#+TITLE: Statistical Mechanics

 #+AUTHOR:    E.T. Jaynes; edited by  Dylan Holmes

 #+EMAIL:     rlm@mit.edu

 #+KEYWORDS: statistical mechanics, thermostatics, thermodynamics, temperature, paradoxes, Jaynes

 #+SETUPFILE: ../../aurellem/org/setup.org

 #+INCLUDE:   ../../aurellem/org/level-0.org

 #+MATHJAX: align:"left" mathml:t path:"http://www.aurellem.org/MathJax/MathJax.js"

 

 # "extensions/eqn-number.js"

 

 #+begin_quote

 *Note:* The following is a typeset version of

  [[../sources/stat.mech.1.pdf][this unpublished book draft]], written by [[http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E.T. Jaynes]].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

  minor changes, e.g. to correct typographical errors, add references, or format equations. The

  content itself is intact. --- Dylan

 #+end_quote

 

 * Development of Thermodynamics

 Our first intuitive, or \ldquo{}subjective\rdquo{} notions of temperature

 arise from the sensations of warmth and cold associated with our

 sense of touch . Yet science has been able to convert this qualitative 

 sensation into an accurately defined quantitative notion,

 which can be applied far beyond the range of our direct experience. 

 Today an experimentalist will report confidently that his

 spin system was at a temperature of 2.51 degrees Kelvin; and a

 theoretician will report with almost as much confidence that the 

 temperature at the center of the sun is about \(2 \times 10^7\) degrees

 Kelvin.

 

 The /fact/ that this has proved possible, and the main technical 

 ideas involved, are assumed already known to the reader;

 and we are not concerned here with repeating standard material

 already available in a dozen other textbooks . However 

 thermodynamics, in spite of its great successes, firmly established

 for over a century, has also produced a great deal of confusion

 and a long list of \ldquo{}paradoxes\rdquo{} centering mostly 

 around the second law and the nature of irreversibility. 

 For this reason and others noted below, we want to dwell here at 

 some length on the /logic/ underlying the development of 

 thermodynamics . Our aim is to emphasize certain points which, 

 in the writer's opinion, are essential for clearing up the 

 confusion and resolving the paradoxes; but which are not 

 sufficiently ernphasized---and indeed in many cases are 

 totally ignored---in other textbooks.

 

 This attention to logic 

 would not be particularly needed if we regarded classical 

 thermodynamics (or, as it is becoming called increasingly, 

 /thermostatics/) as a closed subject, in which the fundamentals 

 are already completely established, and there is

 nothing more to be learned about them. A person who believes

 this will probably prefer a pure axiomatic approach, in which

 the basic laws are simply stated as arbitrary axioms, without

 any attempt to present the evidence for them; and one proceeds

 directly to working out their consequences.

 However, we take the attitude here that thermostatics, for

 all its venerable age, is very far from being a closed subject,

 we still have a great deal to learn about such matters as the

 most general definitions of equilibrium and reversibility, the

 exact range of validity of various statements of the second and

 third laws, the necessary and sufficient conditions for 

 applicability of thermodynamics to special cases such as 

 spin systems, and how thermodynamics can be applied to such 

 systems as putty or polyethylene, which deform under force, 

 but retain a \ldquo{}memory\rdquo{} of their past deformations. 

 Is it possible to apply thermodynamics to a system such as a vibrating quartz crystal? We can by

 no means rule out the possibility that still more laws of 

 thermodynamics exist, as yet undiscovered, which would be 

 useful in such applications.

 

 

 It is only by careful examination of the logic by which

 present thermodynamics was created, asking exactly how much of

 it is mathematical theorems, how much is deducible from the laws

 of mechanics and electrodynamics, and how much rests only on

 empirical evidence, how compelling is present evidence for the

 accuracy and range of validity of its laws; in other words,

 exactly where are the boundaries of present knowledge, that we

 can hope to uncover new things. Clearly, much research is still

 needed in this field, and we shall be able to accomplish only a

 small part of this program in the present review.

 

 

 It will develop that there is an astonishingly close analogy

 with the logic underlying statistical theory in general, where

 again a qualitative feeling that we all have (for the degrees of

 plausibility of various unproved and undisproved assertions) must

 be convertefi into a precisely defined quantitative concept 

 (probability). Our later development of probability theory in 

 Chapter 6,7 will be, to a considerable degree, a paraphrase 

 of our present review of the logic underlying classical 

 thermodynamics.

