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1 #+TITLE: Statistical Mechanics
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2 #+AUTHOR: E.T. Jaynes; edited by Dylan Holmes
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3 #+EMAIL: rlm@mit.edu
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4 #+KEYWORDS: statistical mechanics, thermostatics, thermodynamics, temperature, paradoxes, Jaynes
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5 #+SETUPFILE: ../../aurellem/org/setup.org
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6 #+INCLUDE: ../../aurellem/org/level-0.org
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7 #+MATHJAX: align:"left" mathml:t path:"http://www.aurellem.org/MathJax/MathJax.js"
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8
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9 # "extensions/eqn-number.js"
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10
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11 #+begin_quote
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12 *Note:* The following is a typeset version of
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13 [[../sources/stat.mech.1.pdf][this unpublished book draft]], written by [[http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E.T. Jaynes]]. I have only made
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14 minor changes, e.g. to correct typographical errors, add references, or format equations. The
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15 content itself is intact. --- Dylan
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16 #+end_quote
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17
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18 * Development of Thermodynamics
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19 Our first intuitive, or \ldquo{}subjective\rdquo{} notions of temperature
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20 arise from the sensations of warmth and cold associated with our
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21 sense of touch . Yet science has been able to convert this qualitative
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22 sensation into an accurately defined quantitative notion,
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23 which can be applied far beyond the range of our direct experience.
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24 Today an experimentalist will report confidently that his
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25 spin system was at a temperature of 2.51 degrees Kelvin; and a
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26 theoretician will report with almost as much confidence that the
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27 temperature at the center of the sun is about \(2 \times 10^7\) degrees
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28 Kelvin.
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29
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30 The /fact/ that this has proved possible, and the main technical
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31 ideas involved, are assumed already known to the reader;
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32 and we are not concerned here with repeating standard material
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33 already available in a dozen other textbooks . However
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34 thermodynamics, in spite of its great successes, firmly established
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35 for over a century, has also produced a great deal of confusion
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36 and a long list of \ldquo{}paradoxes\rdquo{} centering mostly
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37 around the second law and the nature of irreversibility.
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38 For this reason and others noted below, we want to dwell here at
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39 some length on the /logic/ underlying the development of
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40 thermodynamics . Our aim is to emphasize certain points which,
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41 in the writer's opinion, are essential for clearing up the
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42 confusion and resolving the paradoxes; but which are not
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43 sufficiently ernphasized---and indeed in many cases are
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44 totally ignored---in other textbooks.
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45
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46 This attention to logic
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47 would not be particularly needed if we regarded classical
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48 thermodynamics (or, as it is becoming called increasingly,
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49 /thermostatics/) as a closed subject, in which the fundamentals
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50 are already completely established, and there is
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51 nothing more to be learned about them. A person who believes
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52 this will probably prefer a pure axiomatic approach, in which
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53 the basic laws are simply stated as arbitrary axioms, without
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54 any attempt to present the evidence for them; and one proceeds
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55 directly to working out their consequences.
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56 However, we take the attitude here that thermostatics, for
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57 all its venerable age, is very far from being a closed subject,
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58 we still have a great deal to learn about such matters as the
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59 most general definitions of equilibrium and reversibility, the
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60 exact range of validity of various statements of the second and
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61 third laws, the necessary and sufficient conditions for
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62 applicability of thermodynamics to special cases such as
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63 spin systems, and how thermodynamics can be applied to such
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64 systems as putty or polyethylene, which deform under force,
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65 but retain a \ldquo{}memory\rdquo{} of their past deformations.
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66 Is it possible to apply thermodynamics to a system such as a vibrating quartz crystal? We can by
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67 no means rule out the possibility that still more laws of
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68 thermodynamics exist, as yet undiscovered, which would be
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69 useful in such applications.
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70
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71
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72 It is only by careful examination of the logic by which
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73 present thermodynamics was created, asking exactly how much of
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74 it is mathematical theorems, how much is deducible from the laws
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75 of mechanics and electrodynamics, and how much rests only on
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76 empirical evidence, how compelling is present evidence for the
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77 accuracy and range of validity of its laws; in other words,
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78 exactly where are the boundaries of present knowledge, that we
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79 can hope to uncover new things. Clearly, much research is still
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80 needed in this field, and we shall be able to accomplish only a
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81 small part of this program in the present review.
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82
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83
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84 It will develop that there is an astonishingly close analogy
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85 with the logic underlying statistical theory in general, where
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86 again a qualitative feeling that we all have (for the degrees of
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87 plausibility of various unproved and undisproved assertions) must
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88 be convertefi into a precisely defined quantitative concept
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89 (probability). Our later development of probability theory in
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90 Chapter 6,7 will be, to a considerable degree, a paraphrase
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91 of our present review of the logic underlying classical
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92 thermodynamics.
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93
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94 ** The Primitive Thermometer.
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95
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96 The earliest stages of our
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97 story are necessarily speculative, since they took place long
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98 before the beginnings of recorded history. But we can hardly
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99 doubt that primitive man learned quickly that objects exposed
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100 to the sun‘s rays or placed near a fire felt different from
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101 those in the shade away from fires; and the same difference was
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102 noted between animal bodies and inanimate objects.
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103
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104
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105 As soon as it was noted that changes in this feeling of
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106 warmth were correlated with other observable changes in the
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107 behavior of objects, such as the boiling and freezing of water,
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108 cooking of meat, melting of fat and wax, etc., the notion of
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109 warmth took its first step away from the purely subjective
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110 toward an objective, physical notion capable of being studied
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111 scientifically.
