ocsenave@0: #+TITLE: Statistical Mechanics
ocsenave@0: #+AUTHOR:    E.T. Jaynes; edited by  Dylan Holmes
ocsenave@0: #+EMAIL:     rlm@mit.edu
ocsenave@0: #+KEYWORDS: statistical mechanics, thermostatics, thermodynamics, temperature, paradoxes, Jaynes
ocsenave@0: #+SETUPFILE: ../../aurellem/org/setup.org
ocsenave@0: #+INCLUDE:   ../../aurellem/org/level-0.org
ocsenave@0: #+MATHJAX: align:"left" mathml:t path:"http://www.aurellem.org/MathJax/MathJax.js"
ocsenave@0: 
ocsenave@0: # "extensions/eqn-number.js"
ocsenave@0: 
ocsenave@0: #+begin_quote
ocsenave@0: *Note:* The following is a typeset version of
ocsenave@0:  [[../sources/stat.mech.1.pdf][this unpublished book draft]], written by [[http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E.T. Jaynes]].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:  minor changes, e.g. to correct typographical errors, add references, or format equations. The
ocsenave@0:  content itself is intact. --- Dylan
ocsenave@0: #+end_quote
ocsenave@0: 
ocsenave@0: * Development of Thermodynamics
ocsenave@0: Our first intuitive, or \ldquo{}subjective\rdquo{} notions of temperature
ocsenave@0: arise from the sensations of warmth and cold associated with our
ocsenave@0: sense of touch . Yet science has been able to convert this qualitative 
ocsenave@0: sensation into an accurately defined quantitative notion,
ocsenave@0: which can be applied far beyond the range of our direct experience. 
ocsenave@0: Today an experimentalist will report confidently that his
ocsenave@0: spin system was at a temperature of 2.51 degrees Kelvin; and a
ocsenave@0: theoretician will report with almost as much confidence that the 
ocsenave@0: temperature at the center of the sun is about \(2 \times 10^7\) degrees
ocsenave@0: Kelvin.
ocsenave@0: 
ocsenave@0: The /fact/ that this has proved possible, and the main technical 
ocsenave@0: ideas involved, are assumed already known to the reader;
ocsenave@0: and we are not concerned here with repeating standard material
ocsenave@0: already available in a dozen other textbooks . However 
ocsenave@0: thermodynamics, in spite of its great successes, firmly established
ocsenave@0: for over a century, has also produced a great deal of confusion
ocsenave@0: and a long list of \ldquo{}paradoxes\rdquo{} centering mostly 
ocsenave@0: around the second law and the nature of irreversibility. 
ocsenave@0: For this reason and others noted below, we want to dwell here at 
ocsenave@0: some length on the /logic/ underlying the development of 
ocsenave@0: thermodynamics . Our aim is to emphasize certain points which, 
ocsenave@0: in the writer's opinion, are essential for clearing up the 
ocsenave@0: confusion and resolving the paradoxes; but which are not 
ocsenave@0: sufficiently ernphasized---and indeed in many cases are 
ocsenave@0: totally ignored---in other textbooks.
ocsenave@0: 
ocsenave@0: This attention to logic 
ocsenave@0: would not be particularly needed if we regarded classical 
ocsenave@0: thermodynamics (or, as it is becoming called increasingly, 
ocsenave@0: /thermostatics/) as a closed subject, in which the fundamentals 
ocsenave@0: are already completely established, and there is
ocsenave@0: nothing more to be learned about them. A person who believes
ocsenave@0: this will probably prefer a pure axiomatic approach, in which
ocsenave@0: the basic laws are simply stated as arbitrary axioms, without
ocsenave@0: any attempt to present the evidence for them; and one proceeds
ocsenave@0: directly to working out their consequences.
ocsenave@0: However, we take the attitude here that thermostatics, for
ocsenave@0: all its venerable age, is very far from being a closed subject,
ocsenave@0: we still have a great deal to learn about such matters as the
ocsenave@0: most general definitions of equilibrium and reversibility, the
ocsenave@0: exact range of validity of various statements of the second and
ocsenave@0: third laws, the necessary and sufficient conditions for 
ocsenave@0: applicability of thermodynamics to special cases such as 
ocsenave@0: spin systems, and how thermodynamics can be applied to such 
ocsenave@0: systems as putty or polyethylene, which deform under force, 
ocsenave@0: but retain a \ldquo{}memory\rdquo{} of their past deformations. 