 

 ** The Primitive Thermometer. 

 

 The earliest stages of our

 story are necessarily speculative, since they took place long

 before the beginnings of recorded history. But we can hardly

 doubt that primitive man learned quickly that objects exposed

 to the sun‘s rays or placed near a fire felt different from

 those in the shade away from fires; and the same difference was

 noted between animal bodies and inanimate objects.

 

 

 As soon as it was noted that changes in this feeling of

 warmth were correlated with other observable changes in the

 behavior of objects, such as the boiling and freezing of water, 

 cooking of meat, melting of fat and wax, etc., the notion of

 warmth took its first step away from the purely subjective 

 toward an objective, physical notion capable of being studied

 scientifically.

 

 One of the most striking manifestations of warmth (but far

 from the earliest discovered) is the almost universal expansion

 of gases, liquids, and solids when heated . This property has

 proved to be a convenient one with which to reduce the notion

 of warmth to something entirely objective. The invention of the

 /thermometer/, in which expansion of a mercury column, or a gas,

 or the bending of a bimetallic strip, etc. is read off on a

 suitable scale, thereby giving us a /number/ with which to work,

 was a necessary prelude to even the crudest study of the physical

 nature of heat. To the best of our knowledge, although the

 necessary technology to do this had been available for at least

 3,000 years, the first person to carry it out in practice was

 Galileo, in 1592.

 

 Later on we will give more precise definitions of the term

 \ldquo{}thermometer.\rdquo{} But at the present stage we 

 are not in a position to do so (as Galileo was not), because 

 the very concepts needed have not yet been developed; 

 more precise definitions can be

 given only after our study has revealed the need for them. In

 deed, our final definition can be given only after the full

 mathematical formalism of statistical mechanics is at hand.

 

 Once a thermometer has been constructed, and the scale

 marked off in a quite arbitrary way (although we will suppose

 that the scale is at least monotonic: i.e., greater warmth always

 corresponds to a greater number), we are ready to begin scien

 tific experiments in thermodynamics. The number read eff from

 any such instrument is called the /empirical temperature/, and we

 denote it by \(t\). Since the exact calibration of the thermometer

 is not specified), any monotonic increasing function 

 \(t‘ = f(t)\) provides an equally good temperature scale for the

 present.

 

 

 ** Thermodynamic Systems.

 

 The \ldquo{}thermodynamic systems\rdquo{} which

 are the objects of our study may be, physically, almost any

 collections of objects. The traditional simplest system with

 which to begin a study of thermodynamics is a volume of gas.

 We shall, however, be concerned from the start also with such

 things as a stretched wire or membrane, an electric cell, a

 polarized dielectric, a paramagnetic body in a magnetic field, etc.

 

 The /thermodynamic state/ of such a system is determined by

 specifying (i.e., measuring) certain macrcoscopic physical 

 properties. Now, any real physical system has many millions of such

 preperties; in order to have a usable theory we cannot require

 that /all/ of them be specified. We see, therefore, that there

 must be a clear distinction between the notions of 

 \ldquo{}thermodynamic system\rdquo{} and \ldquo{}physical

 system.\rdquo{}

 A given /physical/ system may correspond to many different

 /thermodynamic systems/, depending

 on which variables we choose to measure or control; and which

 we decide to leave unmeasured and/or uncontrolled.

 

 

 For example, our physical system might consist of a crystal

 of sodium chloride. For one set of experiments we work with

 temperature, volume, and pressure; and ignore its electrical

 properties. For another set of experiments we work with 

 temperature, electric field, and electric polarization; and 

 ignore the varying stress and strain. The /physical/ system, 

 therefore, corresponds to two entirely different /thermodynamic/ 

 systems. Exactly how much freedom, then, do we have in choosing 

 the variables which shall define the thermodynamic state of our

 system? How many must we choose? What [criteria] determine when

 we have made an adequate choice? These questions cannot be

 answered until we say a little more about what we are trying to

 accomplish by a thermodynamic theory. A mere collection of

 recorded data about our system, as in the [[http://en.wikipedia.org/wiki/CRC_Handbook_of_Chemistry_and_Physics][/Handbook of Physics and

 Chemistry/]], is a very useful thing, but it hardly constitutes

 a theory. In order to construct anything deserving of such a

 name, the primary requirement is that we can recognize some kind

 of reproducible connection between the different properties con

 sidered, so that information about some of them will enable us

 to predict others. And of course, in order that our theory can

 be called thermodynamics (and not some other area of physics),

 it is necessary that the temperature be one of the quantities

 involved in a nontrivial way.