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112
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113 One of the most striking manifestations of warmth (but far
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114 from the earliest discovered) is the almost universal expansion
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115 of gases, liquids, and solids when heated . This property has
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116 proved to be a convenient one with which to reduce the notion
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117 of warmth to something entirely objective. The invention of the
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118 /thermometer/, in which expansion of a mercury column, or a gas,
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119 or the bending of a bimetallic strip, etc. is read off on a
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120 suitable scale, thereby giving us a /number/ with which to work,
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121 was a necessary prelude to even the crudest study of the physical
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122 nature of heat. To the best of our knowledge, although the
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123 necessary technology to do this had been available for at least
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124 3,000 years, the first person to carry it out in practice was
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125 Galileo, in 1592.
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126
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127 Later on we will give more precise definitions of the term
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128 \ldquo{}thermometer.\rdquo{} But at the present stage we
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129 are not in a position to do so (as Galileo was not), because
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130 the very concepts needed have not yet been developed;
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131 more precise definitions can be
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132 given only after our study has revealed the need for them. In
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133 deed, our final definition can be given only after the full
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134 mathematical formalism of statistical mechanics is at hand.
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135
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136 Once a thermometer has been constructed, and the scale
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137 marked off in a quite arbitrary way (although we will suppose
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138 that the scale is at least monotonic: i.e., greater warmth always
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139 corresponds to a greater number), we are ready to begin scien
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140 tific experiments in thermodynamics. The number read eff from
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141 any such instrument is called the /empirical temperature/, and we
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142 denote it by \(t\). Since the exact calibration of the thermometer
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143 is not specified), any monotonic increasing function
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144 \(t‘ = f(t)\) provides an equally good temperature scale for the
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145 present.
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146
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147
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148 ** Thermodynamic Systems.
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149
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150 The \ldquo{}thermodynamic systems\rdquo{} which
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151 are the objects of our study may be, physically, almost any
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152 collections of objects. The traditional simplest system with
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153 which to begin a study of thermodynamics is a volume of gas.
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154 We shall, however, be concerned from the start also with such
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155 things as a stretched wire or membrane, an electric cell, a
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156 polarized dielectric, a paramagnetic body in a magnetic field, etc.
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157
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158 The /thermodynamic state/ of such a system is determined by
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159 specifying (i.e., measuring) certain macrcoscopic physical
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160 properties. Now, any real physical system has many millions of such
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161 preperties; in order to have a usable theory we cannot require
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162 that /all/ of them be specified. We see, therefore, that there
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163 must be a clear distinction between the notions of
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164 \ldquo{}thermodynamic system\rdquo{} and \ldquo{}physical
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165 system.\rdquo{}
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166 A given /physical/ system may correspond to many different
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167 /thermodynamic systems/, depending
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168 on which variables we choose to measure or control; and which
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169 we decide to leave unmeasured and/or uncontrolled.
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170
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171
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172 For example, our physical system might consist of a crystal
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173 of sodium chloride. For one set of experiments we work with
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174 temperature, volume, and pressure; and ignore its electrical
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175 properties. For another set of experiments we work with
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176 temperature, electric field, and electric polarization; and
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177 ignore the varying stress and strain. The /physical/ system,
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178 therefore, corresponds to two entirely different /thermodynamic/
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179 systems. Exactly how much freedom, then, do we have in choosing
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180 the variables which shall define the thermodynamic state of our
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181 system? How many must we choose? What [criteria] determine when
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182 we have made an adequate choice? These questions cannot be
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183 answered until we say a little more about what we are trying to
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184 accomplish by a thermodynamic theory. A mere collection of
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185 recorded data about our system, as in the [[http://en.wikipedia.org/wiki/CRC_Handbook_of_Chemistry_and_Physics][/Handbook of Physics and
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186 Chemistry/]], is a very useful thing, but it hardly constitutes
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187 a theory. In order to construct anything deserving of such a
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188 name, the primary requirement is that we can recognize some kind
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189 of reproducible connection between the different properties con
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190 sidered, so that information about some of them will enable us
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191 to predict others. And of course, in order that our theory can
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192 be called thermodynamics (and not some other area of physics),
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193 it is necessary that the temperature be one of the quantities
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194 involved in a nontrivial way.
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195
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196 The gist of these remarks is that the notion of
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197 \ldquo{}thermodynamic system\rdquo{} is in part
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198 an anthropomorphic one; it is for us to
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199 say which set of variables shall be used. If two different
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200 choices both lead to useful reproducible connections, it is quite
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201 meaningless to say that one choice is any more \ldquo{}correct\rdquo{}
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202 than the other. Recognition of this fact will prove crucial later in
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203 avoiding certain ancient paradoxes.
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204
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205 At this stage we can determine only empirically which other
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206 physical properties need to be introduced before reproducible
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207 connections appear. Once any such connection is established, we
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208 can analyze it with the hope of being able to (1) reduce it to a
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209 /logical/ connection rather than an empirical one; and (2) extend
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210 it to an hypothesis applying beyond the original data, which
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211 enables us to predict further connections capable of being
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212 tested by experiment. Examples of this will be given presently.
