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author Dylan Holmes <ocsenave@gmail.com>
date Sat, 28 Apr 2012 19:32:50 -0500
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1 #+TITLE: Statistical Mechanics
2 #+AUTHOR: E.T. Jaynes; edited by Dylan Holmes
3 #+EMAIL: rlm@mit.edu
4 #+KEYWORDS: statistical mechanics, thermostatics, thermodynamics, temperature, paradoxes, Jaynes
5 #+SETUPFILE: ../../aurellem/org/setup.org
6 #+INCLUDE: ../../aurellem/org/level-0.org
7 #+MATHJAX: align:"left" mathml:t path:"http://www.aurellem.org/MathJax/MathJax.js"
9 # "extensions/eqn-number.js"
11 #+begin_quote
12 *Note:* The following is a typeset version of
13 [[../sources/stat.mech.1.pdf][this unpublished book draft]], written by [[http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E.T. Jaynes]].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
14 minor changes, e.g. to correct typographical errors, add references, or format equations. The
15 content itself is intact. --- Dylan
16 #+end_quote
18 * Development of Thermodynamics
19 Our first intuitive, or \ldquo{}subjective\rdquo{} notions of temperature
20 arise from the sensations of warmth and cold associated with our
21 sense of touch . Yet science has been able to convert this qualitative
22 sensation into an accurately defined quantitative notion,
23 which can be applied far beyond the range of our direct experience.
24 Today an experimentalist will report confidently that his
25 spin system was at a temperature of 2.51 degrees Kelvin; and a
26 theoretician will report with almost as much confidence that the
27 temperature at the center of the sun is about \(2 \times 10^7\) degrees
28 Kelvin.
30 The /fact/ that this has proved possible, and the main technical
31 ideas involved, are assumed already known to the reader;
32 and we are not concerned here with repeating standard material
33 already available in a dozen other textbooks . However
34 thermodynamics, in spite of its great successes, firmly established
35 for over a century, has also produced a great deal of confusion
36 and a long list of \ldquo{}paradoxes\rdquo{} centering mostly
37 around the second law and the nature of irreversibility.
38 For this reason and others noted below, we want to dwell here at
39 some length on the /logic/ underlying the development of
40 thermodynamics . Our aim is to emphasize certain points which,
41 in the writer's opinion, are essential for clearing up the
42 confusion and resolving the paradoxes; but which are not
43 sufficiently ernphasized---and indeed in many cases are
44 totally ignored---in other textbooks.
46 This attention to logic
47 would not be particularly needed if we regarded classical
48 thermodynamics (or, as it is becoming called increasingly,
49 /thermostatics/) as a closed subject, in which the fundamentals
50 are already completely established, and there is
51 nothing more to be learned about them. A person who believes
52 this will probably prefer a pure axiomatic approach, in which
53 the basic laws are simply stated as arbitrary axioms, without
54 any attempt to present the evidence for them; and one proceeds
55 directly to working out their consequences.
56 However, we take the attitude here that thermostatics, for
57 all its venerable age, is very far from being a closed subject,
58 we still have a great deal to learn about such matters as the
59 most general definitions of equilibrium and reversibility, the
60 exact range of validity of various statements of the second and
61 third laws, the necessary and sufficient conditions for
62 applicability of thermodynamics to special cases such as
63 spin systems, and how thermodynamics can be applied to such
64 systems as putty or polyethylene, which deform under force,
65 but retain a \ldquo{}memory\rdquo{} of their past deformations.
66 Is it possible to apply thermodynamics to a system such as a vibrating quartz crystal? We can by
67 no means rule out the possibility that still more laws of
68 thermodynamics exist, as yet undiscovered, which would be
69 useful in such applications.
