10x-x-120">§ </span>1 abgeleitete Gleichung.</p></div>
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212 <!--l. 226--><p class="indent"> Wir betrachten einen Körper, dessen Schwerpunkt sich <br/>längs einer Geraden (<span
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213 class="cmmi-12">X</span>-Achse eines Koordinatensystems) be-<br/>wegen kann. Der Körper sei von einem Gase umgeben und <br/>es herrsche thermisches und mechanisches Gleichgewicht. Nach <br/>der Molekulartheorie wird sich der Körper infolge der Un-<br/>gleichheit der Molekularstöße längs der Geraden in unregel-<br/>mäßiger Weise hin und her bewegen, derart, daß bei dieser <br/>Bewegung kein Punkt der Geraden bevorzugt ist -- voraus-<br/>gesetzt, daß auf den Körper in Richtung der Geraden keine <br/>anderen Kräfte wirken als die Stoßkräfte der Moleküle. Die <br/>Abszisse <span
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214 class="cmmi-12">x </span>des Schwerpunktes ist also ein Parameter des <br/>Systems, welcher die oben für den Parameter <span
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215 class="cmmi-12"><img
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216 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>voraus-<br/>gesetzten Eigenschaften besitzt.
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217 </p><!--l. 242--><p class="indent"> Wir wollen nun eine auf den Körper in Richtung der <br/>Geraden wirkende Kraft
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218 <span
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219 class="cmmi-12">K </span>= <span
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220 class="cmsy-10x-x-120">-</span><span
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221 class="cmmi-12">M x </span>einführen. Dann wird <br/>der Schwerpunkt des Körpers nach der Molekulartheorie <br/>ebenfalls ungeordnete Bewegungen ausführen, ohne sich jedoch
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222 <br/>viel vom Punkte <span
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223 class="cmmi-12">x </span>= 0 zu entfernen, während er nach der <br/>klassischen Thermodynamik im Punkte <span
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224 class="cmmi-12">x </span>= 0 ruhen müßte. <br/>Nach der Molekulartheorie ist (Formel I)
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225 </p>
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226 <center class="par-math-display" >
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227 <img
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228 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_024x.png" alt=" 2 dW = A'e- NRT-M x2-dx, " class="par-math-display" /></center>
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229 <!--l. 254--><p class="nopar">
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230 </p><!--l. 258--><p class="noindent">gleich der Wahrscheinlichkeit dafür, daß in einem zufällig ge-<br/>gewählten Zeitpunkt der Wert der Abszisse <span
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231 class="cmmi-12">x </span>zwischen <span
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232 class="cmmi-12">x </span>und <br/><span
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233 class="cmmi-12">x </span>+ <span
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234 class="cmmi-12">dx </span>liegt. Hieraus findet man den mittleren Abstand des <br/>Schwerpunktes vom Punkte <span
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235 class="cmmi-12">x </span>= 0:
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236 </p>
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237 <center class="par-math-display" >
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238 <img
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239 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_025x.png" alt=" + oo
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240 integral N--Mx2- x2 A'e- RT 2 d x V~ ------ V~ --- - oo R T x2 = -+- oo ----------------= ----- . integral N-Mx2- N M A'e - RT 2 dx - oo
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241 " class="par-math-display" /></center>
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242
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243 <!--l. 272--><p class="nopar">
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244 </p><!--l. 276--><p class="indent"> Damit <img
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245 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_026x.png" alt=" V~ --- x2" class="sqrt" /> genügend groß sei, um der Beobachtung zu-<br/>gänglich zu sein, muß die die Gleichgewichtslage des Körpers <br/><pb/>
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246 </p><!--l. 281--><p class="indent">
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247
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248 </p><!--l. 282--><p class="noindent">bestimmende Kraft sehr klein sein. Setzen wir als untere <br/>Grenze des Beobachtbaren
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249 <img
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250 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_027x.png" alt=" V~ --- x2" class="sqrt" /> = 10<sup ><span
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251 class="cmsy-8">-</span><span
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252 class="cmr-8">4</span></sup> cm, so erhalten wir <br/>für <span
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253 class="cmmi-12">T </span>= 300 <span
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254 class="cmmi-12">M </span>= ca. 5 <span
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255 class="cmmi-12">. </span>10<sup ><span
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256 class="cmsy-8">-</span><span
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257 class="cmr-8">6</span></sup><span
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258 class="cmmi-12">. </span>Damit der Körper mit dem <br/>Mikroskop beobachtbare Schwankungen ausführe, darf also die <br/>auf ihn wirkende Kraft bei einer Elongation von 1 cm nicht <br/>mehr als 5 milliontel Dyn betragen.