 

 The gist of these remarks is that the notion of 

 \ldquo{}thermodynamic system\rdquo{} is in part 

 an anthropomorphic one; it is for us to

 say which set of variables shall be used. If two different

 choices both lead to useful reproducible connections, it is quite

 meaningless to say that one choice is any more \ldquo{}correct\rdquo{} 

 than the other. Recognition of this fact will prove crucial later in

 avoiding certain ancient paradoxes.

 

 At this stage we can determine only empirically which other

 physical properties need to be introduced before reproducible

 connections appear. Once any such connection is established, we

 can analyze it with the hope of being able to (1) reduce it to a

 /logical/ connection rather than an empirical one; and (2) extend

 it to an hypothesis applying beyond the original data, which

 enables us to predict further connections capable of being

 tested by experiment. Examples of this will be given presently.

 

 

 There will remain, however, a few reproducible relations

 which to the best of present knowledge, are not reducible to

 logical relations within the context of classical thermodynamics

 (and. whose demonstration in the wider context of mechanics,

 electrodynamics, and quantum theory remains one of probability

 rather than logical proof); from the standpoint of thermodynamics

 these remain simply statements of empirical fact which must be

 accepted as such without any deeper basis, but without which the

 development of thermodynamics cannot proceed. Because of this

 special status, these relations have become known as the

 \ldquo{}laws\rdquo{}

 of thermodynamics . The most fundamental one is a qualitative

 rather than quantitative relation, the \ldquo{}zero'th law.\rdquo{}

 

 ** Equilibrium; the \ldquo{}Zero‘th Law.\rdquo{}

 

  It is a common experience

 that when objects are placed in contact with each other but

 isolated from their surroundings, they may undergo observable

 changes for a time as a result; one body may become warmer,

 another cooler, the pressure of a gas or volume of a liquid may

 change; stress or magnetization in a solid may change, etc. But

 after a sufficient time, the observable macroscopic properties

 settle down to a steady condition, after which no further changes

 are seen unless there is a new intervention from the outside.

 When this steady condition is reached, the experimentalist says

 that the objects have reached a state of /equilibrium/ with each

 other. Once again, more precise definitions of this term will

 be needed eventually, but they require concepts not yet developed.

 In any event, the criterion just stated is almost the only one

 used in actual laboratory practice to decide when equilibrium

 has been reached.

 

 

 A particular case of equilibrium is encountered when we

 place a thermometer in contact with another body. The reading

 \(t\) of the thermometer may vary at first, but eventually it reach es

 a steady value. Now the number \(t\) read by a thermometer is always.

 by definition, the empirical temperature /of the thermometer/ (more

 precisely, of the sensitive element of the thermometer). When

 this number is constant in time, we say that the thermometer is

 in /thermal equilibrium/ with its surroundings; and we then extend

 the notion of temperature, calling the steady value \(t\) also the

 /temperature of the surroundings/.

 

 We have repeated these elementary facts, well known to every

 child, in order to emphasize this point: Thermodynamics can be

 a theory /only/ of states of equilibrium, because the very 

 procedure by which the temperature of a system is defined by 

 operational means, already presupposes the attainment of 

 equilibrium. Strictly speaking, therefore, classical 

 thermodynamics does not even contain the concept of a 

 \ldquo{}time-varying temperature.\rdquo{}

 

 Of course, to recognize this limitation on conventional

 thermodynamics (best emphasized by calling it instead, 

 thermostatics) in no way rules out the possibility of 

 generalizing the notion of temperature to nonequilibrium states. 

 Indeed, it is clear that one could define any number of 

 time-dependent quantities all of which reduce, in the special 

 case of equilibrium, to the temperature as defined above. 

 Historically, attempts to do this even antedated the discovery 

 of the laws of thermodynamics, as is demonstrated by 

 \ldquo{}Newton's law of cooling.\rdquo{} Therefore, the

 question is not whether generalization is /possible/, but only

 whether it is in any way /useful/; i.e., does the temperature so

 generalized have any connection with other physical properties

 of our system, so that it could help us to predict other things?