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213
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214
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215 There will remain, however, a few reproducible relations
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216 which to the best of present knowledge, are not reducible to
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217 logical relations within the context of classical thermodynamics
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218 (and. whose demonstration in the wider context of mechanics,
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219 electrodynamics, and quantum theory remains one of probability
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220 rather than logical proof); from the standpoint of thermodynamics
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221 these remain simply statements of empirical fact which must be
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222 accepted as such without any deeper basis, but without which the
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223 development of thermodynamics cannot proceed. Because of this
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224 special status, these relations have become known as the
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225 \ldquo{}laws\rdquo{}
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226 of thermodynamics . The most fundamental one is a qualitative
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227 rather than quantitative relation, the \ldquo{}zero'th law.\rdquo{}
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228
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229 ** Equilibrium; the \ldquo{}Zero‘th Law.\rdquo{}
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230
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231 It is a common experience
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232 that when objects are placed in contact with each other but
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233 isolated from their surroundings, they may undergo observable
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234 changes for a time as a result; one body may become warmer,
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235 another cooler, the pressure of a gas or volume of a liquid may
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236 change; stress or magnetization in a solid may change, etc. But
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237 after a sufficient time, the observable macroscopic properties
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238 settle down to a steady condition, after which no further changes
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239 are seen unless there is a new intervention from the outside.
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240 When this steady condition is reached, the experimentalist says
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241 that the objects have reached a state of /equilibrium/ with each
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242 other. Once again, more precise definitions of this term will
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243 be needed eventually, but they require concepts not yet developed.
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244 In any event, the criterion just stated is almost the only one
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245 used in actual laboratory practice to decide when equilibrium
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246 has been reached.
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247
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248
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249 A particular case of equilibrium is encountered when we
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250 place a thermometer in contact with another body. The reading
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251 \(t\) of the thermometer may vary at first, but eventually it reach es
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252 a steady value. Now the number \(t\) read by a thermometer is always.
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253 by definition, the empirical temperature /of the thermometer/ (more
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254 precisely, of the sensitive element of the thermometer). When
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255 this number is constant in time, we say that the thermometer is
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256 in /thermal equilibrium/ with its surroundings; and we then extend
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257 the notion of temperature, calling the steady value \(t\) also the
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258 /temperature of the surroundings/.
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259
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260 We have repeated these elementary facts, well known to every
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261 child, in order to emphasize this point: Thermodynamics can be
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262 a theory /only/ of states of equilibrium, because the very
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263 procedure by which the temperature of a system is defined by
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264 operational means, already presupposes the attainment of
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265 equilibrium. Strictly speaking, therefore, classical
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266 thermodynamics does not even contain the concept of a
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267 \ldquo{}time-varying temperature.\rdquo{}
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268
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269 Of course, to recognize this limitation on conventional
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270 thermodynamics (best emphasized by calling it instead,
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271 thermostatics) in no way rules out the possibility of
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272 generalizing the notion of temperature to nonequilibrium states.
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273 Indeed, it is clear that one could define any number of
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274 time-dependent quantities all of which reduce, in the special
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275 case of equilibrium, to the temperature as defined above.
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276 Historically, attempts to do this even antedated the discovery
|
|
ocsenave@0
|
277 of the laws of thermodynamics, as is demonstrated by
|
|
ocsenave@0
|
278 \ldquo{}Newton's law of cooling.\rdquo{} Therefore, the
|
|
ocsenave@0
|
279 question is not whether generalization is /possible/, but only
|
|
ocsenave@0
|
280 whether it is in any way /useful/; i.e., does the temperature so
|
|
ocsenave@0
|
281 generalized have any connection with other physical properties
|
|
ocsenave@0
|
282 of our system, so that it could help us to predict other things?
|
|
ocsenave@0
|
283 However, to raise such questions takes us far beyond the
|
|
ocsenave@0
|
284 domain of thermostatics; and the general laws of nonequilibrium
|
|
ocsenave@0
|
285 behavior are so much more complicated that it would be virtually
|
|
ocsenave@0
|
286 hopeless to try to unravel them by empirical means alone. For
|
|
ocsenave@0
|
287 example, even if two different kinds of thermometer are calibrated
|
|
ocsenave@0
|
288 so that they agree with each other in equilibrium situations,
|
|
ocsenave@0
|
289 they will not agree in general about the momentary value a
|
|
ocsenave@0
|
290 \ldquo{}time-varying temperature.\rdquo{} To make any real
|
|
ocsenave@0
|
291 progress in this area, we have to supplement empirical observation by the guidance
|
|
ocsenave@0
|
292 of a rather hiqhly-developed theory. The notion of a
|
|
ocsenave@0
|
293 time-dependent temperature is far from simple conceptually, and we
|
|
ocsenave@0
|
294 will find that nothing very helpful can be said about this until
|
|
ocsenave@0
|
295 the full mathematical apparatus of nonequilibrium statistical
|
|
ocsenave@0
|
296 mechanics has been developed.
|
|
ocsenave@0
|
297
|
|
ocsenave@0
|
298 Suppose now that two bodies have the same temperature; i.e.,
|
|
ocsenave@0
|
299 a given thermometer reads the same steady value when in contact
|
|
ocsenave@0
|
300 with either. In order that the statement, \ldquo{}two bodies have the
|
|
ocsenave@1
|
301 same temperature\rdquo{} shall describe a physical property of the bodies,
|
|
ocsenave@0
|
302 and not merely an accidental circumstance due to our having used
|
|
ocsenave@0
|
303 a particular kind of thermometer, it is necessary that /all/
|
|
ocsenave@0
|
304 thermometers agree in assigning equal temperatures to them if
|
|
ocsenave@0
|
305 /any/ thermometer does . Only experiment is competent to determine
|
|
ocsenave@0
|
306 whether this universality property is true. Unfortunately, the
|
|
ocsenave@0
|
307 writer must confess that he is unable to cite any definite
|
|
ocsenave@0
|
308 experiment in which this point was subjected to a careful test.
