annotate org/stat-mech.org @ 0:26acdaf2e8c7

beginit begins.
author Dylan Holmes <ocsenave@gmail.com>
date Sat, 28 Apr 2012 19:32:50 -0500
parents
children 4da2176e4890
rev   line source
ocsenave@0 1 #+TITLE: Statistical Mechanics
ocsenave@0 2 #+AUTHOR: E.T. Jaynes; edited by Dylan Holmes
ocsenave@0 3 #+EMAIL: rlm@mit.edu
ocsenave@0 4 #+KEYWORDS: statistical mechanics, thermostatics, thermodynamics, temperature, paradoxes, Jaynes
ocsenave@0 5 #+SETUPFILE: ../../aurellem/org/setup.org
ocsenave@0 6 #+INCLUDE: ../../aurellem/org/level-0.org
ocsenave@0 7 #+MATHJAX: align:"left" mathml:t path:"http://www.aurellem.org/MathJax/MathJax.js"
ocsenave@0 8
ocsenave@0 9 # "extensions/eqn-number.js"
ocsenave@0 10
ocsenave@0 11 #+begin_quote
ocsenave@0 12 *Note:* The following is a typeset version of
ocsenave@0 13 [[../sources/stat.mech.1.pdf][this unpublished book draft]], written by [[http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E.T. Jaynes]].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
ocsenave@0 14 minor changes, e.g. to correct typographical errors, add references, or format equations. The
ocsenave@0 15 content itself is intact. --- Dylan
ocsenave@0 16 #+end_quote
ocsenave@0 17
ocsenave@0 18 * Development of Thermodynamics
ocsenave@0 19 Our first intuitive, or \ldquo{}subjective\rdquo{} notions of temperature
ocsenave@0 20 arise from the sensations of warmth and cold associated with our
ocsenave@0 21 sense of touch . Yet science has been able to convert this qualitative
ocsenave@0 22 sensation into an accurately defined quantitative notion,
ocsenave@0 23 which can be applied far beyond the range of our direct experience.
ocsenave@0 24 Today an experimentalist will report confidently that his
ocsenave@0 25 spin system was at a temperature of 2.51 degrees Kelvin; and a
ocsenave@0 26 theoretician will report with almost as much confidence that the
ocsenave@0 27 temperature at the center of the sun is about \(2 \times 10^7\) degrees
ocsenave@0 28 Kelvin.
ocsenave@0 29
ocsenave@0 30 The /fact/ that this has proved possible, and the main technical
ocsenave@0 31 ideas involved, are assumed already known to the reader;
ocsenave@0 32 and we are not concerned here with repeating standard material
ocsenave@0 33 already available in a dozen other textbooks . However
ocsenave@0 34 thermodynamics, in spite of its great successes, firmly established
ocsenave@0 35 for over a century, has also produced a great deal of confusion
ocsenave@0 36 and a long list of \ldquo{}paradoxes\rdquo{} centering mostly
ocsenave@0 37 around the second law and the nature of irreversibility.
ocsenave@0 38 For this reason and others noted below, we want to dwell here at
ocsenave@0 39 some length on the /logic/ underlying the development of
ocsenave@0 40 thermodynamics . Our aim is to emphasize certain points which,
ocsenave@0 41 in the writer's opinion, are essential for clearing up the
ocsenave@0 42 confusion and resolving the paradoxes; but which are not
ocsenave@0 43 sufficiently ernphasized---and indeed in many cases are
ocsenave@0 44 totally ignored---in other textbooks.
ocsenave@0 45
ocsenave@0 46 This attention to logic
ocsenave@0 47 would not be particularly needed if we regarded classical
ocsenave@0 48 thermodynamics (or, as it is becoming called increasingly,
ocsenave@0 49 /thermostatics/) as a closed subject, in which the fundamentals
ocsenave@0 50 are already completely established, and there is
ocsenave@0 51 nothing more to be learned about them. A person who believes
ocsenave@0 52 this will probably prefer a pure axiomatic approach, in which
ocsenave@0 53 the basic laws are simply stated as arbitrary axioms, without
ocsenave@0 54 any attempt to present the evidence for them; and one proceeds
ocsenave@0 55 directly to working out their consequences.