ocsenave@0: Is it possible to apply thermodynamics to a system such as a vibrating quartz crystal? We can by
ocsenave@0: no means rule out the possibility that still more laws of 
ocsenave@0: thermodynamics exist, as yet undiscovered, which would be 
ocsenave@0: useful in such applications.
ocsenave@0: 
ocsenave@0: 
ocsenave@0: It is only by careful examination of the logic by which
ocsenave@0: present thermodynamics was created, asking exactly how much of
ocsenave@0: it is mathematical theorems, how much is deducible from the laws
ocsenave@0: of mechanics and electrodynamics, and how much rests only on
ocsenave@0: empirical evidence, how compelling is present evidence for the
ocsenave@0: accuracy and range of validity of its laws; in other words,
ocsenave@0: exactly where are the boundaries of present knowledge, that we
ocsenave@0: can hope to uncover new things. Clearly, much research is still
ocsenave@0: needed in this field, and we shall be able to accomplish only a
ocsenave@0: small part of this program in the present review.
ocsenave@0: 
ocsenave@0: 
ocsenave@0: It will develop that there is an astonishingly close analogy
ocsenave@0: with the logic underlying statistical theory in general, where
ocsenave@0: again a qualitative feeling that we all have (for the degrees of
ocsenave@0: plausibility of various unproved and undisproved assertions) must
ocsenave@0: be convertefi into a precisely defined quantitative concept 
ocsenave@0: (probability). Our later development of probability theory in 
ocsenave@0: Chapter 6,7 will be, to a considerable degree, a paraphrase 
ocsenave@0: of our present review of the logic underlying classical 
ocsenave@0: thermodynamics.
ocsenave@0: 
ocsenave@0: ** The Primitive Thermometer. 
ocsenave@0: 
ocsenave@0: The earliest stages of our
ocsenave@0: story are necessarily speculative, since they took place long
ocsenave@0: before the beginnings of recorded history. But we can hardly
ocsenave@0: doubt that primitive man learned quickly that objects exposed
ocsenave@0: to the sun‘s rays or placed near a fire felt different from
ocsenave@0: those in the shade away from fires; and the same difference was
ocsenave@0: noted between animal bodies and inanimate objects.
ocsenave@0: 
ocsenave@0: 
ocsenave@0: As soon as it was noted that changes in this feeling of
ocsenave@0: warmth were correlated with other observable changes in the
ocsenave@0: behavior of objects, such as the boiling and freezing of water, 
ocsenave@0: cooking of meat, melting of fat and wax, etc., the notion of
ocsenave@0: warmth took its first step away from the purely subjective 
ocsenave@0: toward an objective, physical notion capable of being studied
ocsenave@0: scientifically.
ocsenave@0: 
ocsenave@0: One of the most striking manifestations of warmth (but far
ocsenave@0: from the earliest discovered) is the almost universal expansion
ocsenave@0: of gases, liquids, and solids when heated . This property has
ocsenave@0: proved to be a convenient one with which to reduce the notion
ocsenave@0: of warmth to something entirely objective. The invention of the
ocsenave@0: /thermometer/, in which expansion of a mercury column, or a gas,
ocsenave@0: or the bending of a bimetallic strip, etc. is read off on a
ocsenave@0: suitable scale, thereby giving us a /number/ with which to work,
ocsenave@0: was a necessary prelude to even the crudest study of the physical
ocsenave@0: nature of heat. To the best of our knowledge, although the
ocsenave@0: necessary technology to do this had been available for at least
ocsenave@0: 3,000 years, the first person to carry it out in practice was
ocsenave@0: Galileo, in 1592.
ocsenave@0: 
ocsenave@0: Later on we will give more precise definitions of the term
ocsenave@0: \ldquo{}thermometer.\rdquo{} But at the present stage we 
ocsenave@0: are not in a position to do so (as Galileo was not), because 
ocsenave@0: the very concepts needed have not yet been developed; 
ocsenave@0: more precise definitions can be
ocsenave@0: given only after our study has revealed the need for them. In
ocsenave@0: deed, our final definition can be given only after the full
ocsenave@0: mathematical formalism of statistical mechanics is at hand.
ocsenave@0: 
ocsenave@0: Once a thermometer has been constructed, and the scale
ocsenave@0: marked off in a quite arbitrary way (although we will suppose
ocsenave@0: that the scale is at least monotonic: i.e., greater warmth always
ocsenave@0: corresponds to a greater number), we are ready to begin scien
ocsenave@0: tific experiments in thermodynamics. The number read eff from
ocsenave@0: any such instrument is called the /empirical temperature/, and we
ocsenave@0: denote it by \(t\). Since the exact calibration of the thermometer
ocsenave@0: is not specified), any monotonic increasing function 
ocsenave@0: \(t‘ = f(t)\) provides an equally good temperature scale for the
ocsenave@0: present.