|
|
ocsenave@0
|
309 That equality of temperatures has this absolute meaning, has
|
|
ocsenave@0
|
310 evidently been taken for granted so much that (like absolute
|
|
ocsenave@0
|
311 sirnultaneity in pre-relativity physics) most of us are not even
|
|
ocsenave@0
|
312 consciously aware that we make such an assumption in
|
|
ocsenave@0
|
313 thermodynamics. However, for the present we can only take it as a familiar
|
|
ocsenave@0
|
314 empirical fact that this condition does hold, not because we can
|
|
ocsenave@0
|
315 cite positive evidence for it, but because of the absence of
|
|
ocsenave@0
|
316 negative evidence against it; i.e., we think that, if an
|
|
ocsenave@0
|
317 exception had ever been found, this would have created a sensation in
|
|
ocsenave@0
|
318 physics, and we should have heard of it.
|
|
ocsenave@0
|
319
|
|
ocsenave@0
|
320 We now ask: when two bodies are at the same temperature,
|
|
ocsenave@0
|
321 are they then in thermal equilibrium with each other? Again,
|
|
ocsenave@0
|
322 only experiment is competent to answer this; the general
|
|
ocsenave@0
|
323 conclusion, again supported more by absence of negative evidence
|
|
ocsenave@0
|
324 than by specific positive evidence, is that the relation of
|
|
ocsenave@0
|
325 equilibrium has this property:
|
|
ocsenave@0
|
326 #+begin_quote
|
|
ocsenave@0
|
327 /Two bodies in thermal equilibrium
|
|
ocsenave@0
|
328 with a third body, are thermal equilibrium with each other./
|
|
ocsenave@0
|
329 #+end_quote
|
|
ocsenave@0
|
330
|
|
ocsenave@0
|
331 This empirical fact is usually called the \ldquo{}zero'th law of
|
|
ocsenave@0
|
332 thermodynamics.\rdquo{} Since nothing prevents us from regarding a
|
|
ocsenave@0
|
333 thermometer as the \ldquo{}third body\rdquo{} in the above statement,
|
|
ocsenave@0
|
334 it appears that we may also state the zero'th law as:
|
|
ocsenave@0
|
335 #+begin_quote
|
|
ocsenave@0
|
336 /Two bodies are in thermal equilibrium with each other when they are
|
|
ocsenave@0
|
337 at the same temperature./
|
|
ocsenave@0
|
338 #+end_quote
|
|
ocsenave@0
|
339 Although from the preceding discussion it might appear that
|
|
ocsenave@0
|
340 these two statements of the zero'th law are entirely equivalent
|
|
ocsenave@0
|
341 (and we certainly have no empirical evidence against either), it
|
|
ocsenave@0
|
342 is interesting to note that there are theoretical reasons, arising
|
|
ocsenave@0
|
343 from General Relativity, indicating that while the first
|
|
ocsenave@0
|
344 statement may be universally valid, the second is not. When we
|
|
ocsenave@0
|
345 consider equilibrium in a gravitational field, the verification
|
|
ocsenave@0
|
346 that two bodies have equal temperatures may require transport
|
|
ocsenave@0
|
347 of the thermometer through a gravitational potential difference;
|
|
ocsenave@0
|
348 and this introduces a new element into the discussion. We will
|
|
ocsenave@0
|
349 consider this in more detail in a later Chapter, and show that
|
|
ocsenave@0
|
350 according to General Relativity, equilibrium in a large system
|
|
ocsenave@0
|
351 requires, not that the temperature be uniform at all points, but
|
|
ocsenave@0
|
352 rather that a particular function of temperature and gravitational
|
|
ocsenave@0
|
353 potential be constant (the function is \(T\cdot \exp{(\Phi/c^2})\), where
|
|
ocsenave@0
|
354 \(T\) is the Kelvin temperature to be defined later, and \(\Phi\) is the
|
|
ocsenave@0
|
355 gravitational potential).
|
|
ocsenave@0
|
356
|
|
ocsenave@0
|
357 Of course, this effect is so small that ordinary terrestrial
|
|
ocsenave@0
|
358 experiments would need to have a precision many orders of
|
|
ocsenave@0
|
359 magnitude beyond that presently possible, before one could hope even
|
|
ocsenave@0
|
360 to detect it; and needless to say, it has played no role in the
|
|
ocsenave@0
|
361 development of thermodynamics. For present purposes, therefore,
|
|
ocsenave@0
|
362 we need not distinguish between the two above statements of the
|
|
ocsenave@0
|
363 zero'th law, and we take it as a basic empirical fact that a
|
|
ocsenave@0
|
364 uniform temperature at all points of a system is an essential
|
|
ocsenave@0
|
365 condition for equilibrium. It is an important part of our
|
|
ocsenave@0
|
366 ivestigation to determine whether there are other essential
|
|
ocsenave@0
|
367 conditions as well. In fact, as we will find, there are many
|
|
ocsenave@0
|
368 different kinds of equilibrium; and failure to distinguish between
|
|
ocsenave@0
|
369 them can be a prolific source of paradoxes.
|
|
ocsenave@0
|
370
|
|
ocsenave@0
|
371 ** Equation of State
|
|
ocsenave@0
|
372 Another important reproducible connection is found when
|
|
ocsenave@0
|
373 we consider a thermodynamic system defined by
|
|
ocsenave@0
|
374 three parameters; in addition to the temperature we choose a
|
|
ocsenave@0
|
375 \ldquo{}displacement\rdquo{} and a conjugate \ldquo{}force.\rdquo{}
|
|
ocsenave@0
|
376 Subject to some qualifications given below, we find experimentally
|
|
ocsenave@0
|
377 that these parameters are not independent, but are subject to a constraint.