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259 </p><!--l. 291--><p class="indent"> Wir wollen noch eine theoretische Bemerkung an die ab-<br/>geleitete Gleichung anknüpfen. Der betrachtete Körper trage <br/>eine über einen sehr kleinen Raum verteilte elektrische Ladung <br/>und es sei das den Körper umgebende Gas so verdünnt, daß <br/>der Körper eine durch das umgebende Gas nur schwach modi-<br/>fizierte Sinusschwingung ausführe. Der Körper strahlt dann <br/>elektrische Wellen in den Raum aus und empfängt Energie <br/>aus der Strahlung des umliegenden Raumes; er vermittelt also <br/>einen Energieaustausch zwischen Strahlung und Gas. Wir ge-<br/>langen zu einer Ableitung des Grenzgesetzes der Temperatur-<br/>strahlung, welches für große Wellenlängen und für hohe <br/>Temperaturen zu gelten scheint, indem wir die Bedingung <br/>dafür aufstellen, daß der betrachtete Körper im Durchschnitt
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260 <br/>ebensoviel Strahlung emittiert als absorbiert. Man gelangt <br/>so<sup ><span
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261 class="cmr-8">1</span></sup>) zu der folgenden Formel für die der Schwingungszahl <span
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262 class="cmmi-12"><img
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263 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-17.png" alt="n" class="12x-x-17" /></span> <br/>entsprechende Strahlungsdichte
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264 <span
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265 class="cmmi-12"><img
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266 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /></span><sub ><span
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267 class="cmmi-8"><img
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268 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>:
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269 </p>
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270 <center class="par-math-display" >
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271 <img
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272 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_028x.png" alt=" 2 r = -R-8p-n--T, n N L3 " class="par-math-display" /></center>
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273 <!--l. 314--><p class="nopar">
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274 </p><!--l. 318--><p class="noindent">wobei <span
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275 class="cmmi-12">L </span>die Lichtgeschwindigkeit bedeutet.
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276 </p><!--l. 321--><p class="indent"> Die von Hrn. Planck gegebene Strahlungsformel<sup ><span
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277 class="cmr-8">2</span></sup>) geht <br/>für kleine Periodenzahlen und hohe Temperaturen in diese <br/>Formel über. Aus dem Koeffizienten des Grenzgesetzes läßt <br/>sich die Größe <span
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278 class="cmmi-12">N </span>bestimmen, und man erhält so die Planck-<br/>sche Bestimmung der Elementarquanta. Die Tatsache, daß man
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279 <br/>auf dem angedeuteten Wege nicht zu dem wahren Gesetz der <br/>Strahlung, sondern nur zu einem Grenzgesetz gelangt, scheint <br/>mir in einer elementaren Unvollkommenheit unserer physi-<br/>kalischen Anschauungen ihren Grund zu haben.
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280 </p><!--l. 333--><p class="indent"> Wir wollen nun die Formel (I) noch dazu verwenden, zu <br/>entscheiden, wie klein ein suspendiertes Teilchen sein muß, <br/>----------
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281 </p><!--l. 338--><p class="indent"> 1) Vgl. Ann. d. Phys. <span
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282 class="cmbx-12">17. </span>p. 549. 1905. <span
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283 class="cmsy-10x-x-120">§ </span>1 und 2.
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284 </p><!--l. 340--><p class="indent"> 2) M. Planck, Ann. d. Phys. <span
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285 class="cmbx-12">1. </span>p. 99. 1900. <pb/>
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286 </p><!--l. 344--><p class="indent">
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287
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288 </p><!--l. 345--><p class="noindent">damit es trotz der Wirkung der Schwere dauernd suspendiert <br/>bleibe. Wir können uns dabei auf den Fall beschränken, daß <br/>das Teilchen spezifisch schwerer ist als die Flüssigkeit, da der <br/>entgegengesetzte Fall vollkommen analog ist.