72 It is only by careful examination of the logic by which
73 present thermodynamics was created, asking exactly how much of
74 it is mathematical theorems, how much is deducible from the laws
75 of mechanics and electrodynamics, and how much rests only on
76 empirical evidence, how compelling is present evidence for the
77 accuracy and range of validity of its laws; in other words,
78 exactly where are the boundaries of present knowledge, that we
79 can hope to uncover new things. Clearly, much research is still
80 needed in this field, and we shall be able to accomplish only a
81 small part of this program in the present review.
84 It will develop that there is an astonishingly close analogy
85 with the logic underlying statistical theory in general, where
86 again a qualitative feeling that we all have (for the degrees of
87 plausibility of various unproved and undisproved assertions) must
88 be convertefi into a precisely defined quantitative concept
89 (probability). Our later development of probability theory in
90 Chapter 6,7 will be, to a considerable degree, a paraphrase
91 of our present review of the logic underlying classical
92 thermodynamics.
94 ** The Primitive Thermometer.
96 The earliest stages of our
97 story are necessarily speculative, since they took place long
98 before the beginnings of recorded history. But we can hardly
99 doubt that primitive man learned quickly that objects exposed
100 to the sun‘s rays or placed near a fire felt different from
101 those in the shade away from fires; and the same difference was
102 noted between animal bodies and inanimate objects.
105 As soon as it was noted that changes in this feeling of
106 warmth were correlated with other observable changes in the
107 behavior of objects, such as the boiling and freezing of water,
108 cooking of meat, melting of fat and wax, etc., the notion of
109 warmth took its first step away from the purely subjective
110 toward an objective, physical notion capable of being studied
111 scientifically.
113 One of the most striking manifestations of warmth (but far
114 from the earliest discovered) is the almost universal expansion
115 of gases, liquids, and solids when heated . This property has
116 proved to be a convenient one with which to reduce the notion
117 of warmth to something entirely objective. The invention of the
118 /thermometer/, in which expansion of a mercury column, or a gas,
119 or the bending of a bimetallic strip, etc. is read off on a
120 suitable scale, thereby giving us a /number/ with which to work,
121 was a necessary prelude to even the crudest study of the physical
122 nature of heat. To the best of our knowledge, although the
123 necessary technology to do this had been available for at least
124 3,000 years, the first person to carry it out in practice was
125 Galileo, in 1592.
127 Later on we will give more precise definitions of the term
128 \ldquo{}thermometer.\rdquo{} But at the present stage we
129 are not in a position to do so (as Galileo was not), because
130 the very concepts needed have not yet been developed;
131 more precise definitions can be
132 given only after our study has revealed the need for them. In
133 deed, our final definition can be given only after the full
134 mathematical formalism of statistical mechanics is at hand.
136 Once a thermometer has been constructed, and the scale
137 marked off in a quite arbitrary way (although we will suppose
138 that the scale is at least monotonic: i.e., greater warmth always
139 corresponds to a greater number), we are ready to begin scien
140 tific experiments in thermodynamics. The number read eff from
141 any such instrument is called the /empirical temperature/, and we
142 denote it by \(t\). Since the exact calibration of the thermometer
143 is not specified), any monotonic increasing function
144 \(t‘ = f(t)\) provides an equally good temperature scale for the
145 present.
148 ** Thermodynamic Systems.
150 The \ldquo{}thermodynamic systems\rdquo{} which
151 are the objects of our study may be, physically, almost any
152 collections of objects. The traditional simplest system with
153 which to begin a study of thermodynamics is a volume of gas.
154 We shall, however, be concerned from the start also with such
155 things as a stretched wire or membrane, an electric cell, a
156 polarized dielectric, a paramagnetic body in a magnetic field, etc.
158 The /thermodynamic state/ of such a system is determined by
159 specifying (i.e., measuring) certain macrcoscopic physical
160 properties. Now, any real physical system has many millions of such
161 preperties; in order to have a usable theory we cannot require
162 that /all/ of them be specified. We see, therefore, that there
163 must be a clear distinction between the notions of
164 \ldquo{}thermodynamic system\rdquo{} and \ldquo{}physical
165 system.\rdquo{}
166 A given /physical/ system may correspond to many different
167 /thermodynamic systems/, depending
168 on which variables we choose to measure or control; and which
169 we decide to leave unmeasured and/or uncontrolled.
