1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/org/stat-mech.org	Sat Apr 28 19:32:50 2012 -0500
     1.3 @@ -0,0 +1,560 @@
     1.4 +#+TITLE: Statistical Mechanics
     1.5 +#+AUTHOR:    E.T. Jaynes; edited by  Dylan Holmes
     1.6 +#+EMAIL:     rlm@mit.edu
     1.7 +#+KEYWORDS: statistical mechanics, thermostatics, thermodynamics, temperature, paradoxes, Jaynes
     1.8 +#+SETUPFILE: ../../aurellem/org/setup.org
     1.9 +#+INCLUDE:   ../../aurellem/org/level-0.org
    1.10 +#+MATHJAX: align:"left" mathml:t path:"http://www.aurellem.org/MathJax/MathJax.js"
    1.11 +
    1.12 +# "extensions/eqn-number.js"
    1.13 +
    1.14 +#+begin_quote
    1.15 +*Note:* The following is a typeset version of
    1.16 + [[../sources/stat.mech.1.pdf][this unpublished book draft]], written by [[http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E.T. Jaynes]].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
    1.17 + minor changes, e.g. to correct typographical errors, add references, or format equations. The
    1.18 + content itself is intact. --- Dylan
    1.19 +#+end_quote
    1.20 +
    1.21 +* Development of Thermodynamics
    1.22 +Our first intuitive, or \ldquo{}subjective\rdquo{} notions of temperature
    1.23 +arise from the sensations of warmth and cold associated with our
    1.24 +sense of touch . Yet science has been able to convert this qualitative 
    1.25 +sensation into an accurately defined quantitative notion,
    1.26 +which can be applied far beyond the range of our direct experience. 
    1.27 +Today an experimentalist will report confidently that his
    1.28 +spin system was at a temperature of 2.51 degrees Kelvin; and a
    1.29 +theoretician will report with almost as much confidence that the 
    1.30 +temperature at the center of the sun is about \(2 \times 10^7\) degrees
    1.31 +Kelvin.
    1.32 +
    1.33 +The /fact/ that this has proved possible, and the main technical 
    1.34 +ideas involved, are assumed already known to the reader;
    1.35 +and we are not concerned here with repeating standard material
    1.36 +already available in a dozen other textbooks . However 
    1.37 +thermodynamics, in spite of its great successes, firmly established
    1.38 +for over a century, has also produced a great deal of confusion
    1.39 +and a long list of \ldquo{}paradoxes\rdquo{} centering mostly 
    1.40 +around the second law and the nature of irreversibility. 
    1.41 +For this reason and others noted below, we want to dwell here at 
    1.42 +some length on the /logic/ underlying the development of 
    1.43 +thermodynamics . Our aim is to emphasize certain points which, 
    1.44 +in the writer's opinion, are essential for clearing up the 
    1.45 +confusion and resolving the paradoxes; but which are not 
    1.46 +sufficiently ernphasized---and indeed in many cases are 
    1.47 +totally ignored---in other textbooks.
    1.48 +
    1.49 +This attention to logic 
    1.50 +would not be particularly needed if we regarded classical 
    1.51 +thermodynamics (or, as it is becoming called increasingly, 
    1.52 +/thermostatics/) as a closed subject, in which the fundamentals 
    1.53 +are already completely established, and there is
    1.54 +nothing more to be learned about them. A person who believes
    1.55 +this will probably prefer a pure axiomatic approach, in which
    1.56 +the basic laws are simply stated as arbitrary axioms, without
    1.57 +any attempt to present the evidence for them; and one proceeds
    1.58 +directly to working out their consequences.
    1.59 +However, we take the attitude here that thermostatics, for
    1.60 +all its venerable age, is very far from being a closed subject,
    1.61 +we still have a great deal to learn about such matters as the
    1.62 +most general definitions of equilibrium and reversibility, the
    1.63 +exact range of validity of various statements of the second and
    1.64 +third laws, the necessary and sufficient conditions for 
    1.65 +applicability of thermodynamics to special cases such as 
    1.66 +spin systems, and how thermodynamics can be applied to such 
    1.67 +systems as putty or polyethylene, which deform under force, 
    1.68 +but retain a \ldquo{}memory\rdquo{} of their past deformations. 