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289 </p><!--l. 351--><p class="indent"> Ist <span
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290 class="cmmi-12">v </span>das Volumen des Teilchens, <span
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291 class="cmmi-12"><img
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292 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span>dessen Dichte, <span
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293 class="cmmi-12"><img
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294 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /></span><sub ><span
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295 class="cmr-8">0</span></sub> die <br/>Dichte der Flüssigkeit,
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296 <span
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297 class="cmmi-12">g </span>die Beschleunigung der Schwere und <br/><span
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298 class="cmmi-12">x </span>der vertikale Abstand eines Punktes vom Boden des Ge-<br/>fäßes, so ergibt Gleichung (I)
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299 </p>
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300 <center class="par-math-display" >
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301 <img
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302 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_029x.png" alt=" -RNT-v(r-r0)gx dW = konst.e dx. " class="par-math-display" /></center>
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303 <!--l. 361--><p class="nopar">
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304 </p><!--l. 364--><p class="noindent">Man wird also dann finden, daß suspendierte Teilchen in <br/>einer Flüssigkeit zu schweben vermögen, wenn für Werte <br/>von <span
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305 class="cmmi-12">x</span>, die nicht wegen ihrer Kleinheit sich der Beobachtung <br/>entziehen, die Größe
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306 </p>
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307 <center class="par-math-display" >
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308 <img
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309 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0210x.png" alt="-R--- N T v (r- r0)gx " class="par-math-display" /></center>
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310 <!--l. 373--><p class="nopar">
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311 </p><!--l. 377--><p class="noindent">keinen allzu großen Wert besitzt -- vorausgesetzt, daß an den <br/>Gefäßboden gelangende Teilchen nicht durch irgendwelche Um-<br/>stände an demselben festgehalten werden.
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312 </p>
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313 <div class="center" >
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314
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315 <!--l. 383--><p class="noindent">
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316 </p><!--l. 384--><p class="noindent"><span
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317 class="cmsy-10x-x-120">§ </span>3. Über die von der Wärmebewegung verursachten <br/>Veränderungen des Parameters <span
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318 class="cmmi-12"><img
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319 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" />.</span></p></div>
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320 <!--l. 389--><p class="indent"> Wir kehren wieder zu dem in <span
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321 class="cmsy-10x-x-120">§ </span>1 behandelten allgemeinen <br/>Falle zurück, für den wir Gleichung (I) abgeleitet haben. Der <br/>einfacheren Ausdrucksweise und Vorstellung halber wollen wir <br/>aber nun annehmen, daß eine sehr große Zahl (<span
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322 class="cmmi-12">n</span>) identischer <br/>Systeme von der dort charakterisierten Art vorliege; wir haben <br/>es dann mit Anzahlen statt mit Wahrscheinlichkeiten zu tun. <br/>Gleichung (I) sagt dann aus:
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323 </p><!--l. 398--><p class="indent"> Von <span
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324 class="cmmi-12">N </span>Systemen liegt bei</p>
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325 <table width="100%"
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326 class="equation"><tr><td><a
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327 id="x1-5r3"></a>
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328
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329 <center class="math-display" >
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330 <img
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331 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0211x.png" alt=" -RNT-P dn = f e d a = F (a)d a " class="math-display" /></center></td><td width="5%">(Ia)</td></tr></table>
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332 <!--l. 404--><p class="nopar">
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333 </p><!--l. 407--><p class="noindent">Systemen der Wert des Parameters <span
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334 class="cmmi-12"><img
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335 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>in einem zufällig heraus-<br/>gegriffenen Zeitpunkt zwischen <span
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336 class="cmmi-12"><img
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337 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>und <span
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338 class="cmmi-12"><img
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339 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>+ <span
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340 class="cmmi-12">d<img
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341 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" />.</span>
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342 </p><!--l. 411--><p class="indent"> Diese Beziehung wollen wir dazu benutzen, die Größe der <br/>durch die ungeordneten Wärmevorgänge erzeugten unregel-<br/>mäßigen Veränderungen des Parameters <span
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343 class="cmmi-12"><img
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344 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>zu ermitteln. Zu <br/>diesem Zweck drücken wir in Zeichen aus, daß die Funktion <span
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345 class="cmmi-12">F <img
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346 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>) <br/><pb/>
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347 </p><!--l. 419--><p class="indent">
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348
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349 </p><!--l. 420--><p class="noindent">sich unter der vereinten Wirkung der dem Potential <img
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350 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> ent-<br/>sprechenden Kraft und des ungeordneten Wärmeprozesses sich <br/>innerhalb der Zeitspanne <span
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351 class="cmmi-12">t </span>nicht ändert; <span
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352 class="cmmi-12">t</span>
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353 bedeute hierbei eine <br/>so kleine Zeit, daß die zugehörigen Änderungen der Größen
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354 <span
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355 class="cmmi-12"><img
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356 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span> <br/>der einzelnen Systeme als unendlich kleine Argumentänderungen <br/>der Funktion <span
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357 class="cmmi-12">F</span>
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358 (<span
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359 class="cmmi-12"><img
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360 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>) betrachtet werden können.