ocsenave@0 56 However, we take the attitude here that thermostatics, for
ocsenave@0 57 all its venerable age, is very far from being a closed subject,
ocsenave@0 58 we still have a great deal to learn about such matters as the
ocsenave@0 59 most general definitions of equilibrium and reversibility, the
ocsenave@0 60 exact range of validity of various statements of the second and
ocsenave@0 61 third laws, the necessary and sufficient conditions for
ocsenave@0 62 applicability of thermodynamics to special cases such as
ocsenave@0 63 spin systems, and how thermodynamics can be applied to such
ocsenave@0 64 systems as putty or polyethylene, which deform under force,
ocsenave@0 65 but retain a \ldquo{}memory\rdquo{} of their past deformations.
ocsenave@0 66 Is it possible to apply thermodynamics to a system such as a vibrating quartz crystal? We can by
ocsenave@0 67 no means rule out the possibility that still more laws of
ocsenave@0 68 thermodynamics exist, as yet undiscovered, which would be
ocsenave@0 69 useful in such applications.
ocsenave@0 70
ocsenave@0 71
ocsenave@0 72 It is only by careful examination of the logic by which
ocsenave@0 73 present thermodynamics was created, asking exactly how much of
ocsenave@0 74 it is mathematical theorems, how much is deducible from the laws
ocsenave@0 75 of mechanics and electrodynamics, and how much rests only on
ocsenave@0 76 empirical evidence, how compelling is present evidence for the
ocsenave@0 77 accuracy and range of validity of its laws; in other words,
ocsenave@0 78 exactly where are the boundaries of present knowledge, that we
ocsenave@0 79 can hope to uncover new things. Clearly, much research is still
ocsenave@0 80 needed in this field, and we shall be able to accomplish only a
ocsenave@0 81 small part of this program in the present review.
ocsenave@0 82
ocsenave@0 83
ocsenave@0 84 It will develop that there is an astonishingly close analogy
ocsenave@0 85 with the logic underlying statistical theory in general, where
ocsenave@0 86 again a qualitative feeling that we all have (for the degrees of
ocsenave@0 87 plausibility of various unproved and undisproved assertions) must
ocsenave@0 88 be convertefi into a precisely defined quantitative concept
ocsenave@0 89 (probability). Our later development of probability theory in
ocsenave@0 90 Chapter 6,7 will be, to a considerable degree, a paraphrase
ocsenave@0 91 of our present review of the logic underlying classical
ocsenave@0 92 thermodynamics.
ocsenave@0 93
ocsenave@0 94 ** The Primitive Thermometer.
ocsenave@0 95
ocsenave@0 96 The earliest stages of our
ocsenave@0 97 story are necessarily speculative, since they took place long
ocsenave@0 98 before the beginnings of recorded history. But we can hardly
ocsenave@0 99 doubt that primitive man learned quickly that objects exposed
ocsenave@0 100 to the sun‘s rays or placed near a fire felt different from
ocsenave@0 101 those in the shade away from fires; and the same difference was
ocsenave@0 102 noted between animal bodies and inanimate objects.
ocsenave@0 103
ocsenave@0 104
ocsenave@0 105 As soon as it was noted that changes in this feeling of
ocsenave@0 106 warmth were correlated with other observable changes in the
ocsenave@0 107 behavior of objects, such as the boiling and freezing of water,
ocsenave@0 108 cooking of meat, melting of fat and wax, etc., the notion of
ocsenave@0 109 warmth took its first step away from the purely subjective
ocsenave@0 110 toward an objective, physical notion capable of being studied
ocsenave@0 111 scientifically.