172 For example, our physical system might consist of a crystal
173 of sodium chloride. For one set of experiments we work with
174 temperature, volume, and pressure; and ignore its electrical
175 properties. For another set of experiments we work with
176 temperature, electric field, and electric polarization; and
177 ignore the varying stress and strain. The /physical/ system,
178 therefore, corresponds to two entirely different /thermodynamic/
179 systems. Exactly how much freedom, then, do we have in choosing
180 the variables which shall define the thermodynamic state of our
181 system? How many must we choose? What [criteria] determine when
182 we have made an adequate choice? These questions cannot be
183 answered until we say a little more about what we are trying to
184 accomplish by a thermodynamic theory. A mere collection of
185 recorded data about our system, as in the [[http://en.wikipedia.org/wiki/CRC_Handbook_of_Chemistry_and_Physics][/Handbook of Physics and
186 Chemistry/]], is a very useful thing, but it hardly constitutes
187 a theory. In order to construct anything deserving of such a
188 name, the primary requirement is that we can recognize some kind
189 of reproducible connection between the different properties con
190 sidered, so that information about some of them will enable us
191 to predict others. And of course, in order that our theory can
192 be called thermodynamics (and not some other area of physics),
193 it is necessary that the temperature be one of the quantities
194 involved in a nontrivial way.
196 The gist of these remarks is that the notion of
197 \ldquo{}thermodynamic system\rdquo{} is in part
198 an anthropomorphic one; it is for us to
199 say which set of variables shall be used. If two different
200 choices both lead to useful reproducible connections, it is quite
201 meaningless to say that one choice is any more \ldquo{}correct\rdquo{}
202 than the other. Recognition of this fact will prove crucial later in
203 avoiding certain ancient paradoxes.
205 At this stage we can determine only empirically which other
206 physical properties need to be introduced before reproducible
207 connections appear. Once any such connection is established, we
208 can analyze it with the hope of being able to (1) reduce it to a
209 /logical/ connection rather than an empirical one; and (2) extend
210 it to an hypothesis applying beyond the original data, which
211 enables us to predict further connections capable of being
212 tested by experiment. Examples of this will be given presently.
215 There will remain, however, a few reproducible relations
216 which to the best of present knowledge, are not reducible to
217 logical relations within the context of classical thermodynamics
218 (and. whose demonstration in the wider context of mechanics,
219 electrodynamics, and quantum theory remains one of probability
220 rather than logical proof); from the standpoint of thermodynamics
221 these remain simply statements of empirical fact which must be
222 accepted as such without any deeper basis, but without which the
223 development of thermodynamics cannot proceed. Because of this
224 special status, these relations have become known as the
225 \ldquo{}laws\rdquo{}
226 of thermodynamics . The most fundamental one is a qualitative
227 rather than quantitative relation, the \ldquo{}zero'th law.\rdquo{}
229 ** Equilibrium; the \ldquo{}Zero‘th Law.\rdquo{}
231 It is a common experience
232 that when objects are placed in contact with each other but
233 isolated from their surroundings, they may undergo observable
234 changes for a time as a result; one body may become warmer,
235 another cooler, the pressure of a gas or volume of a liquid may
236 change; stress or magnetization in a solid may change, etc. But
237 after a sufficient time, the observable macroscopic properties
238 settle down to a steady condition, after which no further changes
239 are seen unless there is a new intervention from the outside.
240 When this steady condition is reached, the experimentalist says
241 that the objects have reached a state of /equilibrium/ with each
242 other. Once again, more precise definitions of this term will
243 be needed eventually, but they require concepts not yet developed.