ocsenave@0: 
ocsenave@0: 
ocsenave@0: ** Thermodynamic Systems.
ocsenave@0: 
ocsenave@0: The \ldquo{}thermodynamic systems\rdquo{} which
ocsenave@0: are the objects of our study may be, physically, almost any
ocsenave@0: collections of objects. The traditional simplest system with
ocsenave@0: which to begin a study of thermodynamics is a volume of gas.
ocsenave@0: We shall, however, be concerned from the start also with such
ocsenave@0: things as a stretched wire or membrane, an electric cell, a
ocsenave@0: polarized dielectric, a paramagnetic body in a magnetic field, etc.
ocsenave@0: 
ocsenave@0: The /thermodynamic state/ of such a system is determined by
ocsenave@0: specifying (i.e., measuring) certain macrcoscopic physical 
ocsenave@0: properties. Now, any real physical system has many millions of such
ocsenave@0: preperties; in order to have a usable theory we cannot require
ocsenave@0: that /all/ of them be specified. We see, therefore, that there
ocsenave@0: must be a clear distinction between the notions of 
ocsenave@0: \ldquo{}thermodynamic system\rdquo{} and \ldquo{}physical
ocsenave@0: system.\rdquo{}
ocsenave@0: A given /physical/ system may correspond to many different
ocsenave@0: /thermodynamic systems/, depending
ocsenave@0: on which variables we choose to measure or control; and which
ocsenave@0: we decide to leave unmeasured and/or uncontrolled.
ocsenave@0: 
ocsenave@0: 
ocsenave@0: For example, our physical system might consist of a crystal
ocsenave@0: of sodium chloride. For one set of experiments we work with
ocsenave@0: temperature, volume, and pressure; and ignore its electrical
ocsenave@0: properties. For another set of experiments we work with 
ocsenave@0: temperature, electric field, and electric polarization; and 
ocsenave@0: ignore the varying stress and strain. The /physical/ system, 
ocsenave@0: therefore, corresponds to two entirely different /thermodynamic/ 
ocsenave@0: systems. Exactly how much freedom, then, do we have in choosing 
ocsenave@0: the variables which shall define the thermodynamic state of our
ocsenave@0: system? How many must we choose? What [criteria] determine when
ocsenave@0: we have made an adequate choice? These questions cannot be
ocsenave@0: answered until we say a little more about what we are trying to
ocsenave@0: accomplish by a thermodynamic theory. A mere collection of
ocsenave@0: 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: Chemistry/]], is a very useful thing, but it hardly constitutes
ocsenave@0: a theory. In order to construct anything deserving of such a
ocsenave@0: name, the primary requirement is that we can recognize some kind
ocsenave@0: of reproducible connection between the different properties con
ocsenave@0: sidered, so that information about some of them will enable us
ocsenave@0: to predict others. And of course, in order that our theory can
ocsenave@0: be called thermodynamics (and not some other area of physics),
ocsenave@0: it is necessary that the temperature be one of the quantities
ocsenave@0: involved in a nontrivial way.
ocsenave@0: 
ocsenave@0: The gist of these remarks is that the notion of 
ocsenave@0: \ldquo{}thermodynamic system\rdquo{} is in part 
ocsenave@0: an anthropomorphic one; it is for us to
ocsenave@0: say which set of variables shall be used. If two different
ocsenave@0: choices both lead to useful reproducible connections, it is quite
ocsenave@0: meaningless to say that one choice is any more \ldquo{}correct\rdquo{} 
ocsenave@0: than the other. Recognition of this fact will prove crucial later in
ocsenave@0: avoiding certain ancient paradoxes.
ocsenave@0: 
ocsenave@0: At this stage we can determine only empirically which other
ocsenave@0: physical properties need to be introduced before reproducible
ocsenave@0: connections appear. Once any such connection is established, we
ocsenave@0: can analyze it with the hope of being able to (1) reduce it to a
ocsenave@0: /logical/ connection rather than an empirical one; and (2) extend
ocsenave@0: it to an hypothesis applying beyond the original data, which
ocsenave@0: enables us to predict further connections capable of being
ocsenave@0: tested by experiment. Examples of this will be given presently.