 However, to raise such questions takes us far beyond the

 domain of thermostatics; and the general laws of nonequilibrium

 behavior are so much more complicated that it would be virtually

 hopeless to try to unravel them by empirical means alone. For

 example, even if two different kinds of thermometer are calibrated

 so that they agree with each other in equilibrium situations,

 they will not agree in general about the momentary value a

 \ldquo{}time-varying temperature.\rdquo{} To make any real 

 progress in this area, we have to supplement empirical observation by the guidance

 of a rather hiqhly-developed theory. The notion of a 

 time-dependent temperature is far from simple conceptually, and we

 will find that nothing very helpful can be said about this until

 the full mathematical apparatus of nonequilibrium statistical

 mechanics has been developed.

 

 Suppose now that two bodies have the same temperature; i.e.,

 a given thermometer reads the same steady value when in contact

 with either. In order that the statement, \ldquo{}two bodies have the

 same temperature\rdquo{} shall describe a physical property of the bodies,

 and not merely an accidental circumstance due to our having used

 a particular kind of thermometer, it is necessary that /all/ 

 thermometers agree in assigning equal temperatures to them if 

 /any/ thermometer does . Only experiment is competent to determine

 whether this universality property is true. Unfortunately, the

 writer must confess that he is unable to cite any definite

 experiment in which this point was subjected to a careful test.

 That equality of temperatures has this absolute meaning, has

 evidently been taken for granted so much that (like absolute

 sirnultaneity in pre-relativity physics) most of us are not even

 consciously aware that we make such an assumption in 

 thermodynamics. However, for the present we can only take it as a familiar

 empirical fact that this condition does hold, not because we can

 cite positive evidence for it, but because of the absence of

 negative evidence against it; i.e., we think that, if an 

 exception had ever been found, this would have created a sensation in

 physics, and we should have heard of it.

 

 We now ask: when two bodies are at the same temperature,

 are they then in thermal equilibrium with each other? Again,

 only experiment is competent to answer this; the general 

 conclusion, again supported more by absence of negative evidence

 than by specific positive evidence, is that the relation of

 equilibrium has this property: 

 #+begin_quote 

 /Two bodies in thermal equilibrium

 with a third body, are thermal equilibrium with each other./

 #+end_quote

 

 This empirical fact is usually called the \ldquo{}zero'th law of 

 thermodynamics.\rdquo{} Since nothing prevents us from regarding a 

 thermometer as the \ldquo{}third body\rdquo{} in the above statement, 

 it appears that we may also state the zero'th law as: 

 #+begin_quote

 /Two bodies are in thermal equilibrium with each other when they are

 at the same temperature./

 #+end_quote

 Although from the preceding discussion it might appear that

 these two statements of the zero'th law are entirely equivalent

 (and we certainly have no empirical evidence against either), it

 is interesting to note that there are theoretical reasons, arising

 from General Relativity, indicating that while the first 

 statement may be universally valid, the second is not. When we 

 consider equilibrium in a gravitational field, the verification

 that two bodies have equal temperatures may require transport

 of the thermometer through a gravitational potential difference;

 and this introduces a new element into the discussion. We will

 consider this in more detail in a later Chapter, and show that

 according to General Relativity, equilibrium in a large system

 requires, not that the temperature be uniform at all points, but

 rather that a particular function of temperature and gravitational 

 potential be constant (the function is \(T\cdot \exp{(\Phi/c^2})\), where

 \(T\) is the Kelvin temperature to be defined later, and \(\Phi\) is the

 gravitational potential).

 

 Of course, this effect is so small that ordinary terrestrial

 experiments would need to have a precision many orders of 

 magnitude beyond that presently possible, before one could hope even

 to detect it; and needless to say, it has played no role in the

 development of thermodynamics. For present purposes, therefore,

 we need not distinguish between the two above statements of the

 zero'th law, and we take it as a basic empirical fact that a

 uniform temperature at all points of a system is an essential

 condition for equilibrium. It is an important part of our 

 ivestigation to determine whether there are other essential 

 conditions as well. In fact, as we will find, there are many 

 different kinds of equilibrium; and failure to distinguish between

 them can be a prolific source of paradoxes.