ocsenave@0 112
ocsenave@0 113 One of the most striking manifestations of warmth (but far
ocsenave@0 114 from the earliest discovered) is the almost universal expansion
ocsenave@0 115 of gases, liquids, and solids when heated . This property has
ocsenave@0 116 proved to be a convenient one with which to reduce the notion
ocsenave@0 117 of warmth to something entirely objective. The invention of the
ocsenave@0 118 /thermometer/, in which expansion of a mercury column, or a gas,
ocsenave@0 119 or the bending of a bimetallic strip, etc. is read off on a
ocsenave@0 120 suitable scale, thereby giving us a /number/ with which to work,
ocsenave@0 121 was a necessary prelude to even the crudest study of the physical
ocsenave@0 122 nature of heat. To the best of our knowledge, although the
ocsenave@0 123 necessary technology to do this had been available for at least
ocsenave@0 124 3,000 years, the first person to carry it out in practice was
ocsenave@0 125 Galileo, in 1592.
ocsenave@0 126
ocsenave@0 127 Later on we will give more precise definitions of the term
ocsenave@0 128 \ldquo{}thermometer.\rdquo{} But at the present stage we
ocsenave@0 129 are not in a position to do so (as Galileo was not), because
ocsenave@0 130 the very concepts needed have not yet been developed;
ocsenave@0 131 more precise definitions can be
ocsenave@0 132 given only after our study has revealed the need for them. In
ocsenave@0 133 deed, our final definition can be given only after the full
ocsenave@0 134 mathematical formalism of statistical mechanics is at hand.
ocsenave@0 135
ocsenave@0 136 Once a thermometer has been constructed, and the scale
ocsenave@0 137 marked off in a quite arbitrary way (although we will suppose
ocsenave@0 138 that the scale is at least monotonic: i.e., greater warmth always
ocsenave@0 139 corresponds to a greater number), we are ready to begin scien
ocsenave@0 140 tific experiments in thermodynamics. The number read eff from
ocsenave@0 141 any such instrument is called the /empirical temperature/, and we
ocsenave@0 142 denote it by \(t\). Since the exact calibration of the thermometer
ocsenave@0 143 is not specified), any monotonic increasing function
ocsenave@0 144 \(t‘ = f(t)\) provides an equally good temperature scale for the
ocsenave@0 145 present.
ocsenave@0 146
ocsenave@0 147
ocsenave@0 148 ** Thermodynamic Systems.
ocsenave@0 149
ocsenave@0 150 The \ldquo{}thermodynamic systems\rdquo{} which
ocsenave@0 151 are the objects of our study may be, physically, almost any
ocsenave@0 152 collections of objects. The traditional simplest system with
ocsenave@0 153 which to begin a study of thermodynamics is a volume of gas.
ocsenave@0 154 We shall, however, be concerned from the start also with such
ocsenave@0 155 things as a stretched wire or membrane, an electric cell, a
ocsenave@0 156 polarized dielectric, a paramagnetic body in a magnetic field, etc.
ocsenave@0 157
ocsenave@0 158 The /thermodynamic state/ of such a system is determined by
ocsenave@0 159 specifying (i.e., measuring) certain macrcoscopic physical
ocsenave@0 160 properties. Now, any real physical system has many millions of such
ocsenave@0 161 preperties; in order to have a usable theory we cannot require
ocsenave@0 162 that /all/ of them be specified. We see, therefore, that there
ocsenave@0 163 must be a clear distinction between the notions of
ocsenave@0 164 \ldquo{}thermodynamic system\rdquo{} and \ldquo{}physical
ocsenave@0 165 system.\rdquo{}
ocsenave@0 166 A given /physical/ system may correspond to many different
ocsenave@0 167 /thermodynamic systems/, depending
ocsenave@0 168 on which variables we choose to measure or control; and which
ocsenave@0 169 we decide to leave unmeasured and/or uncontrolled.