    1.69 +Is it possible to apply thermodynamics to a system such as a vibrating quartz crystal? We can by
    1.70 +no means rule out the possibility that still more laws of 
    1.71 +thermodynamics exist, as yet undiscovered, which would be 
    1.72 +useful in such applications.
    1.73 +
    1.74 +
    1.75 +It is only by careful examination of the logic by which
    1.76 +present thermodynamics was created, asking exactly how much of
    1.77 +it is mathematical theorems, how much is deducible from the laws
    1.78 +of mechanics and electrodynamics, and how much rests only on
    1.79 +empirical evidence, how compelling is present evidence for the
    1.80 +accuracy and range of validity of its laws; in other words,
    1.81 +exactly where are the boundaries of present knowledge, that we
    1.82 +can hope to uncover new things. Clearly, much research is still
    1.83 +needed in this field, and we shall be able to accomplish only a
    1.84 +small part of this program in the present review.
    1.85 +
    1.86 +
    1.87 +It will develop that there is an astonishingly close analogy
    1.88 +with the logic underlying statistical theory in general, where
    1.89 +again a qualitative feeling that we all have (for the degrees of
    1.90 +plausibility of various unproved and undisproved assertions) must
    1.91 +be convertefi into a precisely defined quantitative concept 
    1.92 +(probability). Our later development of probability theory in 
    1.93 +Chapter 6,7 will be, to a considerable degree, a paraphrase 
    1.94 +of our present review of the logic underlying classical 
    1.95 +thermodynamics.
    1.96 +
    1.97 +** The Primitive Thermometer. 
    1.98 +
    1.99 +The earliest stages of our
   1.100 +story are necessarily speculative, since they took place long
   1.101 +before the beginnings of recorded history. But we can hardly
   1.102 +doubt that primitive man learned quickly that objects exposed
   1.103 +to the sun‘s rays or placed near a fire felt different from
   1.104 +those in the shade away from fires; and the same difference was
   1.105 +noted between animal bodies and inanimate objects.
   1.106 +
   1.107 +
   1.108 +As soon as it was noted that changes in this feeling of
   1.109 +warmth were correlated with other observable changes in the
   1.110 +behavior of objects, such as the boiling and freezing of water, 
   1.111 +cooking of meat, melting of fat and wax, etc., the notion of
   1.112 +warmth took its first step away from the purely subjective 
   1.113 +toward an objective, physical notion capable of being studied
   1.114 +scientifically.
   1.115 +
   1.116 +One of the most striking manifestations of warmth (but far
   1.117 +from the earliest discovered) is the almost universal expansion
   1.118 +of gases, liquids, and solids when heated . This property has
   1.119 +proved to be a convenient one with which to reduce the notion
   1.120 +of warmth to something entirely objective. The invention of the
   1.121 +/thermometer/, in which expansion of a mercury column, or a gas,
   1.122 +or the bending of a bimetallic strip, etc. is read off on a
   1.123 +suitable scale, thereby giving us a /number/ with which to work,
   1.124 +was a necessary prelude to even the crudest study of the physical
   1.125 +nature of heat. To the best of our knowledge, although the
   1.126 +necessary technology to do this had been available for at least
   1.127 +3,000 years, the first person to carry it out in practice was
   1.128 +Galileo, in 1592.
   1.129 +
   1.130 +Later on we will give more precise definitions of the term
   1.131 +\ldquo{}thermometer.\rdquo{} But at the present stage we 
   1.132 +are not in a position to do so (as Galileo was not), because 
   1.133 +the very concepts needed have not yet been developed; 
   1.134 +more precise definitions can be
   1.135 +given only after our study has revealed the need for them. In
   1.136 +deed, our final definition can be given only after the full
   1.137 +mathematical formalism of statistical mechanics is at hand.
   1.138 +
   1.139 +Once a thermometer has been constructed, and the scale
   1.140 +marked off in a quite arbitrary way (although we will suppose
   1.141 +that the scale is at least monotonic: i.e., greater warmth always
   1.142 +corresponds to a greater number), we are ready to begin scien
   1.143 +tific experiments in thermodynamics. The number read eff from
   1.144 +any such instrument is called the /empirical temperature/, and we
   1.145 +denote it by \(t\). Since the exact calibration of the thermometer
   1.146 +is not specified), any monotonic increasing function 
   1.147 +\(t‘ = f(t)\) provides an equally good temperature scale for the
   1.148 +present.