 

 ** Equation of State

 Another important reproducible connection is found when 

 we consider a thermodynamic system defined by

 three parameters; in addition to the temperature we choose a

 \ldquo{}displacement\rdquo{} and a conjugate \ldquo{}force.\rdquo{} 

 Subject to some qualifications given below, we find experimentally 

 that these parameters are not independent, but are subject to a constraint.

 For example, we cannot vary the equilibrium pressure, volume,

 and temperature of a given mass of gas independently; it is found

 that a given pressure and volume can be realized only at one

 particular temperature, that the gas will assume a given tempera~

 ture and volume only at one particular pressure, etc. Similarly,

 a stretched wire can be made to have arbitrarily assigned tension

 and elongation only if its temperature is suitably chosen, a

 dielectric will assume a state of given temperature and 

 polarization at only one value of the electric field, etc.

 These simplest nontrivial thermodynamic systems (three 

 parameters with one constraint) are said to possess two 

 /degrees of freedom/; for the range of possible equilibrium states is defined

 by specifying any two of the variables arbitrarily, whereupon the

 third, and all others we may introduce, are determined. 

 Mathematically, this is expressed by the existence of a functional

 relationship of the form[fn:: /Edit./: The set of solutions to an equation

 like /f(X,x,t)=/ const. is called a /level set/. Here, Jaynes is

 saying that the quantities /X/, /x/, and /t/ follow a \ldquo{}functional

 rule\rdquo{}, so the set of physically allowed combinations of /X/,

 /x/, and /t/ in equilibrium states can be

 expressed as the level set of a function. 

 

 But not every function expresses a constraint relation; for some

 functions, you can specify two of the variables, and the third will

 still be undetermined. (For example, if f=X^2+x^2+t^2-3,

 the level set /f(X,x,t)=0/ is a sphere, and specifying /x=1/, /t=1/

 leaves you with two potential possibilities for /X/ =\pm 1.)

 

 A function like /f/ has to possess one more propery in order for its

 level set to express a constraint relationship: it must be monotonic in

 each of its variables /X/, /x/, and /t/.

 #the partial derivatives of /f/ exist for every allowed combination of

 #inputs /x/, /X/, and /t/. 

 In other words, the level set has to pass a sort of 

 \ldquo{}vertical line test\rdquo{} for each of its variables.]

 

 #Edit Here, Jaynes

 #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

 #[[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.

 #In order to specify

 

 \begin{equation}

 f(X,x,t) = O

 \end{equation}

 

 where $X$ is a generalized force (pressure, tension, electric or

 magnetic field, etc.), $x$ is the corresponding generalized 

 displacement (volume, elongation, electric or magnetic polarization,

 etc.), and $t$ is the empirical temperature. Equation (1-1) is

 called /the equation of state/.

 

 At the risk of belaboring it, we emphasize once again that

 all of this applies only for a system in equilibrium; for 

 otherwise not only.the temperature, but also some or all of the other

 variables may not be definable. For example, no unique pressure

 can be assigned to a gas which has just suffered a sudden change

 in volume, until the generated sound waves have died out.

 

 Independently of its functional form, the mere fact of the

 /existence/ of an equation of state has certain experimental 

 consequences. For example, suppose that in experiments on oxygen

 gas, in which we control the temperature and pressure 

 independently, we have found that the isothermal compressibility $K$

 varies with temperature, and the thermal expansion coefficient

 \alpha varies with pressure $P$, so that within the accuracy of the data,

 

 \begin{equation}

 \frac{\partial K}{\partial t} = - \frac{\partial \alpha}{\partial P}

 \end{equation}

 

 Is this a particular property of oxygen; or is there reason to

 believe that it holds also for other substances? Does it depend

 on our particular choice of a temperature scale?

 

 In this case, the answer is found at once; for the definitions of $K$,

 \alpha are

 

 \begin{equation}

 K = -\frac{1}{V}\frac{\partial V}{\partial P},\qquad

 \alpha=\frac{1}{V}\frac{\partial V}{\partial t}

 \end{equation}

 

 which is simply a mathematical expression of the fact that the

 volume $V$ is a definite function of $P$ and $t$; i.e., it depends

 only

 on their present values, and not how those values were attained.

 In particular, $V$ does not depend on the direction in the \((P, t)\)

 plane through which the present values were approached; or, as we 

 usually say it, \(dV\) is an /exact differential/.