ocsenave@0: 
ocsenave@0: 
ocsenave@0: There will remain, however, a few reproducible relations
ocsenave@0: which to the best of present knowledge, are not reducible to
ocsenave@0: logical relations within the context of classical thermodynamics
ocsenave@0: (and. whose demonstration in the wider context of mechanics,
ocsenave@0: electrodynamics, and quantum theory remains one of probability
ocsenave@0: rather than logical proof); from the standpoint of thermodynamics
ocsenave@0: these remain simply statements of empirical fact which must be
ocsenave@0: accepted as such without any deeper basis, but without which the
ocsenave@0: development of thermodynamics cannot proceed. Because of this
ocsenave@0: special status, these relations have become known as the
ocsenave@0: \ldquo{}laws\rdquo{}
ocsenave@0: of thermodynamics . The most fundamental one is a qualitative
ocsenave@0: rather than quantitative relation, the \ldquo{}zero'th law.\rdquo{}
ocsenave@0: 
ocsenave@0: ** Equilibrium; the \ldquo{}Zero‘th Law.\rdquo{}
ocsenave@0: 
ocsenave@0:  It is a common experience
ocsenave@0: that when objects are placed in contact with each other but
ocsenave@0: isolated from their surroundings, they may undergo observable
ocsenave@0: changes for a time as a result; one body may become warmer,
ocsenave@0: another cooler, the pressure of a gas or volume of a liquid may
ocsenave@0: change; stress or magnetization in a solid may change, etc. But
ocsenave@0: after a sufficient time, the observable macroscopic properties
ocsenave@0: settle down to a steady condition, after which no further changes
ocsenave@0: are seen unless there is a new intervention from the outside.
ocsenave@0: When this steady condition is reached, the experimentalist says
ocsenave@0: that the objects have reached a state of /equilibrium/ with each
ocsenave@0: other. Once again, more precise definitions of this term will
ocsenave@0: be needed eventually, but they require concepts not yet developed.
ocsenave@0: In any event, the criterion just stated is almost the only one
ocsenave@0: used in actual laboratory practice to decide when equilibrium
ocsenave@0: has been reached.
ocsenave@0: 
ocsenave@0: 
ocsenave@0: A particular case of equilibrium is encountered when we
ocsenave@0: place a thermometer in contact with another body. The reading
ocsenave@0: \(t\) of the thermometer may vary at first, but eventually it reach es
ocsenave@0: a steady value. Now the number \(t\) read by a thermometer is always.
ocsenave@0: by definition, the empirical temperature /of the thermometer/ (more
ocsenave@0: precisely, of the sensitive element of the thermometer). When
ocsenave@0: this number is constant in time, we say that the thermometer is
ocsenave@0: in /thermal equilibrium/ with its surroundings; and we then extend
ocsenave@0: the notion of temperature, calling the steady value \(t\) also the
ocsenave@0: /temperature of the surroundings/.
ocsenave@0: 
ocsenave@0: We have repeated these elementary facts, well known to every
ocsenave@0: child, in order to emphasize this point: Thermodynamics can be
ocsenave@0: a theory /only/ of states of equilibrium, because the very 
ocsenave@0: procedure by which the temperature of a system is defined by 
ocsenave@0: operational means, already presupposes the attainment of 
ocsenave@0: equilibrium. Strictly speaking, therefore, classical 
ocsenave@0: thermodynamics does not even contain the concept of a 
ocsenave@0: \ldquo{}time-varying temperature.\rdquo{}
ocsenave@0: 
ocsenave@0: Of course, to recognize this limitation on conventional
ocsenave@0: thermodynamics (best emphasized by calling it instead, 
ocsenave@0: thermostatics) in no way rules out the possibility of 
ocsenave@0: generalizing the notion of temperature to nonequilibrium states. 
ocsenave@0: Indeed, it is clear that one could define any number of 
ocsenave@0: time-dependent quantities all of which reduce, in the special 
ocsenave@0: case of equilibrium, to the temperature as defined above. 
ocsenave@0: Historically, attempts to do this even antedated the discovery 
ocsenave@0: of the laws of thermodynamics, as is demonstrated by 
ocsenave@0: \ldquo{}Newton's law of cooling.\rdquo{} Therefore, the
ocsenave@0: question is not whether generalization is /possible/, but only
ocsenave@0: whether it is in any way /useful/; i.e., does the temperature so
ocsenave@0: generalized have any connection with other physical properties
ocsenave@0: of our system, so that it could help us to predict other things?