   1.149 +
   1.150 +
   1.151 +** Thermodynamic Systems.
   1.152 +
   1.153 +The \ldquo{}thermodynamic systems\rdquo{} which
   1.154 +are the objects of our study may be, physically, almost any
   1.155 +collections of objects. The traditional simplest system with
   1.156 +which to begin a study of thermodynamics is a volume of gas.
   1.157 +We shall, however, be concerned from the start also with such
   1.158 +things as a stretched wire or membrane, an electric cell, a
   1.159 +polarized dielectric, a paramagnetic body in a magnetic field, etc.
   1.160 +
   1.161 +The /thermodynamic state/ of such a system is determined by
   1.162 +specifying (i.e., measuring) certain macrcoscopic physical 
   1.163 +properties. Now, any real physical system has many millions of such
   1.164 +preperties; in order to have a usable theory we cannot require
   1.165 +that /all/ of them be specified. We see, therefore, that there
   1.166 +must be a clear distinction between the notions of 
   1.167 +\ldquo{}thermodynamic system\rdquo{} and \ldquo{}physical
   1.168 +system.\rdquo{}
   1.169 +A given /physical/ system may correspond to many different
   1.170 +/thermodynamic systems/, depending
   1.171 +on which variables we choose to measure or control; and which
   1.172 +we decide to leave unmeasured and/or uncontrolled.
   1.173 +
   1.174 +
   1.175 +For example, our physical system might consist of a crystal
   1.176 +of sodium chloride. For one set of experiments we work with
   1.177 +temperature, volume, and pressure; and ignore its electrical
   1.178 +properties. For another set of experiments we work with 
   1.179 +temperature, electric field, and electric polarization; and 
   1.180 +ignore the varying stress and strain. The /physical/ system, 
   1.181 +therefore, corresponds to two entirely different /thermodynamic/ 
   1.182 +systems. Exactly how much freedom, then, do we have in choosing 
   1.183 +the variables which shall define the thermodynamic state of our
   1.184 +system? How many must we choose? What [criteria] determine when
   1.185 +we have made an adequate choice? These questions cannot be
   1.186 +answered until we say a little more about what we are trying to
   1.187 +accomplish by a thermodynamic theory. A mere collection of
   1.188 +recorded data about our system, as in the [[http://en.wikipedia.org/wiki/CRC_Handbook_of_Chemistry_and_Physics][/Handbook of Physics and
   1.189 +Chemistry/]], is a very useful thing, but it hardly constitutes
   1.190 +a theory. In order to construct anything deserving of such a
   1.191 +name, the primary requirement is that we can recognize some kind
   1.192 +of reproducible connection between the different properties con
   1.193 +sidered, so that information about some of them will enable us
   1.194 +to predict others. And of course, in order that our theory can
   1.195 +be called thermodynamics (and not some other area of physics),
   1.196 +it is necessary that the temperature be one of the quantities
   1.197 +involved in a nontrivial way.
   1.198 +
   1.199 +The gist of these remarks is that the notion of 
   1.200 +\ldquo{}thermodynamic system\rdquo{} is in part 
   1.201 +an anthropomorphic one; it is for us to
   1.202 +say which set of variables shall be used. If two different
   1.203 +choices both lead to useful reproducible connections, it is quite
   1.204 +meaningless to say that one choice is any more \ldquo{}correct\rdquo{} 
   1.205 +than the other. Recognition of this fact will prove crucial later in
   1.206 +avoiding certain ancient paradoxes.
   1.207 +
   1.208 +At this stage we can determine only empirically which other
   1.209 +physical properties need to be introduced before reproducible
   1.210 +connections appear. Once any such connection is established, we
   1.211 +can analyze it with the hope of being able to (1) reduce it to a
   1.212 +/logical/ connection rather than an empirical one; and (2) extend
   1.213 +it to an hypothesis applying beyond the original data, which
   1.214 +enables us to predict further connections capable of being
   1.215 +tested by experiment. Examples of this will be given presently.