244 In any event, the criterion just stated is almost the only one
245 used in actual laboratory practice to decide when equilibrium
246 has been reached.
249 A particular case of equilibrium is encountered when we
250 place a thermometer in contact with another body. The reading
251 \(t\) of the thermometer may vary at first, but eventually it reach es
252 a steady value. Now the number \(t\) read by a thermometer is always.
253 by definition, the empirical temperature /of the thermometer/ (more
254 precisely, of the sensitive element of the thermometer). When
255 this number is constant in time, we say that the thermometer is
256 in /thermal equilibrium/ with its surroundings; and we then extend
257 the notion of temperature, calling the steady value \(t\) also the
258 /temperature of the surroundings/.
260 We have repeated these elementary facts, well known to every
261 child, in order to emphasize this point: Thermodynamics can be
262 a theory /only/ of states of equilibrium, because the very
263 procedure by which the temperature of a system is defined by
264 operational means, already presupposes the attainment of
265 equilibrium. Strictly speaking, therefore, classical
266 thermodynamics does not even contain the concept of a
267 \ldquo{}time-varying temperature.\rdquo{}
269 Of course, to recognize this limitation on conventional
270 thermodynamics (best emphasized by calling it instead,
271 thermostatics) in no way rules out the possibility of
272 generalizing the notion of temperature to nonequilibrium states.
273 Indeed, it is clear that one could define any number of
274 time-dependent quantities all of which reduce, in the special
275 case of equilibrium, to the temperature as defined above.
276 Historically, attempts to do this even antedated the discovery
277 of the laws of thermodynamics, as is demonstrated by
278 \ldquo{}Newton's law of cooling.\rdquo{} Therefore, the
279 question is not whether generalization is /possible/, but only
280 whether it is in any way /useful/; i.e., does the temperature so
281 generalized have any connection with other physical properties
282 of our system, so that it could help us to predict other things?
283 However, to raise such questions takes us far beyond the
284 domain of thermostatics; and the general laws of nonequilibrium
285 behavior are so much more complicated that it would be virtually
286 hopeless to try to unravel them by empirical means alone. For
287 example, even if two different kinds of thermometer are calibrated
288 so that they agree with each other in equilibrium situations,
289 they will not agree in general about the momentary value a
290 \ldquo{}time-varying temperature.\rdquo{} To make any real
291 progress in this area, we have to supplement empirical observation by the guidance
292 of a rather hiqhly-developed theory. The notion of a
293 time-dependent temperature is far from simple conceptually, and we
294 will find that nothing very helpful can be said about this until
295 the full mathematical apparatus of nonequilibrium statistical
296 mechanics has been developed.
298 Suppose now that two bodies have the same temperature; i.e.,
299 a given thermometer reads the same steady value when in contact
300 with either. In order that the statement, \ldquo{}two bodies have the
301 same temperature\rdquo{} shall describe a physi cal property of the bodies,
302 and not merely an accidental circumstance due to our having used
303 a particular kind of thermometer, it is necessary that /all/
304 thermometers agree in assigning equal temperatures to them if
305 /any/ thermometer does . Only experiment is competent to determine
306 whether this universality property is true. Unfortunately, the
307 writer must confess that he is unable to cite any definite
308 experiment in which this point was subjected to a careful test.
309 That equality of temperatures has this absolute meaning, has
310 evidently been taken for granted so much that (like absolute
311 sirnultaneity in pre-relativity physics) most of us are not even
312 consciously aware that we make such an assumption in
313 thermodynamics. However, for the present we can only take it as a familiar
314 empirical fact that this condition does hold, not because we can
315 cite positive evidence for it, but because of the absence of
316 negative evidence against it; i.e., we think that, if an
317 exception had ever been found, this would have created a sensation in
318 physics, and we should have heard of it.