|
|
ocsenave@0
|
378 For example, we cannot vary the equilibrium pressure, volume,
|
|
ocsenave@0
|
379 and temperature of a given mass of gas independently; it is found
|
|
ocsenave@0
|
380 that a given pressure and volume can be realized only at one
|
|
ocsenave@0
|
381 particular temperature, that the gas will assume a given tempera~
|
|
ocsenave@0
|
382 ture and volume only at one particular pressure, etc. Similarly,
|
|
ocsenave@0
|
383 a stretched wire can be made to have arbitrarily assigned tension
|
|
ocsenave@0
|
384 and elongation only if its temperature is suitably chosen, a
|
|
ocsenave@0
|
385 dielectric will assume a state of given temperature and
|
|
ocsenave@0
|
386 polarization at only one value of the electric field, etc.
|
|
ocsenave@0
|
387 These simplest nontrivial thermodynamic systems (three
|
|
ocsenave@0
|
388 parameters with one constraint) are said to possess two
|
|
ocsenave@0
|
389 /degrees of freedom/; for the range of possible equilibrium states is defined
|
|
ocsenave@0
|
390 by specifying any two of the variables arbitrarily, whereupon the
|
|
ocsenave@0
|
391 third, and all others we may introduce, are determined.
|
|
ocsenave@0
|
392 Mathematically, this is expressed by the existence of a functional
|
|
ocsenave@1
|
393 relationship of the form[fn:: /Edit./: The set of solutions to an equation
|
|
ocsenave@0
|
394 like /f(X,x,t)=/ const. is called a /level set/. Here, Jaynes is
|
|
ocsenave@0
|
395 saying that the quantities /X/, /x/, and /t/ follow a \ldquo{}functional
|
|
ocsenave@0
|
396 rule\rdquo{}, so the set of physically allowed combinations of /X/,
|
|
ocsenave@0
|
397 /x/, and /t/ in equilibrium states can be
|
|
ocsenave@0
|
398 expressed as the level set of a function.
|
|
ocsenave@0
|
399
|
|
ocsenave@0
|
400 But not every function expresses a constraint relation; for some
|
|
ocsenave@0
|
401 functions, you can specify two of the variables, and the third will
|
|
ocsenave@0
|
402 still be undetermined. (For example, if f=X^2+x^2+t^2-3,
|
|
ocsenave@0
|
403 the level set /f(X,x,t)=0/ is a sphere, and specifying /x=1/, /t=1/
|
|
ocsenave@0
|
404 leaves you with two potential possibilities for /X/ =\pm 1.)
|
|
ocsenave@0
|
405
|
|
ocsenave@1
|
406 A function like /f/ has to possess one more propery in order for its
|
|
ocsenave@1
|
407 level set to express a constraint relationship: it must be monotonic in
|
|
ocsenave@0
|
408 each of its variables /X/, /x/, and /t/.
|
|
ocsenave@0
|
409 #the partial derivatives of /f/ exist for every allowed combination of
|
|
ocsenave@0
|
410 #inputs /x/, /X/, and /t/.
|
|
ocsenave@0
|
411 In other words, the level set has to pass a sort of
|
|
ocsenave@0
|
412 \ldquo{}vertical line test\rdquo{} for each of its variables.]
|
|
ocsenave@0
|
413
|
|
ocsenave@0
|
414 #Edit Here, Jaynes
|
|
ocsenave@0
|
415 #is saying that it is possible to express the collection of allowed combinations \(\langle X,x,t \rangle\) of force, quantity, and temperature as a
|
|
ocsenave@0
|
416 #[[http://en.wikipedia.org/wiki/Level_set][level set]] of some function \(f\). However, not all level sets represent constraint relations; consider \(f(X,x,t)=X^2+x^2+t^2-1\)=0.
|
|
ocsenave@0
|
417 #In order to specify
|
|
ocsenave@0
|
418
|
|
ocsenave@0
|
419 \begin{equation}
|
|
ocsenave@0
|
420 f(X,x,t) = O
|
|
ocsenave@0
|
421 \end{equation}
|
|
ocsenave@0
|
422
|
|
ocsenave@0
|
423 where $X$ is a generalized force (pressure, tension, electric or
|
|
ocsenave@0
|
424 magnetic field, etc.), $x$ is the corresponding generalized
|
|
ocsenave@0
|
425 displacement (volume, elongation, electric or magnetic polarization,
|
|
ocsenave@1
|
426 etc.), and $t$ is the empirical temperature. Equation (1-1) is
|
|
ocsenave@0
|
427 called /the equation of state/.
|
|
ocsenave@0
|
428
|
|
ocsenave@0
|
429 At the risk of belaboring it, we emphasize once again that
|
|
ocsenave@0
|
430 all of this applies only for a system in equilibrium; for
|
|
ocsenave@0
|
431 otherwise not only.the temperature, but also some or all of the other
|
|
ocsenave@0
|
432 variables may not be definable. For example, no unique pressure
|
|
ocsenave@0
|
433 can be assigned to a gas which has just suffered a sudden change
|
|
ocsenave@0
|
434 in volume, until the generated sound waves have died out.
|
|
ocsenave@0
|
435
|
|
ocsenave@0
|
436 Independently of its functional form, the mere fact of the
|
|
ocsenave@0
|
437 /existence/ of an equation of state has certain experimental
|
|
ocsenave@0
|
438 consequences. For example, suppose that in experiments on oxygen
|
|
ocsenave@0
|
439 gas, in which we control the temperature and pressure
|
|
ocsenave@0
|
440 independently, we have found that the isothermal compressibility $K$
|
|
ocsenave@0
|
441 varies with temperature, and the thermal expansion coefficient
|
|
ocsenave@0
|
442 \alpha varies with pressure $P$, so that within the accuracy of the data,
|
|
ocsenave@0
|
443
|
|
ocsenave@0
|
444 \begin{equation}
|
|
ocsenave@0
|
445 \frac{\partial K}{\partial t} = - \frac{\partial \alpha}{\partial P}
|
|
ocsenave@0
|
446 \end{equation}
|
|
ocsenave@0
|
447
|
|
ocsenave@0
|
448 Is this a particular property of oxygen; or is there reason to
|
|
ocsenave@0
|
449 believe that it holds also for other substances? Does it depend
|
|
ocsenave@0
|
450 on our particular choice of a temperature scale?