   1.216 +
   1.217 +
   1.218 +There will remain, however, a few reproducible relations
   1.219 +which to the best of present knowledge, are not reducible to
   1.220 +logical relations within the context of classical thermodynamics
   1.221 +(and. whose demonstration in the wider context of mechanics,
   1.222 +electrodynamics, and quantum theory remains one of probability
   1.223 +rather than logical proof); from the standpoint of thermodynamics
   1.224 +these remain simply statements of empirical fact which must be
   1.225 +accepted as such without any deeper basis, but without which the
   1.226 +development of thermodynamics cannot proceed. Because of this
   1.227 +special status, these relations have become known as the
   1.228 +\ldquo{}laws\rdquo{}
   1.229 +of thermodynamics . The most fundamental one is a qualitative
   1.230 +rather than quantitative relation, the \ldquo{}zero'th law.\rdquo{}
   1.231 +
   1.232 +** Equilibrium; the \ldquo{}Zero‘th Law.\rdquo{}
   1.233 +
   1.234 + It is a common experience
   1.235 +that when objects are placed in contact with each other but
   1.236 +isolated from their surroundings, they may undergo observable
   1.237 +changes for a time as a result; one body may become warmer,
   1.238 +another cooler, the pressure of a gas or volume of a liquid may
   1.239 +change; stress or magnetization in a solid may change, etc. But
   1.240 +after a sufficient time, the observable macroscopic properties
   1.241 +settle down to a steady condition, after which no further changes
   1.242 +are seen unless there is a new intervention from the outside.
   1.243 +When this steady condition is reached, the experimentalist says
   1.244 +that the objects have reached a state of /equilibrium/ with each
   1.245 +other. Once again, more precise definitions of this term will
   1.246 +be needed eventually, but they require concepts not yet developed.
   1.247 +In any event, the criterion just stated is almost the only one
   1.248 +used in actual laboratory practice to decide when equilibrium
   1.249 +has been reached.
   1.250 +
   1.251 +
   1.252 +A particular case of equilibrium is encountered when we
   1.253 +place a thermometer in contact with another body. The reading
   1.254 +\(t\) of the thermometer may vary at first, but eventually it reach es
   1.255 +a steady value. Now the number \(t\) read by a thermometer is always.
   1.256 +by definition, the empirical temperature /of the thermometer/ (more
   1.257 +precisely, of the sensitive element of the thermometer). When
   1.258 +this number is constant in time, we say that the thermometer is
   1.259 +in /thermal equilibrium/ with its surroundings; and we then extend
   1.260 +the notion of temperature, calling the steady value \(t\) also the
   1.261 +/temperature of the surroundings/.
   1.262 +
   1.263 +We have repeated these elementary facts, well known to every
   1.264 +child, in order to emphasize this point: Thermodynamics can be
   1.265 +a theory /only/ of states of equilibrium, because the very 
   1.266 +procedure by which the temperature of a system is defined by 
   1.267 +operational means, already presupposes the attainment of 
   1.268 +equilibrium. Strictly speaking, therefore, classical 
   1.269 +thermodynamics does not even contain the concept of a 
   1.270 +\ldquo{}time-varying temperature.\rdquo{}
   1.271 +
   1.272 +Of course, to recognize this limitation on conventional
   1.273 +thermodynamics (best emphasized by calling it instead, 
   1.274 +thermostatics) in no way rules out the possibility of 
   1.275 +generalizing the notion of temperature to nonequilibrium states. 
   1.276 +Indeed, it is clear that one could define any number of 
   1.277 +time-dependent quantities all of which reduce, in the special 
   1.278 +case of equilibrium, to the temperature as defined above. 
   1.279 +Historically, attempts to do this even antedated the discovery 
   1.280 +of the laws of thermodynamics, as is demonstrated by 
   1.281 +\ldquo{}Newton's law of cooling.\rdquo{} Therefore, the
   1.282 +question is not whether generalization is /possible/, but only
   1.283 +whether it is in any way /useful/; i.e., does the temperature so
   1.284 +generalized have any connection with other physical properties
   1.285 +of our system, so that it could help us to predict other things?