 

 Therefore, although at first glance the relation (1-2) appears

 nontrivial and far from obvious, a trivial mathematical analysis

 convinces us that it must hold regardless of our particular

 temperature scale, and that it is true not only of oxygen; it must

 hold for any substance, or mixture of substances, which possesses a

 definite, reproducible equation of state \(f(P,V,t)=0\).

 

 But this understanding also enables us to predict situations in which

 (1-2) will /not/ hold. Equation (1-2), as we have just learned, expresses

 the fact that an equation of state exists involving only the three

 variables \((P,V,t)\). Now suppose we try to apply it to a liquid such

 as nitrobenzene. The nitrobenzene molecule has a large electric dipole

 moment; and so application of an electric field (as in the

 [[http://en.wikipedia.org/wiki/Kerr_effect][electro-optical Kerr cell]]) causes an alignment of molecules which, as

 accurate measurements will verify, changes the pressure at a given

 temperature and volume. Therefore, there can no longer exist any

 unique equation of state involving \((P, V, t)\) only; with

 sufficiently accurate measurements, nitrobenzene must be regarded as a

 thermodynamic system with at least three degrees of freedom, and the

 general equation of state must have at least a complicated a form as

 \(f(P,V,t,E) = 0\).

 

 But if we introduce a varying electric field $E$ into the discussion,

 the resulting varying electric polarization $M$ also becomes a new

 thermodynamic variable capable of being measured. Experimentally, it

 is easiest to control temperature, pressure, and electric field

 independently, and of course we find that both the volume and

 polarization are then determined; i.e., there must exist functional

 relations of the form \(V = V(P,t,E)\), \(M = M(P,t,E)\), or in more

 symmetrical form

 

 \begin{equation}

 f(V,P,t,E) = 0 \qquad g(M,P,t,E)=0.

 \end{equation}

 

 In other words, if we regard nitrobenzene as a thermodynamic system of

 three degrees of freedom (i.e., having specified three parameters

 arbitrarily, all others are then determined), it must possess two

 independent equations of state.

 

 Similarly, a thermodynamic system with four degrees of freedom,

 defined by the termperature and three pairs of conjugate forces and

 displacements, will have three independent equations of state, etc.

 

 Now, returning to our original question, if nitrobenzene possesses

 this extra electrical degree of freedom, under what circumstances do

 we exprect to find a reproducible equation of state involving

 \((p,V,t)\) only? Evidently, if $E$ is held constant, then the first

 of equations (1-5) becomes such an equation of state, involving $E$ as

 a fixed parameter; we would find many different equations of state of

 the form \(f(P,V,t) = 0\) with a different function $f$ for each

 different value of the electric field. Likewise, if \(M\) is held

 constant, we can eliminate \(E\) between equations (1-5) and find a

 relation \(h(P,V,t,M)=0\), which is an equation of state for

 \((P,V,t)\) containing \(M\) as a fixed parameter.

 

 More generally, if an electrical constraint is imposed on the system

 (for example, by connecting an external charged capacitor to the

 electrodes) so that \(M\) is determined by \(E\); i.e., there is a

 functional relation of the form 

 

 \begin{equation}

 g(M,E) = \text{const.}

 \end{equation}

 

 then (1-5) and (1-6) constitute three simultaneous equations, from

 which both \(E\) and \(M\) may be eliminated mathematically, leading

 to a relation of the form \(h(P,V,t;q)=0\), which is an equation of

 state for \((P,V,t)\) involving the fixed parameter \(q\).

 

 We see, then, that as long as a fixed constraint of the form (1-6) is

 imposed on the electrical degree of freedom, we can still observe a

 reproducible equation of state for nitrobenzene, considered as a

 thermodynamic system of only two degrees of freedom. If, however, this

 electrical constraint is removed, so that as we vary $P$ and $t$, the

 values of $E$ and $M$ vary in an uncontrolled way over a

 /two-dimensional/ region of the \((E, M)\) plane, then we will find no

 definite equation of state involving only \((P,V,t)\).

 

 This may be stated more colloqually as follows: even though a system

 has three degrees of freedom, we can still consider only the variables

 belonging to two of them, and we will find a definite equation of

 state, /provided/ that in the course of the experiments, the unused

 degree of freedom is not \ldquo{}tampered with\rdquo{} in an

 uncontrolled way.