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361 </p><!--l. 430--><p class="indent"> Trägt man auf einer Geraden von einem bestimmten Null-<br/>punkte aus den Größen <span
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362 class="cmmi-12"><img
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363 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>numerisch gleiche Strecken ab, so <br/>entspricht jedem System ein Punkt (<span
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364 class="cmmi-12"><img
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365 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>) auf dieser Geraden. <br/><span
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366 class="cmmi-12">F</span>(<span
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367 class="cmmi-12"><img
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368 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>) ist die Lagerungsdichte der Systempunkte (<span
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369 class="cmmi-12"><img
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370 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>) auf der <br/>Geraden. Durch einen beliebigen Punkt (<span
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371 class="cmmi-12"><img
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372 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span><sub ><span
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373 class="cmr-8">0</span></sub>) der Geraden <br/>müssen nun während der Zeit <span
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374 class="cmmi-12">t </span>genau soviele Systempunkte <br/>in dem einen Sinne hindurchwandern, wie in dem anderen <br/>Sinne.
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375 </p><!--l. 440--><p class="indent"> Die dem Potential <img
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376 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> entsprechende Kraft bewirke eine <br/>Änderung von <span
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377 class="cmmi-12"><img
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378 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>von der Größe
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379 </p>
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380 <center class="par-math-display" >
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381 <img
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382 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0212x.png" alt=" @ P D1 = - B ----t, @ a " class="par-math-display" /></center>
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383 <!--l. 448--><p class="nopar">
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384 </p><!--l. 452--><p class="noindent">wobei <span
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385 class="cmmi-12">B </span>von <span
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386 class="cmmi-12"><img
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387 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>unabhängig sei, d. h. die Änderungsgeschwindig-<br/>keit von <span
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388 class="cmmi-12"><img
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389 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>sei proportional der wirkenden Kraft und unabhängig <br/>vom Werte des Parameters. Den Faktor <span
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390 class="cmmi-12">B </span>nennen wir die <br/>,,Beweglichkeit des Systems in bezug auf
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391 <span
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392 class="cmmi-12"><img
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393 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>&#8220;.
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394 </p><!--l. 458--><p class="indent"> Würde also die äußere Kraft wirken, ohne daß der un-<br/>regelmäßige molekulare Wärmeprozeß die Größen <span
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395 class="cmmi-12"><img
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396 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>änderte, <br/>so gingen durch den Punkt (<span
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397 class="cmmi-12"><img
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398 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span><sub ><span
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399 class="cmr-8">0</span></sub>) während der Zeit <span
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400 class="cmmi-12">t</span>
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401 </p>
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402 <center class="par-math-display" >
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403 <img
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404 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0213x.png" alt=" ( ) n = B @-P- .t.F (a ) 1 @ a a=a 0 0 " class="par-math-display" /></center>
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405 <!--l. 468--><p class="nopar">
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406 </p><!--l. 471--><p class="noindent">Systempunkte nach der negativen Seite hindurch.
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407 </p><!--l. 474--><p class="indent"> Es sei ferner die Wahrscheinlichkeit dafür, daß der Para-<br/>meter <span
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408 class="cmmi-12"><img
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409 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>eines Systems infolge des ungeordneten Wärmeprozesses <br/>innerhalb der Zeit <span
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410 class="cmmi-12">t</span>
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411 eine Änderung erfahre, deren Wert zwischen <img
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412 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> <br/>und <img
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413 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> + <span
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414 class="cmmi-12">d </span><img
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415 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> liegt, gleich
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416 <span
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417 class="cmmi-12"><img
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418 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-20.png" alt="y" class="12x-x-20" /></span>(<img
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419 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" />)<span
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420 class="cmmi-12">, </span>wobei <span
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421 class="cmmi-12"><img
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422 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-20.png" alt="y" class="12x-x-20" /></span>(<img
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423 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" />) = <span
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424 class="cmmi-12"><img
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425 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-20.png" alt="y" class="12x-x-20" /></span>(<span
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426 class="cmsy-10x-x-120">-</span><img
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427 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" />) und <span
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428 class="cmmi-12"><img
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429 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-20.png" alt="y" class="12x-x-20" /></span> <br/>von <span
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430 class="cmmi-12"><img
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431 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>unabhängig sei. Die Anzahl der infolge des ungeord-<br/>neten Wärmeprozesses durch den Punkt (<span
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432 class="cmmi-12"><img
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433 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span><sub ><span
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434 class="cmr-8">0</span></sub>) während der <br/>Zeit <span
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435 class="cmmi-12">t </span>nach der positiven Seite hin wandernden Systempunkte <br/>ist dann:
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436 </p>
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437 <center class="par-math-display" >
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438 <img
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439 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0214x.png" alt=" D = oo
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440 integral
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441 n2 = F(a0 - D) x(D) d D , D = 0 " class="par-math-display" /></center>
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442 <!--l. 491--><p class="nopar"> <pb/>
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443 </p><!--l. 498--><p class="indent">
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444
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445 </p><!--l. 499--><p class="noindent">wenn
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446 </p>
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447 <center class="par-math-display" >
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448 <img
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449 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0215x.png" alt=" integral oo
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450 y(D) d D = x(D)
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451 D " class="par-math-display" /></center>
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452 <!--l. 506--><p class="nopar">
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453 </p><!--l. 510--><p class="noindent">gesetzt wird. Die Anzahl der nach der negativen Seite infolge <br/>des ungeordneten Wärmeprozesses wandernden Systempunkte ist:
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454 </p>
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455 <center class="par-math-display" >
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456 <img
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457 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0216x.png" alt=" integral oo
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458 n3 = F (a0 + D) x (D) d D .