320 We now ask: when two bodies are at the same temperature,
321 are they then in thermal equilibrium with each other? Again,
322 only experiment is competent to answer this; the general
323 conclusion, again supported more by absence of negative evidence
324 than by specific positive evidence, is that the relation of
325 equilibrium has this property:
326 #+begin_quote
327 /Two bodies in thermal equilibrium
328 with a third body, are thermal equilibrium with each other./
329 #+end_quote
331 This empirical fact is usually called the \ldquo{}zero'th law of
332 thermodynamics.\rdquo{} Since nothing prevents us from regarding a
333 thermometer as the \ldquo{}third body\rdquo{} in the above statement,
334 it appears that we may also state the zero'th law as:
335 #+begin_quote
336 /Two bodies are in thermal equilibrium with each other when they are
337 at the same temperature./
338 #+end_quote
339 Although from the preceding discussion it might appear that
340 these two statements of the zero'th law are entirely equivalent
341 (and we certainly have no empirical evidence against either), it
342 is interesting to note that there are theoretical reasons, arising
343 from General Relativity, indicating that while the first
344 statement may be universally valid, the second is not. When we
345 consider equilibrium in a gravitational field, the verification
346 that two bodies have equal temperatures may require transport
347 of the thermometer through a gravitational potential difference;
348 and this introduces a new element into the discussion. We will
349 consider this in more detail in a later Chapter, and show that
350 according to General Relativity, equilibrium in a large system
351 requires, not that the temperature be uniform at all points, but
352 rather that a particular function of temperature and gravitational
353 potential be constant (the function is \(T\cdot \exp{(\Phi/c^2})\), where
354 \(T\) is the Kelvin temperature to be defined later, and \(\Phi\) is the
355 gravitational potential).
357 Of course, this effect is so small that ordinary terrestrial
358 experiments would need to have a precision many orders of
359 magnitude beyond that presently possible, before one could hope even
360 to detect it; and needless to say, it has played no role in the
361 development of thermodynamics. For present purposes, therefore,
362 we need not distinguish between the two above statements of the
363 zero'th law, and we take it as a basic empirical fact that a
364 uniform temperature at all points of a system is an essential
365 condition for equilibrium. It is an important part of our
366 ivestigation to determine whether there are other essential
367 conditions as well. In fact, as we will find, there are many
368 different kinds of equilibrium; and failure to distinguish between
369 them can be a prolific source of paradoxes.
371 ** Equation of State
372 Another important reproducible connection is found when
373 we consider a thermodynamic system defined by
374 three parameters; in addition to the temperature we choose a
375 \ldquo{}displacement\rdquo{} and a conjugate \ldquo{}force.\rdquo{}
376 Subject to some qualifications given below, we find experimentally
377 that these parameters are not independent, but are subject to a constraint.
378 For example, we cannot vary the equilibrium pressure, volume,
379 and temperature of a given mass of gas independently; it is found
380 that a given pressure and volume can be realized only at one
381 particular temperature, that the gas will assume a given tempera~
382 ture and volume only at one particular pressure, etc. Similarly,
383 a stretched wire can be made to have arbitrarily assigned tension
384 and elongation only if its temperature is suitably chosen, a
385 dielectric will assume a state of given temperature and
386 polarization at only one value of the electric field, etc.
387 These simplest nontrivial thermodynamic systems (three
388 parameters with one constraint) are said to possess two
389 /degrees of freedom/; for the range of possible equilibrium states is defined
390 by specifying any two of the variables arbitrarily, whereupon the
391 third, and all others we may introduce, are determined.
392 Mathematically, this is expressed by the existence of a functional
393 relationship of the form[fn::Edit: The set of solutions to an equation
394 like /f(X,x,t)=/ const. is called a /level set/. Here, Jaynes is
395 saying that the quantities /X/, /x/, and /t/ follow a \ldquo{}functional
396 rule\rdquo{}, so the set of physically allowed combinations of /X/,
397 /x/, and /t/ in equilibrium states can be
398 expressed as the level set of a function.