   1.286 +However, to raise such questions takes us far beyond the
   1.287 +domain of thermostatics; and the general laws of nonequilibrium
   1.288 +behavior are so much more complicated that it would be virtually
   1.289 +hopeless to try to unravel them by empirical means alone. For
   1.290 +example, even if two different kinds of thermometer are calibrated
   1.291 +so that they agree with each other in equilibrium situations,
   1.292 +they will not agree in general about the momentary value a
   1.293 +\ldquo{}time-varying temperature.\rdquo{} To make any real 
   1.294 +progress in this area, we have to supplement empirical observation by the guidance
   1.295 +of a rather hiqhly-developed theory. The notion of a 
   1.296 +time-dependent temperature is far from simple conceptually, and we
   1.297 +will find that nothing very helpful can be said about this until
   1.298 +the full mathematical apparatus of nonequilibrium statistical
   1.299 +mechanics has been developed.
   1.300 +
   1.301 +Suppose now that two bodies have the same temperature; i.e.,
   1.302 +a given thermometer reads the same steady value when in contact
   1.303 +with either. In order that the statement, \ldquo{}two bodies have the
   1.304 +same temperature\rdquo{} shall describe a physi cal property of the bodies,
   1.305 +and not merely an accidental circumstance due to our having used
   1.306 +a particular kind of thermometer, it is necessary that /all/ 
   1.307 +thermometers agree in assigning equal temperatures to them if 
   1.308 +/any/ thermometer does . Only experiment is competent to determine
   1.309 +whether this universality property is true. Unfortunately, the
   1.310 +writer must confess that he is unable to cite any definite
   1.311 +experiment in which this point was subjected to a careful test.
   1.312 +That equality of temperatures has this absolute meaning, has
   1.313 +evidently been taken for granted so much that (like absolute
   1.314 +sirnultaneity in pre-relativity physics) most of us are not even
   1.315 +consciously aware that we make such an assumption in 
   1.316 +thermodynamics. However, for the present we can only take it as a familiar
   1.317 +empirical fact that this condition does hold, not because we can
   1.318 +cite positive evidence for it, but because of the absence of
   1.319 +negative evidence against it; i.e., we think that, if an 
   1.320 +exception had ever been found, this would have created a sensation in
   1.321 +physics, and we should have heard of it.
   1.322 +
   1.323 +We now ask: when two bodies are at the same temperature,
   1.324 +are they then in thermal equilibrium with each other? Again,
   1.325 +only experiment is competent to answer this; the general 
   1.326 +conclusion, again supported more by absence of negative evidence
   1.327 +than by specific positive evidence, is that the relation of
   1.328 +equilibrium has this property: 
   1.329 +#+begin_quote 
   1.330 +/Two bodies in thermal equilibrium
   1.331 +with a third body, are thermal equilibrium with each other./
   1.332 +#+end_quote
   1.333 +
   1.334 +This empirical fact is usually called the \ldquo{}zero'th law of 
   1.335 +thermodynamics.\rdquo{} Since nothing prevents us from regarding a 
   1.336 +thermometer as the \ldquo{}third body\rdquo{} in the above statement, 
   1.337 +it appears that we may also state the zero'th law as: 
   1.338 +#+begin_quote
   1.339 +/Two bodies are in thermal equilibrium with each other when they are
   1.340 +at the same temperature./
   1.341 +#+end_quote
   1.342 +Although from the preceding discussion it might appear that
   1.343 +these two statements of the zero'th law are entirely equivalent
   1.344 +(and we certainly have no empirical evidence against either), it
   1.345 +is interesting to note that there are theoretical reasons, arising
   1.346 +from General Relativity, indicating that while the first 
   1.347 +statement may be universally valid, the second is not. When we 
   1.348 +consider equilibrium in a gravitational field, the verification
   1.349 +that two bodies have equal temperatures may require transport
   1.350 +of the thermometer through a gravitational potential difference;
   1.351 +and this introduces a new element into the discussion. We will
   1.352 +consider this in more detail in a later Chapter, and show that
   1.353 +according to General Relativity, equilibrium in a large system
   1.354 +requires, not that the temperature be uniform at all points, but
   1.355 +rather that a particular function of temperature and gravitational 
   1.356 +potential be constant (the function is \(T\cdot \exp{(\Phi/c^2})\), where
   1.357 +\(T\) is the Kelvin temperature to be defined later, and \(\Phi\) is the
   1.358 +gravitational potential).