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459 D " class="par-math-display" /></center>
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460 <!--l. 518--><p class="nopar">
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461 </p><!--l. 522--><p class="indent"> Der mathematische Ausdruck für die Unveränderlichkeit <br/>der Funktion <span
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462 class="cmmi-12">F </span>ist also:</p>
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463 <center class="par-math-display" >
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464 <img
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465 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0217x.png" alt="- n1 + n2 - n3 = 0 . " class="par-math-display" /></center>
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466 <!--l. 528--><p class="nopar">
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467 </p><!--l. 531--><p class="noindent">Setzt man die für <span
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468 class="cmmi-12">n</span><sub ><span
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469 class="cmr-8">1</span></sub>, <span
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470 class="cmmi-12">n</span><sub ><span
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471 class="cmr-8">2</span></sub>, <span
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472 class="cmmi-12">n</span><sub ><span
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473 class="cmr-8">3</span></sub> gefundenen Ausdrücke ein und <br/>berücksichtigt, daß <img
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474 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" />
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475 unendlich klein ist bez. daß <span
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476 class="cmmi-12"><img
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477 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-20.png" alt="y" class="12x-x-20" /></span>(<img
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478 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" />) nur <br/>für unendlich kleine Werte von <img
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479 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> von 0 verschieden ist, so <br/>erhält man hieraus nach einfacher Rechnung:
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480 </p>
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481 <center class="par-math-display" >
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482 <img
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483 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0218x.png" alt=" ( ) --- B @-P- F (a0)t + 1F '(a0)D2 = 0. @ a a=a0 2 " class="par-math-display" /></center>
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484 <!--l. 544--><p class="nopar">
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485 </p><!--l. 547--><p class="noindent">Hierbei bedeutet
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486 </p>
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487 <center class="par-math-display" >
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488 <img
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489 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0219x.png" alt=" integral + oo
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490 2 2 D = D y (D) d D - oo
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491 " class="par-math-display" /></center>
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492 <!--l. 554--><p class="nopar">
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493 </p><!--l. 558--><p class="noindent">den Mittelwert der Quadrate der durch den unregelmäßigen <br/>Wärmeprozeß während der Zeit <span
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494 class="cmmi-12">t </span>hervorgerufenen Änderungen <br/>der Größen <span
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495 class="cmmi-12"><img
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496 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>Aus dieser Beziehung erhält man unter Be-<br/>rücksichtigung von Gleichung (Ia):
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497 </p>
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498 <table width="100%"
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499 class="equation"><tr><td><a
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500 id="x1-6r3"></a>
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501 <center class="math-display" >
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502 <img
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503 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0220x.png" alt=" V~ ----- V~ --2- 2R-- V~ ------ D = N . B T t . " class="math-display" /></center></td><td width="5%">(II)</td></tr></table>
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504 <!--l. 568--><p class="nopar">
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505 </p><!--l. 571--><p class="noindent">Hierbei bedeutet <span
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506 class="cmmi-12">R </span>die Konstante der Gasgleichung (8,31.10<sup ><span
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507 class="cmr-8">7</span></sup>), <br/><span
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508 class="cmmi-12">N </span>die Anzahl der wirklichen Moleküle in einem Grammolekül <br/>(ca. 4. 10<sup ><span
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509 class="cmr-8">23</span></sup>), <span
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510 class="cmmi-12">B </span>die ,,Beweglichkeit des Systems in bezug auf <br/>den Parameter <span
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511 class="cmmi-12"><img
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512 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>&#8220;, <span
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513 class="cmmi-12">T </span>die absolute Temperatur, <span
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514 class="cmmi-12">t </span>die Zeit,
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515 <br/>innerhalb welcher die durch den ungeordneten Wärmeprozeß <br/>hervorgerufenen Änderungen von <span
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516 class="cmmi-12"><img
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517 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>stattfinden.
