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  [[../sources/stat.mech.1.pdf][this unpublished book draft]], written by [[http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E.T. Jaynes]].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

 

  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 1.1) 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

 thermal interaction, which can be carried out as soon as a primitive

 thermometer is at hand. In fact these experiments, which we summarize

 so quickly, required a very long time for their first performance, and

 the essential conclusions of this Section were first arrived at only

 about 1760---more than 160 years after Galileo's invention of the

 thermometer---[[http://web.lemoyne.edu/~giunta/blackheat.html][by Joseph Black]], who was Professor of Chemistry at

 Glasgow University. Black's analysis of calorimetric experiments

 initiated by G. D. Fahrenheit before 1736 led to the first recognition

 of the distinction between temperature and heat, and prepared the way

 for the work of his better-known pupil, James Watt.

 

 We first observe that if two bodies at different temperatures are

 separated by walls of various materials, they sometimes maintain their

 temperature difference for a long time, and sometimes reach thermal

 equilibrium very quickly. The differences in behavior observed must be

 ascribed to the different properties of the separating walls, since

 nothing else is changed. Materials such as wood, asbestos, porous

 ceramics (and most of all, modern porous plastics like styrofoam), are

 able to sustain a temperature difference for a long time; a wall of an

 imaginary material with this property idealized to the point where a

 temperature difference is maintained indefinitely is called an

 /adiabatic wall/. A very close approximation to a perfect adiabatic

 wall is realized by the Dewar flask (thermos bottle), of which the

 walls consist of two layers of glass separated by a vacuum, with the

 surfaces silvered like a mirror. In such a container, as we all know,

 liquids may be maintained hot or cold for days.

 

 On the other hand, a thin wall of copper or silver is hardly able to

 sustain any temperature difference at all; two bodies separated by

 such a partition come to thermal equilibrium very quickly. Such a wall

 is called /diathermic/. It is found in general that the best

 diathermic materials are the metals and good electrical conductors,

 while electrical insulators make fairly good adiabatic walls. There

 are good theoretical reasons for this rule; a particular case of it is

 given by the [[http://en.wikipedia.org/wiki/Wiedemann_franz_law][Wiedemann-Franz law]] of solid-state theory.

 

 Since a body surrounded by an adiabatic wall is able to maintain its

 temperature independently of the temperature of its surroundings, an

 adiabatic wall provides a means of thermally /isolating/ a system from

 the rest of the universe; it is to be expected, therefore, that the

 laws of thermal interaction between two systems will assume the

 simplest form if they are enclosed in a common adiabatic container,

 and that the best way of carrying out experiments on thermal

 peroperties of substances is to so enclose them. Such an apparatus, in

 which systems are made to interact inside an adiabatic container

 supplied with a thermometer, is called a /calorimeter/.

 

 Let us imagine that we have a calorimeter in which there is initially

 a volume $V_W$ of water at a temperature $t_1$, and suspended above it

 a volume $V_I$ of some other substance (say, iron) at temperature

 $t_2$. When we drop the iron into the water, they interact thermally

 (and the exact nature of this interaction is one of the things we hope

 to learn now), the temperature of both changing until they are in

 thermal equilibrium at a final temperature $t_0$.

 

 Now we repeat the experiment with different initial temperatures

 $t_1^\prime$ and $t_2^\prime$, so that a new equilibrium is reached at

 temperature $t_0^\prime$. It is found that, if the temperature

 differences are sufficiently small (and in practice this is not a

 serious limitation if we use a mercury thermometer calibrated with

 uniformly spaced degree marks on a capillary of uniform bore), then

 whatever the values of $t_1^\prime$, $t_2^\prime$, $t_1$, $t_2$, the

 final temperatures $t_0^\prime$, $t_0$ will adjust themselves so that

 the following relation holds:

 

 \begin{equation}

 \frac{t_2 - t_0}{t_0 - t_1} = \frac{t_2^\prime -

 t_0^\prime}{t_0^\prime - t_1^\prime}

 \end{equation} 

 

 in other words, the /ratio/ of the temperature changes of the iron and

 water is independent of the initial temperatures used.