|
|
ocsenave@0
|
451
|
|
ocsenave@0
|
452 In this case, the answer is found at once; for the definitions of $K$,
|
|
ocsenave@0
|
453 \alpha are
|
|
ocsenave@0
|
454
|
|
ocsenave@0
|
455 \begin{equation}
|
|
ocsenave@0
|
456 K = -\frac{1}{V}\frac{\partial V}{\partial P},\qquad
|
|
ocsenave@0
|
457 \alpha=\frac{1}{V}\frac{\partial V}{\partial t}
|
|
ocsenave@0
|
458 \end{equation}
|
|
ocsenave@0
|
459
|
|
ocsenave@0
|
460 which is simply a mathematical expression of the fact that the
|
|
ocsenave@0
|
461 volume $V$ is a definite function of $P$ and $t$; i.e., it depends
|
|
ocsenave@0
|
462 only
|
|
ocsenave@0
|
463 on their present values, and not how those values were attained.
|
|
ocsenave@0
|
464 In particular, $V$ does not depend on the direction in the \((P, t)\)
|
|
ocsenave@0
|
465 plane through which the present values were approached; or, as we
|
|
ocsenave@0
|
466 usually say it, \(dV\) is an /exact differential/.
|
|
ocsenave@0
|
467
|
|
ocsenave@1
|
468 Therefore, although at first glance the relation (1-2) appears
|
|
ocsenave@0
|
469 nontrivial and far from obvious, a trivial mathematical analysis
|
|
ocsenave@0
|
470 convinces us that it must hold regardless of our particular
|
|
ocsenave@0
|
471 temperature scale, and that it is true not only of oxygen; it must
|
|
ocsenave@0
|
472 hold for any substance, or mixture of substances, which possesses a
|
|
ocsenave@0
|
473 definite, reproducible equation of state \(f(P,V,t)=0\).
|
|
ocsenave@0
|
474
|
|
ocsenave@0
|
475 But this understanding also enables us to predict situations in which
|
|
ocsenave@1
|
476 (1-2) will /not/ hold. Equation (1-2), as we have just learned, expresses
|
|
ocsenave@0
|
477 the fact that an equation of state exists involving only the three
|
|
ocsenave@0
|
478 variables \((P,V,t)\). Now suppose we try to apply it to a liquid such
|
|
ocsenave@0
|
479 as nitrobenzene. The nitrobenzene molecule has a large electric dipole
|
|
ocsenave@0
|
480 moment; and so application of an electric field (as in the
|
|
ocsenave@0
|
481 [[http://en.wikipedia.org/wiki/Kerr_effect][electro-optical Kerr cell]]) causes an alignment of molecules which, as
|
|
ocsenave@0
|
482 accurate measurements will verify, changes the pressure at a given
|
|
ocsenave@0
|
483 temperature and volume. Therefore, there can no longer exist any
|
|
ocsenave@0
|
484 unique equation of state involving \((P, V, t)\) only; with
|
|
ocsenave@0
|
485 sufficiently accurate measurements, nitrobenzene must be regarded as a
|
|
ocsenave@0
|
486 thermodynamic system with at least three degrees of freedom, and the
|
|
ocsenave@0
|
487 general equation of state must have at least a complicated a form as
|
|
ocsenave@0
|
488 \(f(P,V,t,E) = 0\).
|
|
ocsenave@0
|
489
|
|
ocsenave@0
|
490 But if we introduce a varying electric field $E$ into the discussion,
|
|
ocsenave@0
|
491 the resulting varying electric polarization $M$ also becomes a new
|
|
ocsenave@0
|
492 thermodynamic variable capable of being measured. Experimentally, it
|
|
ocsenave@0
|
493 is easiest to control temperature, pressure, and electric field
|
|
ocsenave@0
|
494 independently, and of course we find that both the volume and
|
|
ocsenave@0
|
495 polarization are then determined; i.e., there must exist functional
|
|
ocsenave@0
|
496 relations of the form \(V = V(P,t,E)\), \(M = M(P,t,E)\), or in more
|
|
ocsenave@0
|
497 symmetrical form
|
|
ocsenave@0
|
498
|
|
ocsenave@0
|
499 \begin{equation}
|
|
ocsenave@0
|
500 f(V,P,t,E) = 0 \qquad g(M,P,t,E)=0.
|
|
ocsenave@0
|
501 \end{equation}
|
|
ocsenave@0
|
502
|
|
ocsenave@0
|
503 In other words, if we regard nitrobenzene as a thermodynamic system of
|
|
ocsenave@0
|
504 three degrees of freedom (i.e., having specified three parameters
|
|
ocsenave@0
|
505 arbitrarily, all others are then determined), it must possess two
|
|
ocsenave@0
|
506 independent equations of state.
|
|
ocsenave@0
|
507
|
|
ocsenave@0
|
508 Similarly, a thermodynamic system with four degrees of freedom,
|
|
ocsenave@0
|
509 defined by the termperature and three pairs of conjugate forces and
|
|
ocsenave@0
|
510 displacements, will have three independent equations of state, etc.