ocsenave@0: However, to raise such questions takes us far beyond the
ocsenave@0: domain of thermostatics; and the general laws of nonequilibrium
ocsenave@0: behavior are so much more complicated that it would be virtually
ocsenave@0: hopeless to try to unravel them by empirical means alone. For
ocsenave@0: example, even if two different kinds of thermometer are calibrated
ocsenave@0: so that they agree with each other in equilibrium situations,
ocsenave@0: they will not agree in general about the momentary value a
ocsenave@0: \ldquo{}time-varying temperature.\rdquo{} To make any real 
ocsenave@0: progress in this area, we have to supplement empirical observation by the guidance
ocsenave@0: of a rather hiqhly-developed theory. The notion of a 
ocsenave@0: time-dependent temperature is far from simple conceptually, and we
ocsenave@0: will find that nothing very helpful can be said about this until
ocsenave@0: the full mathematical apparatus of nonequilibrium statistical
ocsenave@0: mechanics has been developed.
ocsenave@0: 
ocsenave@0: Suppose now that two bodies have the same temperature; i.e.,
ocsenave@0: a given thermometer reads the same steady value when in contact
ocsenave@0: with either. In order that the statement, \ldquo{}two bodies have the
ocsenave@0: same temperature\rdquo{} shall describe a physi cal property of the bodies,
ocsenave@0: and not merely an accidental circumstance due to our having used
ocsenave@0: a particular kind of thermometer, it is necessary that /all/ 
ocsenave@0: thermometers agree in assigning equal temperatures to them if 
ocsenave@0: /any/ thermometer does . Only experiment is competent to determine
ocsenave@0: whether this universality property is true. Unfortunately, the
ocsenave@0: writer must confess that he is unable to cite any definite
ocsenave@0: experiment in which this point was subjected to a careful test.
ocsenave@0: That equality of temperatures has this absolute meaning, has
ocsenave@0: evidently been taken for granted so much that (like absolute
ocsenave@0: sirnultaneity in pre-relativity physics) most of us are not even
ocsenave@0: consciously aware that we make such an assumption in 
ocsenave@0: thermodynamics. However, for the present we can only take it as a familiar
ocsenave@0: empirical fact that this condition does hold, not because we can
ocsenave@0: cite positive evidence for it, but because of the absence of
ocsenave@0: negative evidence against it; i.e., we think that, if an 
ocsenave@0: exception had ever been found, this would have created a sensation in
ocsenave@0: physics, and we should have heard of it.
ocsenave@0: 
ocsenave@0: We now ask: when two bodies are at the same temperature,
ocsenave@0: are they then in thermal equilibrium with each other? Again,
ocsenave@0: only experiment is competent to answer this; the general 
ocsenave@0: conclusion, again supported more by absence of negative evidence
ocsenave@0: than by specific positive evidence, is that the relation of
ocsenave@0: equilibrium has this property: 
ocsenave@0: #+begin_quote 
ocsenave@0: /Two bodies in thermal equilibrium
ocsenave@0: with a third body, are thermal equilibrium with each other./
ocsenave@0: #+end_quote
ocsenave@0: 
ocsenave@0: This empirical fact is usually called the \ldquo{}zero'th law of 
ocsenave@0: thermodynamics.\rdquo{} Since nothing prevents us from regarding a 
ocsenave@0: thermometer as the \ldquo{}third body\rdquo{} in the above statement, 
ocsenave@0: it appears that we may also state the zero'th law as: 
ocsenave@0: #+begin_quote
ocsenave@0: /Two bodies are in thermal equilibrium with each other when they are
ocsenave@0: at the same temperature./
ocsenave@0: #+end_quote
ocsenave@0: Although from the preceding discussion it might appear that
ocsenave@0: these two statements of the zero'th law are entirely equivalent
ocsenave@0: (and we certainly have no empirical evidence against either), it
ocsenave@0: is interesting to note that there are theoretical reasons, arising
ocsenave@0: from General Relativity, indicating that while the first 
ocsenave@0: statement may be universally valid, the second is not. When we 
ocsenave@0: consider equilibrium in a gravitational field, the verification
ocsenave@0: that two bodies have equal temperatures may require transport
ocsenave@0: of the thermometer through a gravitational potential difference;
ocsenave@0: and this introduces a new element into the discussion. We will
ocsenave@0: consider this in more detail in a later Chapter, and show that
ocsenave@0: according to General Relativity, equilibrium in a large system
ocsenave@0: requires, not that the temperature be uniform at all points, but
ocsenave@0: rather that a particular function of temperature and gravitational 
ocsenave@0: potential be constant (the function is \(T\cdot \exp{(\Phi/c^2})\), where
ocsenave@0: \(T\) is the Kelvin temperature to be defined later, and \(\Phi\) is the
ocsenave@0: gravitational potential).