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518 </p>
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519 <div class="center" >
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520
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521 <!--l. 582--><p class="noindent">
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522 </p><!--l. 583--><p class="noindent"><span
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523 class="cmsy-10x-x-120">§ </span>4. Anwendung der abgeleiteten Gleichung auf die <br/>Brownsche Bewegung.</p></div>
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524 <!--l. 588--><p class="indent"> Wir berechnen nun mit Hilfe der Gleichungen (II) zu-<br/>nächst die mittlere Verschiebung, die ein kugelförmiger, in <br/><pb/>
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525 </p><!--l. 593--><p class="indent">
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526
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527 </p><!--l. 594--><p class="noindent">einer Flüssigkeit suspendierter Körper während der Zeit <span
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528 class="cmmi-12">t </span>in <br/>einer bestimmten Richtung (<span
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529 class="cmmi-12">X</span>-Richtung eines Koordinaten-<br/>systems) erleidet. Zu diesem Zweck haben wir in jene Glei-<br/>chung den entsprechenden Wert für <span
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530 class="cmmi-12">B</span>
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531 einzusetzen.
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532 </p><!--l. 600--><p class="indent"> Wirkt auf eine Kugel vom Radius <span
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533 class="cmmi-12">P</span>, die in einer Flüssig-<br/>keit vom Reibungskoeffizienten <span
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534 class="cmmi-12">k </span>suspendiert ist, eine Kraft <span
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535 class="cmmi-12">K</span>, <br/>so bewegt sie sich mit der Geschwindigkeit<sup ><span
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536 class="cmr-8">1</span></sup>) <span
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537 class="cmmi-12">K/</span>6 <span
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538 class="cmmi-12"><img
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539 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-19.png" alt="p" class="12x-x-19" /> k P. </span>Es ist <br/>also zu setzen
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540 </p>
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541 <center class="par-math-display" >
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542 <img
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543 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0221x.png" alt="B = --1----, 6p kP " class="par-math-display" /></center>
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544 <!--l. 609--><p class="nopar">
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545 </p><!--l. 613--><p class="noindent">so daß man -- in Übereinstimmung mit der oben zitierten <br/>Arbeit -- für die mittlere Verschiebung der suspendierten <br/>Kugel in Richtung der X-Achse den Wert erhält:
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546 </p>
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547 <center class="par-math-display" >
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548 <img
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549 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0222x.png" alt=" V~ ---- V~ ------------ --2 V~ - R-T----1--- D x = t N 3 pk P . " class="par-math-display" /></center>
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550 <!--l. 622--><p class="nopar">
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551 </p><!--l. 626--><p class="indent"> Wir behandeln zweitens den Fall, daß die betrachtete <br/>Kugel in der Flüssigkeit um einen ihrer Durchmesser (ohne <br/>Lagerreibung) frei drehbar gelagert sei und fragen nach der <br/>mittleren Drehung <img
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552 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0223x.png" alt=" V~ ---- D2r" class="sqrt" /> der Kugel während der Zeit <span
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553 class="cmmi-12">t </span>infolge <br/>des ungeordneten Wärmeprozesses.
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554 </p><!--l. 633--><p class="indent"> Wirkt auf eine Kugel vom Radius <span
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555 class="cmmi-12">P</span>, die in einer Flüssig-<br/>keit vom Reibungskoeffizienten <span
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556 class="cmmi-12">k </span>drehbar gelagert ist, das <br/>Drehmoment <span
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557 class="cmmi-12">D</span>, so dreht sie sich mit der Winkelgeschwindigkeit<sup ><span
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558 class="cmr-8">1</span></sup>)
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559 </p>
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560 <center class="par-math-display" >
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561 <img
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562 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0224x.png" alt=" D y = --------. 8 pk P 3 " class="par-math-display" /></center>
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563 <!--l. 641--><p class="nopar">
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564 </p><!--l. 644--><p class="noindent">Es ist also zu setzen:
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565 </p>
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566 <center class="par-math-display" >
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567 <img
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568 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0225x.png" alt=" ---1---- B = 8 pk P 3 . " class="par-math-display" /></center>
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569
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570 <!--l. 650--><p class="nopar">
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571 </p><!--l. 653--><p class="noindent">Man erhält also:
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572 </p>
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573 <center class="par-math-display" >
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574 <img
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575 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0226x.png" alt=" V~ ------------- V~ ---- V~ R T 1 D2r = t ----------3-. N 4p k P " class="par-math-display" /></center>
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576 <!--l. 660--><p class="nopar">
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577 </p><!--l. 664--><p class="indent"> Die durch die Molekularbewegung erzeugte Drehbewegung <br/>sinkt also mit wachsendem <span
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578 class="cmmi-12">P </span>viel rascher als die fortschreitende <br/>Bewegung.