ocsenave@0 170
ocsenave@0 171
ocsenave@0 172 For example, our physical system might consist of a crystal
ocsenave@0 173 of sodium chloride. For one set of experiments we work with
ocsenave@0 174 temperature, volume, and pressure; and ignore its electrical
ocsenave@0 175 properties. For another set of experiments we work with
ocsenave@0 176 temperature, electric field, and electric polarization; and
ocsenave@0 177 ignore the varying stress and strain. The /physical/ system,
ocsenave@0 178 therefore, corresponds to two entirely different /thermodynamic/
ocsenave@0 179 systems. Exactly how much freedom, then, do we have in choosing
ocsenave@0 180 the variables which shall define the thermodynamic state of our
ocsenave@0 181 system? How many must we choose? What [criteria] determine when
ocsenave@0 182 we have made an adequate choice? These questions cannot be
ocsenave@0 183 answered until we say a little more about what we are trying to
ocsenave@0 184 accomplish by a thermodynamic theory. A mere collection of
ocsenave@0 185 recorded data about our system, as in the [[http://en.wikipedia.org/wiki/CRC_Handbook_of_Chemistry_and_Physics][/Handbook of Physics and
ocsenave@0 186 Chemistry/]], is a very useful thing, but it hardly constitutes
ocsenave@0 187 a theory. In order to construct anything deserving of such a
ocsenave@0 188 name, the primary requirement is that we can recognize some kind
ocsenave@0 189 of reproducible connection between the different properties con
ocsenave@0 190 sidered, so that information about some of them will enable us
ocsenave@0 191 to predict others. And of course, in order that our theory can
ocsenave@0 192 be called thermodynamics (and not some other area of physics),
ocsenave@0 193 it is necessary that the temperature be one of the quantities
ocsenave@0 194 involved in a nontrivial way.
ocsenave@0 195
ocsenave@0 196 The gist of these remarks is that the notion of
ocsenave@0 197 \ldquo{}thermodynamic system\rdquo{} is in part
ocsenave@0 198 an anthropomorphic one; it is for us to
ocsenave@0 199 say which set of variables shall be used. If two different
ocsenave@0 200 choices both lead to useful reproducible connections, it is quite
ocsenave@0 201 meaningless to say that one choice is any more \ldquo{}correct\rdquo{}
ocsenave@0 202 than the other. Recognition of this fact will prove crucial later in
ocsenave@0 203 avoiding certain ancient paradoxes.
ocsenave@0 204
ocsenave@0 205 At this stage we can determine only empirically which other
ocsenave@0 206 physical properties need to be introduced before reproducible
ocsenave@0 207 connections appear. Once any such connection is established, we
ocsenave@0 208 can analyze it with the hope of being able to (1) reduce it to a
ocsenave@0 209 /logical/ connection rather than an empirical one; and (2) extend
ocsenave@0 210 it to an hypothesis applying beyond the original data, which
ocsenave@0 211 enables us to predict further connections capable of being
ocsenave@0 212 tested by experiment. Examples of this will be given presently.