   1.359 +
   1.360 +Of course, this effect is so small that ordinary terrestrial
   1.361 +experiments would need to have a precision many orders of 
   1.362 +magnitude beyond that presently possible, before one could hope even
   1.363 +to detect it; and needless to say, it has played no role in the
   1.364 +development of thermodynamics. For present purposes, therefore,
   1.365 +we need not distinguish between the two above statements of the
   1.366 +zero'th law, and we take it as a basic empirical fact that a
   1.367 +uniform temperature at all points of a system is an essential
   1.368 +condition for equilibrium. It is an important part of our 
   1.369 +ivestigation to determine whether there are other essential 
   1.370 +conditions as well. In fact, as we will find, there are many 
   1.371 +different kinds of equilibrium; and failure to distinguish between
   1.372 +them can be a prolific source of paradoxes.
   1.373 +
   1.374 +** Equation of State
   1.375 +Another important reproducible connection is found when 
   1.376 +we consider a thermodynamic system defined by
   1.377 +three parameters; in addition to the temperature we choose a
   1.378 +\ldquo{}displacement\rdquo{} and a conjugate \ldquo{}force.\rdquo{} 
   1.379 +Subject to some qualifications given below, we find experimentally 
   1.380 +that these parameters are not independent, but are subject to a constraint.
   1.381 +For example, we cannot vary the equilibrium pressure, volume,
   1.382 +and temperature of a given mass of gas independently; it is found
   1.383 +that a given pressure and volume can be realized only at one
   1.384 +particular temperature, that the gas will assume a given tempera~
   1.385 +ture and volume only at one particular pressure, etc. Similarly,
   1.386 +a stretched wire can be made to have arbitrarily assigned tension
   1.387 +and elongation only if its temperature is suitably chosen, a
   1.388 +dielectric will assume a state of given temperature and 
   1.389 +polarization at only one value of the electric field, etc.
   1.390 +These simplest nontrivial thermodynamic systems (three 
   1.391 +parameters with one constraint) are said to possess two 
   1.392 +/degrees of freedom/; for the range of possible equilibrium states is defined
   1.393 +by specifying any two of the variables arbitrarily, whereupon the
   1.394 +third, and all others we may introduce, are determined. 
   1.395 +Mathematically, this is expressed by the existence of a functional
   1.396 +relationship of the form[fn::Edit: The set of solutions to an equation
   1.397 +like /f(X,x,t)=/ const. is called a /level set/. Here, Jaynes is
   1.398 +saying that the quantities /X/, /x/, and /t/ follow a \ldquo{}functional
   1.399 +rule\rdquo{}, so the set of physically allowed combinations of /X/,
   1.400 +/x/, and /t/ in equilibrium states can be
   1.401 +expressed as the level set of a function. 
   1.402 +
   1.403 +But not every function expresses a constraint relation; for some
   1.404 +functions, you can specify two of the variables, and the third will
   1.405 +still be undetermined. (For example, if f=X^2+x^2+t^2-3,
   1.406 +the level set /f(X,x,t)=0/ is a sphere, and specifying /x=1/, /t=1/
   1.407 +leaves you with two potential possibilities for /X/ =\pm 1.)
   1.408 +
   1.409 +A function like /f/ has to possess one more propery in order to
   1.410 +express a constraint relationship: it must be monotonic in
   1.411 +each of its variables /X/, /x/, and /t/.
   1.412 +#the partial derivatives of /f/ exist for every allowed combination of
   1.413 +#inputs /x/, /X/, and /t/. 
   1.414 +In other words, the level set has to pass a sort of 
   1.415 +\ldquo{}vertical line test\rdquo{} for each of its variables.]
   1.416 +
   1.417 +#Edit Here, Jaynes
   1.418 +#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
   1.419 +#[[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.