ocsenave@0: 
ocsenave@0: Of course, this effect is so small that ordinary terrestrial
ocsenave@0: experiments would need to have a precision many orders of 
ocsenave@0: magnitude beyond that presently possible, before one could hope even
ocsenave@0: to detect it; and needless to say, it has played no role in the
ocsenave@0: development of thermodynamics. For present purposes, therefore,
ocsenave@0: we need not distinguish between the two above statements of the
ocsenave@0: zero'th law, and we take it as a basic empirical fact that a
ocsenave@0: uniform temperature at all points of a system is an essential
ocsenave@0: condition for equilibrium. It is an important part of our 
ocsenave@0: ivestigation to determine whether there are other essential 
ocsenave@0: conditions as well. In fact, as we will find, there are many 
ocsenave@0: different kinds of equilibrium; and failure to distinguish between
ocsenave@0: them can be a prolific source of paradoxes.
ocsenave@0: 
ocsenave@0: ** Equation of State
ocsenave@0: Another important reproducible connection is found when 
ocsenave@0: we consider a thermodynamic system defined by
ocsenave@0: three parameters; in addition to the temperature we choose a
ocsenave@0: \ldquo{}displacement\rdquo{} and a conjugate \ldquo{}force.\rdquo{} 
ocsenave@0: Subject to some qualifications given below, we find experimentally 
ocsenave@0: that these parameters are not independent, but are subject to a constraint.
ocsenave@0: For example, we cannot vary the equilibrium pressure, volume,
ocsenave@0: and temperature of a given mass of gas independently; it is found
ocsenave@0: that a given pressure and volume can be realized only at one
ocsenave@0: particular temperature, that the gas will assume a given tempera~
ocsenave@0: ture and volume only at one particular pressure, etc. Similarly,
ocsenave@0: a stretched wire can be made to have arbitrarily assigned tension
ocsenave@0: and elongation only if its temperature is suitably chosen, a
ocsenave@0: dielectric will assume a state of given temperature and 
ocsenave@0: polarization at only one value of the electric field, etc.
ocsenave@0: These simplest nontrivial thermodynamic systems (three 
ocsenave@0: parameters with one constraint) are said to possess two 
ocsenave@0: /degrees of freedom/; for the range of possible equilibrium states is defined
ocsenave@0: by specifying any two of the variables arbitrarily, whereupon the
ocsenave@0: third, and all others we may introduce, are determined. 
ocsenave@0: Mathematically, this is expressed by the existence of a functional
ocsenave@0: relationship of the form[fn::Edit: The set of solutions to an equation
ocsenave@0: like /f(X,x,t)=/ const. is called a /level set/. Here, Jaynes is
ocsenave@0: saying that the quantities /X/, /x/, and /t/ follow a \ldquo{}functional
ocsenave@0: rule\rdquo{}, so the set of physically allowed combinations of /X/,
ocsenave@0: /x/, and /t/ in equilibrium states can be
ocsenave@0: expressed as the level set of a function. 
ocsenave@0: 
ocsenave@0: But not every function expresses a constraint relation; for some
ocsenave@0: functions, you can specify two of the variables, and the third will
ocsenave@0: still be undetermined. (For example, if f=X^2+x^2+t^2-3,
ocsenave@0: the level set /f(X,x,t)=0/ is a sphere, and specifying /x=1/, /t=1/
ocsenave@0: leaves you with two potential possibilities for /X/ =\pm 1.)
ocsenave@0: 
ocsenave@0: A function like /f/ has to possess one more propery in order to
ocsenave@0: express a constraint relationship: it must be monotonic in
ocsenave@0: each of its variables /X/, /x/, and /t/.
ocsenave@0: #the partial derivatives of /f/ exist for every allowed combination of
ocsenave@0: #inputs /x/, /X/, and /t/. 
ocsenave@0: In other words, the level set has to pass a sort of 
ocsenave@0: \ldquo{}vertical line test\rdquo{} for each of its variables.]
ocsenave@0: 
ocsenave@0: #Edit Here, Jaynes
ocsenave@0: #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: #[[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: #In order to specify
ocsenave@0: 
ocsenave@0: \begin{equation}
ocsenave@0: f(X,x,t) = O
ocsenave@0: \end{equation}
ocsenave@0: 
ocsenave@0: where $X$ is a generalized force (pressure, tension, electric or
ocsenave@0: magnetic field, etc.), $x$ is the corresponding generalized 
ocsenave@0: displacement (volume, elongation, electric or magnetic polarization,
ocsenave@0: etc.), and $t$ is the empirical temperature. Equation (1) is
ocsenave@0: called /the equation of state/.