 

 We now vary the amounts of iron and water used in the calorimeter. It

 is found that the ratio (1-8), although always independent of the

 starting temperatures, does depend on the relative amounts of iron and

 water. It is, in fact, proportional to the mass $M_W$ of water and

 inversely proportional to the mass $M_I$ of iron, so that 

 

 \begin{equation}

 \frac{t_2-t_0}{t_0-t_1} = \frac{M_W}{K_I M_I}

 \end{equation}

 

 where $K_I$ is a constant.

 

 We next repeat the above experiments using a different material in

 place of the iron (say, copper). We find again a relation

 

 \begin{equation}

 \frac{t_2-t_0}{t_0-t_1} = \frac{M_W}{K_C \cdot M_C}

 \end{equation}

 

 where $M_C$ is the mass of copper; but the constant $K_C$ is different

 from the previous $K_I$. In fact, we see that the constant $K_I$ is a

 new physical property of the substance iron, while $K_C$ is a physical

 property of copper. The number $K$ is called the /specific heat/ of a

 substance, and it is seen that according to this definition, the

 specific heat of water is unity.

 

 We now have enough experimental facts to begin speculating about their

 interpretation, as was first done in the 18th century. First, note

 that equation (1-9) can be put into a neater form that is symmetrical

 between the two substances. We write $\Delta t_I = t_0 - t_2$, $\Delta

 t_W = t_0 - t_1$ for the temperature changes of iron and water

 respectively, and define $K_W \equiv 1$ for water. Equation (1-9) then

 becomes

 

 \begin{equation}

 K_W M_W \Delta t_W + K_I M_I \Delta t_I = 0

 \end{equation}

 

 The form of this equation suggests a new experiment; we go back into

 the laboratory, and find $n$ substances for which the specific heats

 \(\{K_1,\ldots K_n\}\) have been measured previously. Taking masses

 \(\{M_1, \ldots, M_n\}\) of these substances, we heat them to $n$

 different temperatures \(\{t_1,\ldots, t_n\}\) and throw them all into

 the calorimeter at once. After they have all come to thermal

 equilibrium at temperature $t_0$, we find the differences $\Delta t_j

 = t_0 - t_j$. Just as we suspected, it turns out that regardless of

 the $K$'s, $M$'s, and $t$'s chosen, the relation

 \begin{equation}

 \sum_{j=0}^n K_j M_j \Delta t_j = 0

 \end{equation}

 is always satisfied. This sort of process is an old story in

 scientific investigations; although the great theoretician Boltzmann

 is said to have remarked: \ldquo{}Elegance is for tailors\rdquo{}, it

 remains true that the attempt to reduce equations to the most

 symmetrical form has often suggested important generalizations of

 physical laws, and is a great aid to memory. Witness Maxwell's

 \ldquo{}displacement current\rdquo{}, which was needed to fill in a

 gap and restore the symmetry of the electromagnetic equations; as soon

 as it was put in, the equations predicted the existence of

 electromagnetic waves. In the present case, the search for a rather

 rudimentary form of \ldquo{}elegance\rdquo{} has also been fruitful,

 for we recognize that (1-12) has the standard form of a /conservation

 law/; it defines a new quantity which is conserved in thermal

 interactions of the type just studied.

 

 The similarity of (1-12) to conservation laws in general may be seen

 as follows. Let $A$ be some quantity that is conserved; the \(i\)th

 system has an amount of it $A_i$. Now when the systems interact such

 that some $A$ is transferred between them, the amount of $A$ in the

 \(i\)th system is changed by a net amount \(\Delta A_i = (A_i)_{final} -

 (A_i)_{initial}\); and the fact that there is no net change in the

 total amount of $A$ is expressed by the equation \(\sum_i \Delta

 A_i = 0\). Thus, the law of conservation of matter in a chemical

 reaction is expressed by \(\sum_i \Delta M_i = 0\), where $M_i$ is the

 mass of the \(i\)th chemical component.

 

 What is this new conserved quantity? Mathematically, it can be defined

 as $Q_i = K_i\cdot M_i \cdot t_i$; whereupon (1-12) becomes 

 

 \begin{equation}

 \sum_i \Delta Q_i = 0

 \end{equation}

 

 and at this point we can correct a slight quantitative inaccuracy. As

 noted, the above relations hold accurately only when the temperature

 differences are sufficiently small; i.e., they are really only

 differential laws. On sufficiently accurate measurements one find that

 the specific heats $K_i$ depend on temperature; if we then adopt the

 integral definition of $\Delta Q_i$,

 \begin{equation}

 \Delta Q_i = \int_{t_{i}}^{t_0} K_i(t) M_i dt

 \end{equation}

 

 the conservation law (1-13) will be found to hold in calorimetric

 experiments with liquids and solids, to any accuracy now feasible. And

 of course, from the manner in which the $K_i(t)$ are defined, this

 relation will hold however our thermometers are calibrated.