400 But not every function expresses a constraint relation; for some
401 functions, you can specify two of the variables, and the third will
402 still be undetermined. (For example, if f=X^2+x^2+t^2-3,
403 the level set /f(X,x,t)=0/ is a sphere, and specifying /x=1/, /t=1/
404 leaves you with two potential possibilities for /X/ =\pm 1.)
406 A function like /f/ has to possess one more propery in order to
407 express a constraint relationship: it must be monotonic in
408 each of its variables /X/, /x/, and /t/.
409 #the partial derivatives of /f/ exist for every allowed combination of
410 #inputs /x/, /X/, and /t/.
411 In other words, the level set has to pass a sort of
412 \ldquo{}vertical line test\rdquo{} for each of its variables.]
414 #Edit Here, Jaynes
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
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.
417 #In order to specify
419 \begin{equation}
420 f(X,x,t) = O
421 \end{equation}
423 where $X$ is a generalized force (pressure, tension, electric or
424 magnetic field, etc.), $x$ is the corresponding generalized
425 displacement (volume, elongation, electric or magnetic polarization,
426 etc.), and $t$ is the empirical temperature. Equation (1) is
427 called /the equation of state/.
429 At the risk of belaboring it, we emphasize once again that
430 all of this applies only for a system in equilibrium; for
431 otherwise not only.the temperature, but also some or all of the other
432 variables may not be definable. For example, no unique pressure
433 can be assigned to a gas which has just suffered a sudden change
434 in volume, until the generated sound waves have died out.
436 Independently of its functional form, the mere fact of the
437 /existence/ of an equation of state has certain experimental
438 consequences. For example, suppose that in experiments on oxygen
439 gas, in which we control the temperature and pressure
440 independently, we have found that the isothermal compressibility $K$
441 varies with temperature, and the thermal expansion coefficient
442 \alpha varies with pressure $P$, so that within the accuracy of the data,
444 \begin{equation}
445 \frac{\partial K}{\partial t} = - \frac{\partial \alpha}{\partial P}
446 \end{equation}
448 Is this a particular property of oxygen; or is there reason to
449 believe that it holds also for other substances? Does it depend
450 on our particular choice of a temperature scale?
452 In this case, the answer is found at once; for the definitions of $K$,
453 \alpha are
455 \begin{equation}
456 K = -\frac{1}{V}\frac{\partial V}{\partial P},\qquad
457 \alpha=\frac{1}{V}\frac{\partial V}{\partial t}
458 \end{equation}
460 which is simply a mathematical expression of the fact that the
461 volume $V$ is a definite function of $P$ and $t$; i.e., it depends
462 only
463 on their present values, and not how those values were attained.
464 In particular, $V$ does not depend on the direction in the \((P, t)\)
465 plane through which the present values were approached; or, as we
466 usually say it, \(dV\) is an /exact differential/.
468 Therefore, although at first glance the relation (2) appears
469 nontrivial and far from obvious, a trivial mathematical analysis
470 convinces us that it must hold regardless of our particular
471 temperature scale, and that it is true not only of oxygen; it must
472 hold for any substance, or mixture of substances, which possesses a
473 definite, reproducible equation of state \(f(P,V,t)=0\).
475 But this understanding also enables us to predict situations in which
476 (2) will /not/ hold. Equation (2), as we have just learned, expresses
477 the fact that an equation of state exists involving only the three
478 variables \((P,V,t)\). Now suppose we try to apply it to a liquid such
479 as nitrobenzene. The nitrobenzene molecule has a large electric dipole
480 moment; and so application of an electric field (as in the
481 [[http://en.wikipedia.org/wiki/Kerr_effect][electro-optical Kerr cell]]) causes an alignment of molecules which, as
482 accurate measurements will verify, changes the pressure at a given
483 temperature and volume. Therefore, there can no longer exist any
484 unique equation of state involving \((P, V, t)\) only; with
485 sufficiently accurate measurements, nitrobenzene must be regarded as a
486 thermodynamic system with at least three degrees of freedom, and the
487 general equation of state must have at least a complicated a form as
488 \(f(P,V,t,E) = 0\).