ocsenave@0: 
ocsenave@0: At the risk of belaboring it, we emphasize once again that
ocsenave@0: all of this applies only for a system in equilibrium; for 
ocsenave@0: otherwise not only.the temperature, but also some or all of the other
ocsenave@0: variables may not be definable. For example, no unique pressure
ocsenave@0: can be assigned to a gas which has just suffered a sudden change
ocsenave@0: in volume, until the generated sound waves have died out.
ocsenave@0: 
ocsenave@0: Independently of its functional form, the mere fact of the
ocsenave@0: /existence/ of an equation of state has certain experimental 
ocsenave@0: consequences. For example, suppose that in experiments on oxygen
ocsenave@0: gas, in which we control the temperature and pressure 
ocsenave@0: independently, we have found that the isothermal compressibility $K$
ocsenave@0: varies with temperature, and the thermal expansion coefficient
ocsenave@0: \alpha varies with pressure $P$, so that within the accuracy of the data,
ocsenave@0: 
ocsenave@0: \begin{equation}
ocsenave@0: \frac{\partial K}{\partial t} = - \frac{\partial \alpha}{\partial P}
ocsenave@0: \end{equation}
ocsenave@0: 
ocsenave@0: Is this a particular property of oxygen; or is there reason to
ocsenave@0: believe that it holds also for other substances? Does it depend
ocsenave@0: on our particular choice of a temperature scale?
ocsenave@0: 
ocsenave@0: In this case, the answer is found at once; for the definitions of $K$,
ocsenave@0: \alpha are
ocsenave@0: 
ocsenave@0: \begin{equation}
ocsenave@0: K = -\frac{1}{V}\frac{\partial V}{\partial P},\qquad
ocsenave@0: \alpha=\frac{1}{V}\frac{\partial V}{\partial t}
ocsenave@0: \end{equation}
ocsenave@0: 
ocsenave@0: which is simply a mathematical expression of the fact that the
ocsenave@0: volume $V$ is a definite function of $P$ and $t$; i.e., it depends
ocsenave@0: only
ocsenave@0: on their present values, and not how those values were attained.
ocsenave@0: In particular, $V$ does not depend on the direction in the \((P, t)\)
ocsenave@0: plane through which the present values were approached; or, as we 
ocsenave@0: usually say it, \(dV\) is an /exact differential/.
ocsenave@0: 
ocsenave@0: Therefore, although at first glance the relation (2) appears
ocsenave@0: nontrivial and far from obvious, a trivial mathematical analysis
ocsenave@0: convinces us that it must hold regardless of our particular
ocsenave@0: temperature scale, and that it is true not only of oxygen; it must
ocsenave@0: hold for any substance, or mixture of substances, which possesses a
ocsenave@0: definite, reproducible equation of state \(f(P,V,t)=0\).
ocsenave@0: 
ocsenave@0: But this understanding also enables us to predict situations in which
ocsenave@0: (2) will /not/ hold. Equation (2), as we have just learned, expresses
ocsenave@0: the fact that an equation of state exists involving only the three
ocsenave@0: variables \((P,V,t)\). Now suppose we try to apply it to a liquid such
ocsenave@0: as nitrobenzene. The nitrobenzene molecule has a large electric dipole
ocsenave@0: moment; and so application of an electric field (as in the
ocsenave@0: [[http://en.wikipedia.org/wiki/Kerr_effect][electro-optical Kerr cell]]) causes an alignment of molecules which, as
ocsenave@0: accurate measurements will verify, changes the pressure at a given
ocsenave@0: temperature and volume. Therefore, there can no longer exist any
ocsenave@0: unique equation of state involving \((P, V, t)\) only; with
ocsenave@0: sufficiently accurate measurements, nitrobenzene must be regarded as a
ocsenave@0: thermodynamic system with at least three degrees of freedom, and the
ocsenave@0: general equation of state must have at least a complicated a form as
ocsenave@0: \(f(P,V,t,E) = 0\).