|
|
ocsenave@0
|
511
|
|
ocsenave@0
|
512 Now, returning to our original question, if nitrobenzene possesses
|
|
ocsenave@0
|
513 this extra electrical degree of freedom, under what circumstances do
|
|
ocsenave@0
|
514 we exprect to find a reproducible equation of state involving
|
|
ocsenave@0
|
515 \((p,V,t)\) only? Evidently, if $E$ is held constant, then the first
|
|
ocsenave@0
|
516 of equations (1-5) becomes such an equation of state, involving $E$ as
|
|
ocsenave@0
|
517 a fixed parameter; we would find many different equations of state of
|
|
ocsenave@0
|
518 the form \(f(P,V,t) = 0\) with a different function $f$ for each
|
|
ocsenave@0
|
519 different value of the electric field. Likewise, if \(M\) is held
|
|
ocsenave@0
|
520 constant, we can eliminate \(E\) between equations (1-5) and find a
|
|
ocsenave@0
|
521 relation \(h(P,V,t,M)=0\), which is an equation of state for
|
|
ocsenave@0
|
522 \((P,V,t)\) containing \(M\) as a fixed parameter.
|
|
ocsenave@0
|
523
|
|
ocsenave@0
|
524 More generally, if an electrical constraint is imposed on the system
|
|
ocsenave@0
|
525 (for example, by connecting an external charged capacitor to the
|
|
ocsenave@0
|
526 electrodes) so that \(M\) is determined by \(E\); i.e., there is a
|
|
ocsenave@0
|
527 functional relation of the form
|
|
ocsenave@0
|
528
|
|
ocsenave@0
|
529 \begin{equation}
|
|
ocsenave@0
|
530 g(M,E) = \text{const.}
|
|
ocsenave@0
|
531 \end{equation}
|
|
ocsenave@0
|
532
|
|
ocsenave@0
|
533 then (1-5) and (1-6) constitute three simultaneous equations, from
|
|
ocsenave@0
|
534 which both \(E\) and \(M\) may be eliminated mathematically, leading
|
|
ocsenave@0
|
535 to a relation of the form \(h(P,V,t;q)=0\), which is an equation of
|
|
ocsenave@0
|
536 state for \((P,V,t)\) involving the fixed parameter \(q\).
|
|
ocsenave@0
|
537
|
|
ocsenave@0
|
538 We see, then, that as long as a fixed constraint of the form (1-6) is
|
|
ocsenave@0
|
539 imposed on the electrical degree of freedom, we can still observe a
|
|
ocsenave@0
|
540 reproducible equation of state for nitrobenzene, considered as a
|
|
ocsenave@0
|
541 thermodynamic system of only two degrees of freedom. If, however, this
|
|
ocsenave@0
|
542 electrical constraint is removed, so that as we vary $P$ and $t$, the
|
|
ocsenave@0
|
543 values of $E$ and $M$ vary in an uncontrolled way over a
|
|
ocsenave@0
|
544 /two-dimensional/ region of the \((E, M)\) plane, then we will find no
|
|
ocsenave@0
|
545 definite equation of state involving only \((P,V,t)\).
|
|
ocsenave@0
|
546
|
|
ocsenave@0
|
547 This may be stated more colloqually as follows: even though a system
|
|
ocsenave@0
|
548 has three degrees of freedom, we can still consider only the variables
|
|
ocsenave@0
|
549 belonging to two of them, and we will find a definite equation of
|
|
ocsenave@0
|
550 state, /provided/ that in the course of the experiments, the unused
|
|
ocsenave@0
|
551 degree of freedom is not \ldquo{}tampered with\rdquo{} in an
|
|
ocsenave@0
|
552 uncontrolled way.
|
|
ocsenave@0
|
553
|
|
ocsenave@0
|
554 We have already emphasized that any physical system corresponds to
|
|
ocsenave@0
|
555 many different thermodynamic systems, depending on which variables we
|
|
ocsenave@0
|
556 choose to control and measure. In fact, it is easy to see that any
|
|
ocsenave@0
|
557 physical system has, for all practical purposes, an /arbitrarily
|
|
ocsenave@0
|
558 large/ number of degrees of freedom. In the case of nitrobenzene, for
|
|
ocsenave@0
|
559 example, we may impose any variety of nonuniform electric fields on
|
|
ocsenave@1
|
560 our sample. Suppose we place $(n+1)$ different electrodes, labelled
|
|
ocsenave@1
|
561 \(\{e_0,e_1, e_2 \ldots e_n\}\) in contact with the liquid in various
|
|
ocsenave@1
|
562 positions. Regarding \(e_0\) as the \ldquo{}ground\rdquo{}, maintained
|
|
ocsenave@1
|
563 at zero potential, we can then impose $n$ different potentials
|
|
ocsenave@1
|
564 \(\{v_1, \ldots, v_n\}\) on the other electrodes independently, and we
|
|
ocsenave@1
|
565 can also measure the $n$ different conjugate displacements, as the
|
|
ocsenave@1
|
566 charges \(\{q_1,\ldots, q_n\}\) accumulated on electrodes
|
|
ocsenave@1
|
567 \(\{e_1,\ldots e_n\}\). Together with the pressure (understood as the
|
|
ocsenave@1
|
568 pressure measured at one given position), volume, and temperature, our
|
|
ocsenave@1
|
569 sample of nitrobenzene is now a thermodynamic system of $(n+1)$
|
|
ocsenave@1
|
570 degrees of freedom. This number may be as large as we please, limited
|
|
ocsenave@1
|
571 only by our patience in constructing the apparatus needed to control
|
|
ocsenave@1
|
572 or measure all these quantities.