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579 </p><!--l. 668--><p class="indent"> Für <span
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580 class="cmmi-12">P </span>= 0<span
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581 class="cmmi-12">, </span>5 mm und Wasser von 17<sup > <span
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582 class="cmr-8">0</span></sup> liefert die Formel <br/>für den im Mittel in einer Sekunde zurückgelegten Winkel <br/>etwa 11 Bogensekunden, in der Stunde ca. 11 Bogenminuten. <br/>----------
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583 </p><!--l. 674--><p class="indent"> 1) Vgl. G. Kirchhoff. Vorles. über Mechanik. 26. Vorl. <pb/>
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584 </p><!--l. 679--><p class="indent">
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585
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586 </p><!--l. 680--><p class="noindent">Für <span
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587 class="cmmi-12">P </span>= 0<span
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588 class="cmmi-12">, </span>5 Mikron und Wasser von 17<sup > <span
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589 class="cmr-8">0</span></sup> erhält man für <br/><span
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590 class="cmmi-12">t </span>= 1 Sekunde ca. 100 Winkelgrade.
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591 </p><!--l. 684--><p class="indent"> Bei einem frei schwebenden suspendierten Teilchen finden <br/>drei voneinander unabhängige derartige Drehbewegungen statt.
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592 </p><!--l. 688--><p class="indent"> Die für <img
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593 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0227x.png" alt=" V~ ---2 D" class="sqrt" /> entwickelte Formel ließe sich noch auf andere <br/>Fälle anwenden. Setzt man z. B. für <span
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594 class="cmmi-12">B </span>den reziproken elek-<br/>trischen Widerstand eines geschlossenen Stromkreises ein, so <br/>gibt sie an, wieviel Elektrizität im Durchschnitt während der <br/>Zeit <span
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595 class="cmmi-12">t </span>durch irgend einen Leiterquerschnitt geht, welche Be-<br/>ziehung abermals mit dem Grenzgesetz der Strahlung des <br/>schwarzen Körpers für große Wellenlängen und hohe Tem-<br/>peraturen zusammenhängt. Da ich jedoch keine durch das <br/>Experiment kontrollierbare Konsequenz mehr habe auffinden <br/>können, scheint mir die Behandlung weiterer Spezialfälle unnütz.
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596 </p>
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597 <div class="center" >
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598
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599 <!--l. 703--><p class="noindent">
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600 </p><!--l. 704--><p class="noindent"><span
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601 class="cmsy-10x-x-120">§ </span>5. Über die Gültigkeitsgrenze der Formel für<img
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602 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0228x.png" alt=" V~ ---- D2" class="sqrt" />.</p></div>
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603 <!--l. 709--><p class="indent"> Es ist klar, daß die Formel (II) nicht für beliebig kleine <br/>Zeiten gültig sein kann. Die mittlere Veränderungsgeschwindig-<br/>keit von <span
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604 class="cmmi-12"><img
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605 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>infolge des Wärmeprozesses
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606 </p>
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607 <center class="par-math-display" >
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608 <img
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609 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0229x.png" alt=" V~ ---- V~ --------- ---D2-= 2-R-T-B- .1 V~ --- t N t " class="par-math-display" /></center>
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610 <!--l. 719--><p class="nopar">
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611 </p><!--l. 723--><p class="noindent">wird nämlich für unendlich kleine Zeitdauer <span
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612 class="cmmi-12">t </span>unendlich groß, <br/>was offenbar unmöglich ist, denn es müßte sich ja sonst jeder <br/>suspendierte Körper mit unendlich großer Momentangeschwindig-<br/>keit bewegen. Der Grund liegt daran, daß wir in unserer <br/>Entwickelung implizite angenommen haben, daß der Vorgang
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613 <br/>während der Zeit <span
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614 class="cmmi-12">t </span>als von dem Vorgange in den unmittelbar <br/>vorangehenden Zeiten unabhängiges Ereignis aufzufassen sei. <br/>Diese Annahme trifft aber um so weniger zu, je kleiner die <br/>Zeiten <span
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615 class="cmmi-12">t </span>gewählt werden. Wäre nämlich zur Zeit
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616 <span
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617 class="cmmi-12">z </span>= 0
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618 </p>
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619 <center class="par-math-display" >
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620 <img
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621 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0230x.png" alt="da- d t = b0 " class="par-math-display" /></center>
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622 <!--l. 739--><p class="nopar">
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623
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624 </p><!--l. 743--><p class="noindent">der Momentanwert der Änderungsgeschwindigkeit, und würde <br/>die Änderungsgeschwindigkeit <span
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625 class="cmmi-12"><img
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626 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>in einem gewissen darauf <br/>folgenden Zeitintervall durch den ungeordneten thermischen <br/>Prozeß nicht beeinflußt, sondern die Änderung von <span