ocsenave@0: 
ocsenave@0: But if we introduce a varying electric field $E$ into the discussion,
ocsenave@0: the resulting varying electric polarization $M$ also becomes a new
ocsenave@0: thermodynamic variable capable of being measured. Experimentally, it
ocsenave@0: is easiest to control temperature, pressure, and electric field
ocsenave@0: independently, and of course we find that both the volume and
ocsenave@0: polarization are then determined; i.e., there must exist functional
ocsenave@0: relations of the form \(V = V(P,t,E)\), \(M = M(P,t,E)\), or in more
ocsenave@0: symmetrical form
ocsenave@0: 
ocsenave@0: \begin{equation}
ocsenave@0: f(V,P,t,E) = 0 \qquad g(M,P,t,E)=0.
ocsenave@0: \end{equation}
ocsenave@0: 
ocsenave@0: In other words, if we regard nitrobenzene as a thermodynamic system of
ocsenave@0: three degrees of freedom (i.e., having specified three parameters
ocsenave@0: arbitrarily, all others are then determined), it must possess two
ocsenave@0: independent equations of state.
ocsenave@0: 
ocsenave@0: Similarly, a thermodynamic system with four degrees of freedom,
ocsenave@0: defined by the termperature and three pairs of conjugate forces and
ocsenave@0: displacements, will have three independent equations of state, etc.
ocsenave@0: 
ocsenave@0: Now, returning to our original question, if nitrobenzene possesses
ocsenave@0: this extra electrical degree of freedom, under what circumstances do
ocsenave@0: we exprect to find a reproducible equation of state involving
ocsenave@0: \((p,V,t)\) only? Evidently, if $E$ is held constant, then the first
ocsenave@0: of equations (1-5) becomes such an equation of state, involving $E$ as
ocsenave@0: a fixed parameter; we would find many different equations of state of
ocsenave@0: the form \(f(P,V,t) = 0\) with a different function $f$ for each
ocsenave@0: different value of the electric field. Likewise, if \(M\) is held
ocsenave@0: constant, we can eliminate \(E\) between equations (1-5) and find a
ocsenave@0: relation \(h(P,V,t,M)=0\), which is an equation of state for
ocsenave@0: \((P,V,t)\) containing \(M\) as a fixed parameter.
ocsenave@0: 
ocsenave@0: More generally, if an electrical constraint is imposed on the system
ocsenave@0: (for example, by connecting an external charged capacitor to the
ocsenave@0: electrodes) so that \(M\) is determined by \(E\); i.e., there is a
ocsenave@0: functional relation of the form 
ocsenave@0: 
ocsenave@0: \begin{equation}
ocsenave@0: g(M,E) = \text{const.}
ocsenave@0: \end{equation}
ocsenave@0: 
ocsenave@0: then (1-5) and (1-6) constitute three simultaneous equations, from
ocsenave@0: which both \(E\) and \(M\) may be eliminated mathematically, leading
ocsenave@0: to a relation of the form \(h(P,V,t;q)=0\), which is an equation of
ocsenave@0: state for \((P,V,t)\) involving the fixed parameter \(q\).
ocsenave@0: 
ocsenave@0: We see, then, that as long as a fixed constraint of the form (1-6) is
ocsenave@0: imposed on the electrical degree of freedom, we can still observe a
ocsenave@0: reproducible equation of state for nitrobenzene, considered as a
ocsenave@0: thermodynamic system of only two degrees of freedom. If, however, this
ocsenave@0: electrical constraint is removed, so that as we vary $P$ and $t$, the
ocsenave@0: values of $E$ and $M$ vary in an uncontrolled way over a
ocsenave@0: /two-dimensional/ region of the \((E, M)\) plane, then we will find no
ocsenave@0: definite equation of state involving only \((P,V,t)\).
ocsenave@0: 
ocsenave@0: This may be stated more colloqually as follows: even though a system
ocsenave@0: has three degrees of freedom, we can still consider only the variables
ocsenave@0: belonging to two of them, and we will find a definite equation of
ocsenave@0: state, /provided/ that in the course of the experiments, the unused
ocsenave@0: degree of freedom is not \ldquo{}tampered with\rdquo{} in an
ocsenave@0: uncontrolled way.
ocsenave@0: 
ocsenave@0: We have already emphasized that any physical system corresponds to
ocsenave@0: many different thermodynamic systems, depending on which variables we
ocsenave@0: choose to control and measure. In fact, it is easy to see that any
ocsenave@0: physical system has, for all practical purposes, an /arbitrarily
ocsenave@0: large/ number of  degrees of freedom. In the case of nitrobenzene, for
ocsenave@0: example, we may impose any variety of nonuniform electric fields on
ocsenave@0: our sample. Suppose we place $(n+1)$