 

 Evidently, the stage is now set for a \ldquo{}new\rdquo{} physical

 theory to account for these facts. In the 17th century, both Francis

 Bacon and Isaac Newton had expressed their opinions that heat was a

 form of motion; but they had no supporting factual evidence. By the

 latter part of the 18th century, one had definite factual evidence

 which seemed to make this view untenable; by the calorimetric

 \ldquo{}mixing\rdquo{} experiments just described, Joseph Black had

 recognized the distinction between temperature $t$ as a measure of

 \ldquo{}hotness\rdquo{}, and heat $Q$ as a measure of /quantity/ of

 something, and introduced the notion of heat capacity. He also

 recognized the latent heats of freezing and vaporization. To account

 for the conservation laws thus discovered, the theory then suggested

 itself, naturally and almost inevitably, that heat was /fluid/,

 indestructable and uncreatable, which had no appreciable weight and

 was attracted differently by different kinds of matter. In 1787,

 Lavoisier invented the name \ldquo{}caloric\rdquo{} for this fluid.

 

 Looking down today from our position of superior knowledge (i.e.,

 hindsight) we perhaps need to be reminded that the caloric theory was

 a perfectly respectable scientific theory, fully deserving of serious

 consideration; for it accounted quantitatively for a large body of

 experimental fact, and made new predictions capable of being tested by

 experiment.

 

 One of these predictions was the possibility of accounting for the

 thermal expansion of bodies when heated; perhaps the increase in

 volume was just a measure of the volume of caloric fluid

 absorbed. This view met with some disappointment as a result of

 experiments which showed that different materials, on absorbing the

 same quantity of heat, expanded by different amounts. Of course, this

 in itself was not enough to overthrow the caloric theory, because one

 could suppose that the caloric fluid was compressible, and was held

 under different pressure in different media.

 

 Another difficulty that seemed increasingly serious by the end of the

 18th century was the failure of all attempts to weigh this fluid. Many

 careful experiments were carried out, by Boyle, Fordyce, Rumford and

 others (and continued by Landolt almost into the 20th century), with

 balances capable of detecting a change of weight of one part in a

 million; and no change could be detected on the melting of ice,

 heating of substances, or carrying out of chemical reactions. But even

 this is not really a conclusive argument against the caloric theory,

 since there is no /a priori/ reason why the fluid should be dense

 enough to weigh with balances (of course, we know today from

 Einstein's $E=mc^2$ that small changes in weight should indeed exist

 in these experiments; but to measure them would require balances about

 10^7 times more sensitive than were available).

 

 Since the caloric theory derives entirely from the empirical

 conservation law (1-33), it can be refuted conclusively only by

 exhibiting new experimental facts revealing situations in which (1-13)

 is /not/ valid. The first such case was [[http://www.chemteam.info/Chem-History/Rumford-1798.html][found by Count Rumford (1798)]],

 who was in charge of boring cannon in the Munich arsenal, and noted

 that the cannon and chips became hot as a result of the cutting. He

 found that heat could be produced indefinitely, as long as the boring

 was continued, without any compensating cooling of any other part of

 the system. Here, then, was a clear case in which caloric was /not/

 conserved, as in (1-13); but could be created at will. Rumford wrote

 that he could not conceive of anything that could be produced

 indefinitely by the expenditure of work, \ldquo{}except it be /motion/\rdquo{}.

 

 But even this was not enough to cause abandonment of the caloric

 theory; for while Rumford's observations accomplished the negative

 purpose of showing that the conservation law (1-13) is not universally

 valid, they failed to accomplish the positive one of showing what

 specific law should replace it (although he produced a good hint, not

 sufficiently appreciated at the time, in his crude measurements of the

 rate of heat production due to the work of one horse). Within the

 range of the original calorimetric experiments, (1-13) was still

 valid, and a theory successful in a restricted domain is better than

 no theory at all; so Rumford's work had very little impact on the

 actual development of thermodynamics.