   1.420 +#In order to specify
   1.421 +
   1.422 +\begin{equation}
   1.423 +f(X,x,t) = O
   1.424 +\end{equation}
   1.425 +
   1.426 +where $X$ is a generalized force (pressure, tension, electric or
   1.427 +magnetic field, etc.), $x$ is the corresponding generalized 
   1.428 +displacement (volume, elongation, electric or magnetic polarization,
   1.429 +etc.), and $t$ is the empirical temperature. Equation (1) is
   1.430 +called /the equation of state/.
   1.431 +
   1.432 +At the risk of belaboring it, we emphasize once again that
   1.433 +all of this applies only for a system in equilibrium; for 
   1.434 +otherwise not only.the temperature, but also some or all of the other
   1.435 +variables may not be definable. For example, no unique pressure
   1.436 +can be assigned to a gas which has just suffered a sudden change
   1.437 +in volume, until the generated sound waves have died out.
   1.438 +
   1.439 +Independently of its functional form, the mere fact of the
   1.440 +/existence/ of an equation of state has certain experimental 
   1.441 +consequences. For example, suppose that in experiments on oxygen
   1.442 +gas, in which we control the temperature and pressure 
   1.443 +independently, we have found that the isothermal compressibility $K$
   1.444 +varies with temperature, and the thermal expansion coefficient
   1.445 +\alpha varies with pressure $P$, so that within the accuracy of the data,
   1.446 +
   1.447 +\begin{equation}
   1.448 +\frac{\partial K}{\partial t} = - \frac{\partial \alpha}{\partial P}
   1.449 +\end{equation}
   1.450 +
   1.451 +Is this a particular property of oxygen; or is there reason to
   1.452 +believe that it holds also for other substances? Does it depend
   1.453 +on our particular choice of a temperature scale?
   1.454 +
   1.455 +In this case, the answer is found at once; for the definitions of $K$,
   1.456 +\alpha are
   1.457 +
   1.458 +\begin{equation}
   1.459 +K = -\frac{1}{V}\frac{\partial V}{\partial P},\qquad
   1.460 +\alpha=\frac{1}{V}\frac{\partial V}{\partial t}
   1.461 +\end{equation}
   1.462 +
   1.463 +which is simply a mathematical expression of the fact that the
   1.464 +volume $V$ is a definite function of $P$ and $t$; i.e., it depends
   1.465 +only
   1.466 +on their present values, and not how those values were attained.
   1.467 +In particular, $V$ does not depend on the direction in the \((P, t)\)
   1.468 +plane through which the present values were approached; or, as we 
   1.469 +usually say it, \(dV\) is an /exact differential/.
   1.470 +
   1.471 +Therefore, although at first glance the relation (2) appears
   1.472 +nontrivial and far from obvious, a trivial mathematical analysis
   1.473 +convinces us that it must hold regardless of our particular
   1.474 +temperature scale, and that it is true not only of oxygen; it must
   1.475 +hold for any substance, or mixture of substances, which possesses a
   1.476 +definite, reproducible equation of state \(f(P,V,t)=0\).
   1.477 +
   1.478 +But this understanding also enables us to predict situations in which
   1.479 +(2) will /not/ hold. Equation (2), as we have just learned, expresses
   1.480 +the fact that an equation of state exists involving only the three
   1.481 +variables \((P,V,t)\). Now suppose we try to apply it to a liquid such
   1.482 +as nitrobenzene. The nitrobenzene molecule has a large electric dipole
   1.483 +moment; and so application of an electric field (as in the
   1.484 +[[http://en.wikipedia.org/wiki/Kerr_effect][electro-optical Kerr cell]]) causes an alignment of molecules which, as
   1.485 +accurate measurements will verify, changes the pressure at a given
   1.486 +temperature and volume. Therefore, there can no longer exist any
   1.487 +unique equation of state involving \((P, V, t)\) only; with
   1.488 +sufficiently accurate measurements, nitrobenzene must be regarded as a
   1.489 +thermodynamic system with at least three degrees of freedom, and the
   1.490 +general equation of state must have at least a complicated a form as
   1.491 +\(f(P,V,t,E) = 0\).