ocsenave@0 213
ocsenave@0 214
ocsenave@0 215 There will remain, however, a few reproducible relations
ocsenave@0 216 which to the best of present knowledge, are not reducible to
ocsenave@0 217 logical relations within the context of classical thermodynamics
ocsenave@0 218 (and. whose demonstration in the wider context of mechanics,
ocsenave@0 219 electrodynamics, and quantum theory remains one of probability
ocsenave@0 220 rather than logical proof); from the standpoint of thermodynamics
ocsenave@0 221 these remain simply statements of empirical fact which must be
ocsenave@0 222 accepted as such without any deeper basis, but without which the
ocsenave@0 223 development of thermodynamics cannot proceed. Because of this
ocsenave@0 224 special status, these relations have become known as the
ocsenave@0 225 \ldquo{}laws\rdquo{}
ocsenave@0 226 of thermodynamics . The most fundamental one is a qualitative
ocsenave@0 227 rather than quantitative relation, the \ldquo{}zero'th law.\rdquo{}
ocsenave@0 228
ocsenave@0 229 ** Equilibrium; the \ldquo{}Zero‘th Law.\rdquo{}
ocsenave@0 230
ocsenave@0 231 It is a common experience
ocsenave@0 232 that when objects are placed in contact with each other but
ocsenave@0 233 isolated from their surroundings, they may undergo observable
ocsenave@0 234 changes for a time as a result; one body may become warmer,
ocsenave@0 235 another cooler, the pressure of a gas or volume of a liquid may
ocsenave@0 236 change; stress or magnetization in a solid may change, etc. But
ocsenave@0 237 after a sufficient time, the observable macroscopic properties
ocsenave@0 238 settle down to a steady condition, after which no further changes
ocsenave@0 239 are seen unless there is a new intervention from the outside.
ocsenave@0 240 When this steady condition is reached, the experimentalist says
ocsenave@0 241 that the objects have reached a state of /equilibrium/ with each
ocsenave@0 242 other. Once again, more precise definitions of this term will
ocsenave@0 243 be needed eventually, but they require concepts not yet developed.
ocsenave@0 244 In any event, the criterion just stated is almost the only one
ocsenave@0 245 used in actual laboratory practice to decide when equilibrium
ocsenave@0 246 has been reached.
ocsenave@0 247
ocsenave@0 248
ocsenave@0 249 A particular case of equilibrium is encountered when we
ocsenave@0 250 place a thermometer in contact with another body. The reading
ocsenave@0 251 \(t\) of the thermometer may vary at first, but eventually it reach es
ocsenave@0 252 a steady value. Now the number \(t\) read by a thermometer is always.
ocsenave@0 253 by definition, the empirical temperature /of the thermometer/ (more
ocsenave@0 254 precisely, of the sensitive element of the thermometer). When
ocsenave@0 255 this number is constant in time, we say that the thermometer is
ocsenave@0 256 in /thermal equilibrium/ with its surroundings; and we then extend
ocsenave@0 257 the notion of temperature, calling the steady value \(t\) also the
ocsenave@0 258 /temperature of the surroundings/.
ocsenave@0 259
ocsenave@0 260 We have repeated these elementary facts, well known to every
ocsenave@0 261 child, in order to emphasize this point: Thermodynamics can be
ocsenave@0 262 a theory /only/ of states of equilibrium, because the very
ocsenave@0 263 procedure by which the temperature of a system is defined by
ocsenave@0 264 operational means, already presupposes the attainment of
ocsenave@0 265 equilibrium. Strictly speaking, therefore, classical
ocsenave@0 266 thermodynamics does not even contain the concept of a
ocsenave@0 267 \ldquo{}time-varying temperature.\rdquo{}
ocsenave@0 268
ocsenave@0 269 Of course, to recognize this limitation on conventional
ocsenave@0 270 thermodynamics (best emphasized by calling it instead,
ocsenave@0 271 thermostatics) in no way rules out the possibility of
ocsenave@0 272 generalizing the notion of temperature to nonequilibrium states.
ocsenave@0 273 Indeed, it is clear that one could define any number of
ocsenave@0 274 time-dependent quantities all of which reduce, in the special
ocsenave@0 275 case of equilibrium, to the temperature as defined above.
ocsenave@0 276 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@0 301 same temperature\rdquo{} shall describe a physi cal 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@0 393 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@0 406 A function like /f/ has to possess one more propery in order to
ocsenave@0 407 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@0 426 etc.), and $t$ is the empirical temperature. Equation (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@0 468 Therefore, although at first glance the relation (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@0 476 (2) will /not/ hold. Equation (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@0 560 our sample. Suppose we place $(n+1)$