 

 We have already emphasized that any physical system corresponds to

 many different thermodynamic systems, depending on which variables we

 choose to control and measure. In fact, it is easy to see that any

 physical system has, for all practical purposes, an /arbitrarily

 large/ number of  degrees of freedom. In the case of nitrobenzene, for

 example, we may impose any variety of nonuniform electric fields on

 our sample. Suppose we place $(n+1)$ different electrodes, labelled

 \(\{e_0,e_1, e_2 \ldots e_n\}\) in contact with the liquid in various

 positions. Regarding \(e_0\) as the \ldquo{}ground\rdquo{}, maintained

 at zero potential, we can then impose $n$ different potentials

 \(\{v_1, \ldots, v_n\}\) on the other electrodes independently, and we

 can also measure the $n$ different conjugate displacements, as the

 charges \(\{q_1,\ldots, q_n\}\) accumulated on electrodes 

 \(\{e_1,\ldots e_n\}\). Together with the pressure (understood as the

 pressure measured at one given position), volume, and temperature, our

 sample of nitrobenzene is now a thermodynamic system of $(n+1)$

 degrees of freedom. This number may be as large as we please, limited

 only by our patience in constructing the apparatus needed to control

 or measure all these quantities.

 

 We leave it as an exercise for the reader (Problem 1) to find the most

 general condition on the variables \(\{v_1, q_1, v_2, q_2, \ldots

 v_n,q_n\}\) which will ensure that a definite equation of state

 $f(P,V,t)=0$ is observed in spite of all these new degrees of

 freedom. The simplest special case of this relation is, evidently, to

 ground all electrodes, thereby inposing the conditions $v_1 = v_2 =

 \ldots = v_n = 0$. Equally well (if we regard nitrobenzene as having

 negligible electrical conductivity) we may open-circuit all

 electrodes, thereby imposing the conditions \(q_i = \text{const.}\) In

 the latter case, in addition to an equation of state of the form

 \(f(P,V,t)=0\), which contains these constants as fixed parameters,

 there are \(n\) additional equations of state of the form $v_i =

 v_i(P,t)$. But if we choose to ignore these voltages, there will be no

 contradiction in considering our nitrobenzene to be a thermodynamic

 system of two degrees of freedom, involving only the variables

 \(P,V,t\).

 

 Similarly, if our system of interest is a crystal, we may impose on it

 a wide variety of nonuniform stress fields; each component of the

 stress tensor $T_{ij}$ may bary with position. We might expand each of

 these functions in a complete orthonormal set of functions

 \(\phi_k(x,y,z)\):

 

 \begin{equation}

 T_{ij}(x,y,z) = \sum_k a_{ijk} \phi_k(x,y,z)

 \end{equation}

 

 and with a sufficiently complicated system of levers which in various

 ways squeeze and twist the crystal, we might vary each of the first

 1,000 expansion coefficients $a_{ijk}$ independently, and measure the

 conjugate displacements $q_{ijk}$. Our crystal is then a thermodynamic

 system of over 1,000 degrees of freedom. 

 

 The notion of \ldquo{}numbers of degrees of freedom\rdquo{} is

 therefore not a /physical property/ of any system; it is entirely

 anthropomorphic, since any physical system may be regarded as a

 thermodynamic system with any number of degrees of freedom we please.

 

 If new thermodynamic variables are always introduced in pairs,

 consisting of a \ldquo{}force\rdquo{} and conjugate

 \ldquo{}displacement\rdquo{}, then a thermodynamic system of $n$

 degrees of freedom must possess $(n-1)$ independent equations of

 state, so that specifying $n$ quantities suffices to determine all

 others.

 

 This raises an interesting question; whether the scheme of classifying

 thermodynamic variables in conjugate pairs is the most general

 one. Why, for example, is it not natural to introduce three related

 variables at a time? To the best of the writer's knowledge, this is an

 open question; there seems to be no fundamental reason why variables

 /must/ always be introduced in conjugate pairs, but there seems to be

 no known case in which a different scheme suggests itself as more

 appropriate.

 

 ** Heat

 We are now in a position to consider the results and interpretation of

 a number of elementary experiments involving

 

 

 * Appendix

 

 | Generalized Force  | Generalized Displacement |

 |--------------------+--------------------------|

 | force              | displacement             |

 | pressure           | volume                   |

 | electric potential | charge                   |