490 But if we introduce a varying electric field $E$ into the discussion,
491 the resulting varying electric polarization $M$ also becomes a new
492 thermodynamic variable capable of being measured. Experimentally, it
493 is easiest to control temperature, pressure, and electric field
494 independently, and of course we find that both the volume and
495 polarization are then determined; i.e., there must exist functional
496 relations of the form \(V = V(P,t,E)\), \(M = M(P,t,E)\), or in more
497 symmetrical form
499 \begin{equation}
500 f(V,P,t,E) = 0 \qquad g(M,P,t,E)=0.
501 \end{equation}
503 In other words, if we regard nitrobenzene as a thermodynamic system of
504 three degrees of freedom (i.e., having specified three parameters
505 arbitrarily, all others are then determined), it must possess two
506 independent equations of state.
508 Similarly, a thermodynamic system with four degrees of freedom,
509 defined by the termperature and three pairs of conjugate forces and
510 displacements, will have three independent equations of state, etc.
512 Now, returning to our original question, if nitrobenzene possesses
513 this extra electrical degree of freedom, under what circumstances do
514 we exprect to find a reproducible equation of state involving
515 \((p,V,t)\) only? Evidently, if $E$ is held constant, then the first
516 of equations (1-5) becomes such an equation of state, involving $E$ as
517 a fixed parameter; we would find many different equations of state of
518 the form \(f(P,V,t) = 0\) with a different function $f$ for each
519 different value of the electric field. Likewise, if \(M\) is held
520 constant, we can eliminate \(E\) between equations (1-5) and find a
521 relation \(h(P,V,t,M)=0\), which is an equation of state for
522 \((P,V,t)\) containing \(M\) as a fixed parameter.
524 More generally, if an electrical constraint is imposed on the system
525 (for example, by connecting an external charged capacitor to the
526 electrodes) so that \(M\) is determined by \(E\); i.e., there is a
527 functional relation of the form
529 \begin{equation}
530 g(M,E) = \text{const.}
531 \end{equation}
533 then (1-5) and (1-6) constitute three simultaneous equations, from
534 which both \(E\) and \(M\) may be eliminated mathematically, leading
535 to a relation of the form \(h(P,V,t;q)=0\), which is an equation of
536 state for \((P,V,t)\) involving the fixed parameter \(q\).
538 We see, then, that as long as a fixed constraint of the form (1-6) is
539 imposed on the electrical degree of freedom, we can still observe a
540 reproducible equation of state for nitrobenzene, considered as a
541 thermodynamic system of only two degrees of freedom. If, however, this
542 electrical constraint is removed, so that as we vary $P$ and $t$, the
543 values of $E$ and $M$ vary in an uncontrolled way over a
544 /two-dimensional/ region of the \((E, M)\) plane, then we will find no
545 definite equation of state involving only \((P,V,t)\).
547 This may be stated more colloqually as follows: even though a system
548 has three degrees of freedom, we can still consider only the variables
549 belonging to two of them, and we will find a definite equation of
550 state, /provided/ that in the course of the experiments, the unused
551 degree of freedom is not \ldquo{}tampered with\rdquo{} in an
552 uncontrolled way.
554 We have already emphasized that any physical system corresponds to
555 many different thermodynamic systems, depending on which variables we
556 choose to control and measure. In fact, it is easy to see that any
557 physical system has, for all practical purposes, an /arbitrarily
558 large/ number of degrees of freedom. In the case of nitrobenzene, for
559 example, we may impose any variety of nonuniform electric fields on
560 our sample. Suppose we place $(n+1)$