|
|
ocsenave@1
|
573
|
|
ocsenave@1
|
574 We leave it as an exercise for the reader (Problem 1) to find the most
|
|
ocsenave@1
|
575 general condition on the variables \(\{v_1, q_1, v_2, q_2, \ldots
|
|
ocsenave@1
|
576 v_n,q_n\}\) which will ensure that a definite equation of state
|
|
ocsenave@1
|
577 $f(P,V,t)=0$ is observed in spite of all these new degrees of
|
|
ocsenave@1
|
578 freedom. The simplest special case of this relation is, evidently, to
|
|
ocsenave@1
|
579 ground all electrodes, thereby inposing the conditions $v_1 = v_2 =
|
|
ocsenave@1
|
580 \ldots = v_n = 0$. Equally well (if we regard nitrobenzene as having
|
|
ocsenave@1
|
581 negligible electrical conductivity) we may open-circuit all
|
|
ocsenave@1
|
582 electrodes, thereby imposing the conditions \(q_i = \text{const.}\) In
|
|
ocsenave@1
|
583 the latter case, in addition to an equation of state of the form
|
|
ocsenave@1
|
584 \(f(P,V,t)=0\), which contains these constants as fixed parameters,
|
|
ocsenave@1
|
585 there are \(n\) additional equations of state of the form $v_i =
|
|
ocsenave@1
|
586 v_i(P,t)$. But if we choose to ignore these voltages, there will be no
|
|
ocsenave@1
|
587 contradiction in considering our nitrobenzene to be a thermodynamic
|
|
ocsenave@1
|
588 system of two degrees of freedom, involving only the variables
|
|
ocsenave@1
|
589 \(P,V,t\).
|
|
ocsenave@1
|
590
|
|
ocsenave@1
|
591 Similarly, if our system of interest is a crystal, we may impose on it
|
|
ocsenave@1
|
592 a wide variety of nonuniform stress fields; each component of the
|
|
ocsenave@1
|
593 stress tensor $T_{ij}$ may bary with position. We might expand each of
|
|
ocsenave@1
|
594 these functions in a complete orthonormal set of functions
|
|
ocsenave@1
|
595 \(\phi_k(x,y,z)\):
|
|
ocsenave@1
|
596
|
|
ocsenave@1
|
597 \begin{equation}
|
|
ocsenave@1
|
598 T_{ij}(x,y,z) = \sum_k a_{ijk} \phi_k(x,y,z)
|
|
ocsenave@1
|
599 \end{equation}
|
|
ocsenave@1
|
600
|
|
ocsenave@1
|
601 and with a sufficiently complicated system of levers which in various
|
|
ocsenave@1
|
602 ways squeeze and twist the crystal, we might vary each of the first
|
|
ocsenave@1
|
603 1,000 expansion coefficients $a_{ijk}$ independently, and measure the
|
|
ocsenave@1
|
604 conjugate displacements $q_{ijk}$. Our crystal is then a thermodynamic
|
|
ocsenave@1
|
605 system of over 1,000 degrees of freedom.
|
|
ocsenave@1
|
606
|
|
ocsenave@1
|
607 The notion of \ldquo{}numbers of degrees of freedom\rdquo{} is
|
|
ocsenave@1
|
608 therefore not a /physical property/ of any system; it is entirely
|
|
ocsenave@1
|
609 anthropomorphic, since any physical system may be regarded as a
|
|
ocsenave@1
|
610 thermodynamic system with any number of degrees of freedom we please.
|
|
ocsenave@1
|
611
|
|
ocsenave@1
|
612 If new thermodynamic variables are always introduced in pairs,
|
|
ocsenave@1
|
613 consisting of a \ldquo{}force\rdquo{} and conjugate
|
|
ocsenave@1
|
614 \ldquo{}displacement\rdquo{}, then a thermodynamic system of $n$
|
|
ocsenave@1
|
615 degrees of freedom must possess $(n-1)$ independent equations of
|
|
ocsenave@1
|
616 state, so that specifying $n$ quantities suffices to determine all
|
|
ocsenave@1
|
617 others.
|
|
ocsenave@1
|
618
|
|
ocsenave@1
|
619 This raises an interesting question; whether the scheme of classifying
|
|
ocsenave@1
|
620 thermodynamic variables in conjugate pairs is the most general
|
|
ocsenave@1
|
621 one. Why, for example, is it not natural to introduce three related
|
|
ocsenave@1
|
622 variables at a time? To the best of the writer's knowledge, this is an
|
|
ocsenave@1
|
623 open question; there seems to be no fundamental reason why variables
|
|
ocsenave@1
|
624 /must/ always be introduced in conjugate pairs, but there seems to be
|
|
ocsenave@1
|
625 no known case in which a different scheme suggests itself as more
|
|
ocsenave@1
|
626 appropriate.
|
|
ocsenave@1
|
627
|
|
ocsenave@1
|
628 ** Heat
|
|
ocsenave@1
|
629 We are now in a position to consider the results and interpretation of
|
|
ocsenave@1
|
630 a number of elementary experiments involving
|
|
ocsenave@1
|
631
|
|
ocsenave@1
|
632
|
|
ocsenave@1
|
633 * Appendix
|
|
ocsenave@1
|
634
|
|
ocsenave@1
|
635 | Generalized Force | Generalized Displacement |
|
|
ocsenave@1
|
636 |--------------------+--------------------------|
|
|
ocsenave@1
|
637 | force | displacement |
|
|
ocsenave@1
|
638 | pressure | volume |
|
|
ocsenave@1
|
639 | electric potential | charge |
|