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627 class="cmmi-12"><img
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628 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>lediglich <br/><pb/>
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629 </p><!--l. 751--><p class="indent">
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630
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631 </p><!--l. 752--><p class="noindent">durch den passiven Widerstand (1/<span
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632 class="cmmi-12">B</span>) bestimmt, so würde <br/>für <span
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633 class="cmmi-12">d<img
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634 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> /dz </span>die Beziehung gelten:
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635 </p>
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636 <center class="par-math-display" >
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637 <img
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638 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0231x.png" alt=" d b b - m --- = --. d z B " class="par-math-display" /></center>
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639 <!--l. 759--><p class="nopar">
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640 </p><!--l. 762--><p class="noindent"><span
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641 class="cmmi-12"><img
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642 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>ist hierbei durch die Festsetzung definiert, daß <span
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643 class="cmmi-12"><img
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644 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>(<span
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645 class="cmmi-12"><img
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646 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /></span><sup ><span
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647 class="cmr-8">2</span></sup><span
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648 class="cmmi-12">/</span>2) die <br/>der Änderungsgeschwindigkeit <span
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649 class="cmmi-12"><img
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650 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /> </span>entsprechende Energie sein <br/>soll. In dem Falle der Translationsbewegung der suspendierten <br/>Kugel wäre also z. B. <span
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651 class="cmmi-12"><img
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652 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /> </span>(<span
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653 class="cmmi-12"><img
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654 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /></span><sup ><span
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655 class="cmr-8">2</span></sup> <span
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656 class="cmmi-12">/</span>2) die kinetische Energie der Kugel <br/>samt der kinetischen Energie der mitbewegten Flüssigkeit. <br/>Durch Integration folgt:
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657 </p>
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658 <center class="par-math-display" >
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659 <img
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660 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/Einst_Zurth_de_1906_0232x.png" alt=" -z- b = b0e- mB . " class="par-math-display" /></center>
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661 <!--l. 774--><p class="nopar">
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662 </p><!--l. 778--><p class="indent"> Aus diesem Resultat folgert man, daß die Formel (II) nur <br/>für Zeitintervalle gilt, welche groß sind gegen <span
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663 class="cmmi-12"><img
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664 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" />B</span>.
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665 </p><!--l. 782--><p class="indent"> Für Körperchen von 1 Mikron Durchmesser und von der <br/>Dichte <span
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666 class="cmmi-12"><img
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667 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Zurth_de_1906_02/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span>= 1 in Wasser von Zimmertemperatur ist die untere <br/>Grenze der Gültigkeit der Formel (II) ca. 10<sup ><span
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668 class="cmsy-8">-</span><span
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669 class="cmr-8">7</span></sup> Sekunden; <br/>diese untere Grenze für die Zeitintervalle wächst proportional <br/>dem Quadrat des Radius des Körperchens. Beides gilt sowohl <br/>für die fortschreitende wie für die Rotationsbewegung der
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670 <br/>Teilchen.
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671 </p><!--l. 792--><p class="indent"> Bern, Dezember 1905.
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672 </p>
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673 <div class="center" >
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674
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675 <!--l. 795--><p class="noindent">
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676 </p><!--l. 796--><p class="noindent">(Eingegangen 19. Dezember 1905.)</p></div>
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677 <div class="center" >
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678
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679 <!--l. 801--><p class="noindent">
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680 </p><!--l. 802--><p class="noindent">----------</p></div>
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681
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682 </body></html>
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683
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684
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685