   1.492 +
   1.493 +But if we introduce a varying electric field $E$ into the discussion,
   1.494 +the resulting varying electric polarization $M$ also becomes a new
   1.495 +thermodynamic variable capable of being measured. Experimentally, it
   1.496 +is easiest to control temperature, pressure, and electric field
   1.497 +independently, and of course we find that both the volume and
   1.498 +polarization are then determined; i.e., there must exist functional
   1.499 +relations of the form \(V = V(P,t,E)\), \(M = M(P,t,E)\), or in more
   1.500 +symmetrical form
   1.501 +
   1.502 +\begin{equation}
   1.503 +f(V,P,t,E) = 0 \qquad g(M,P,t,E)=0.
   1.504 +\end{equation}
   1.505 +
   1.506 +In other words, if we regard nitrobenzene as a thermodynamic system of
   1.507 +three degrees of freedom (i.e., having specified three parameters
   1.508 +arbitrarily, all others are then determined), it must possess two
   1.509 +independent equations of state.
   1.510 +
   1.511 +Similarly, a thermodynamic system with four degrees of freedom,
   1.512 +defined by the termperature and three pairs of conjugate forces and
   1.513 +displacements, will have three independent equations of state, etc.
   1.514 +
   1.515 +Now, returning to our original question, if nitrobenzene possesses
   1.516 +this extra electrical degree of freedom, under what circumstances do
   1.517 +we exprect to find a reproducible equation of state involving
   1.518 +\((p,V,t)\) only? Evidently, if $E$ is held constant, then the first
   1.519 +of equations (1-5) becomes such an equation of state, involving $E$ as
   1.520 +a fixed parameter; we would find many different equations of state of
   1.521 +the form \(f(P,V,t) = 0\) with a different function $f$ for each
   1.522 +different value of the electric field. Likewise, if \(M\) is held
   1.523 +constant, we can eliminate \(E\) between equations (1-5) and find a
   1.524 +relation \(h(P,V,t,M)=0\), which is an equation of state for
   1.525 +\((P,V,t)\) containing \(M\) as a fixed parameter.
   1.526 +
   1.527 +More generally, if an electrical constraint is imposed on the system
   1.528 +(for example, by connecting an external charged capacitor to the
   1.529 +electrodes) so that \(M\) is determined by \(E\); i.e., there is a
   1.530 +functional relation of the form 
   1.531 +
   1.532 +\begin{equation}
   1.533 +g(M,E) = \text{const.}
   1.534 +\end{equation}
   1.535 +
   1.536 +then (1-5) and (1-6) constitute three simultaneous equations, from
   1.537 +which both \(E\) and \(M\) may be eliminated mathematically, leading
   1.538 +to a relation of the form \(h(P,V,t;q)=0\), which is an equation of
   1.539 +state for \((P,V,t)\) involving the fixed parameter \(q\).
   1.540 +
   1.541 +We see, then, that as long as a fixed constraint of the form (1-6) is
   1.542 +imposed on the electrical degree of freedom, we can still observe a
   1.543 +reproducible equation of state for nitrobenzene, considered as a
   1.544 +thermodynamic system of only two degrees of freedom. If, however, this
   1.545 +electrical constraint is removed, so that as we vary $P$ and $t$, the
   1.546 +values of $E$ and $M$ vary in an uncontrolled way over a
   1.547 +/two-dimensional/ region of the \((E, M)\) plane, then we will find no
   1.548 +definite equation of state involving only \((P,V,t)\).
   1.549 +
   1.550 +This may be stated more colloqually as follows: even though a system
   1.551 +has three degrees of freedom, we can still consider only the variables
   1.552 +belonging to two of them, and we will find a definite equation of
   1.553 +state, /provided/ that in the course of the experiments, the unused
   1.554 +degree of freedom is not \ldquo{}tampered with\rdquo{} in an
   1.555 +uncontrolled way.
   1.556 +
   1.557 +We have already emphasized that any physical system corresponds to
   1.558 +many different thermodynamic systems, depending on which variables we
   1.559 +choose to control and measure. In fact, it is easy to see that any
   1.560 +physical system has, for all practical purposes, an /arbitrarily
   1.561 +large/ number of  degrees of freedom. In the case of nitrobenzene, for
   1.562 +example, we may impose any variety of nonuniform electric fields on
   1.563 +our sample. Suppose we place $(n+1)$   

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