 

 (This situation is a recurrent one in science, and today physics offers

 another good example. It is recognized by all that our present quantum

 field theory is unsatisfactory on logical, conceptual, and

 mathematical grounds; yet it also contains some important truth, and

 no responsible person has suggested that it be abandoned. Once again,

 a semi-satisfactory theory is better than none at all, and we will

 continue to teach it and to use it until we have something better to

 put in its place.)

 

 # what is "the specific heat of a gas at constant pressure/volume"?

 # changed t for temperature below from capital T to lowercase t.

 Another failure of the conservation law (1-13) was [[http://web.lemoyne.edu/~giunta/mayer.html][noted in 1842]] by

 R. Mayer, a German physician, who pointed out that the data already

 available showed that the specific heat of a gas at constant pressure, 

 C_p, was greater than at constant volume $C_v$. He surmised that the

 difference was due to the work done in expansion of the gas against

 atmospheric pressure, when measuring $C_p$. Supposing that the

 difference $\Delta Q = (C_p - C_v)\Delta t$ calories, in the heat

 required to raise the temperature by $\Delta t$ was actually a

 measure of amount of energy, he could estimate from the amount

 $P\Delta V$ ergs of work done the amount of mechanical energy (number

 of ergs) corresponding to a calorie of heat; but again his work had

 very little impact on the development of thermodynamics, because he

 merely offered this notion as an interpretation of the data without

 performing or suggesting any new experiments to check his hypothesis

 further.

 

 Up to the point, then, one has the experimental fact that a

 conservation law (1-13) exists whenever purely thermal interactions

 were involved; but in processes involving mechanical work, the

 conservation law broke down.

 

 ** The First Law

 Corresponding to the partially valid law of \ldquo{}conservation of

 heat\rdquo{}, there had long been known another partially valid

 conservation law in mechanics. The principle of conservation of

 mechanical energy had been given by Leibnitz in 1693 in noting that,

 according to the laws of Newtonian mechanics, one could define

 potential and kinetic energy so that in mechanical processes they were

 interconverted into each other, the total energy remaining

 constant. But this too was not universally valid---the mechanical

 energy was conserved only in the absence of frictional forces. In

 processes involving friction, the mechanical energy seemed to

 disappear.

 

 So we had a law of conservation of heat, which broke down whenever

 mechanical work was done; and a law of conservation of mechanical

 energy, which broke down when frictional forces were present. If, as

 Mayer had suggested, heat was itself a form of energy, then one had

 the possibility of accounting for both of these failures in a new law

 of conservation of /total/ (mechanical + heat) energy. On one hand,

 the difference $C_p-C_v$ of heat capacities of gases would be

 accounted for by the mechanical work done in expansion; on the other

 hand, the disappearance of mechanical energy would be accounted for by

 the heat produced by friction.

 

 But to establish this requires more than just suggesting the idea and

 illustrating its application in one or two cases --- if this is really

 a new conservation law adequate to replace the two old ones, it must

 be shown to be valid for /all/ substances and /all/ kinds of

 interaction. For example, if one calorie of heat corresponded to $E$

 ergs of mechanical energy in the gas experiments, but to a different

 amoun $E^\prime$ in heat produced by friction, then there would be no

 universal conservation law. This \ldquo{}first law\rdquo{} of

 thermodynamics must therefore take the form:

 #+begin_quote

 There exists a /universal/ mechanical equivalent of heat, so that the

 total (mechanical energy) + (heat energy) remeains constant in all

 physical processes.

 #+end_quote

 

 It was James Prescott Joule who provided the [[http://www.chemteam.info/Chem-History/Joule-Heat-1845.html][first experimental data]]

 indicating this universality, and providing the first accurate

 numerical value of this mechanical equivalent. The calorie had been

 defined as the amount of heat required to raise the temperature of one

 gram of water by one degree Centigrade (more precisely, to raise it

 from 14.5 to 15.5$^\circ C$). Joule measured the heating of a number

 of different liquids due to mechanical stirring and electrical

 heating, and established that, within the experimental accuracy (about

 one percent) a /calorie/ of heat always corresponded to the same

 amount of energy. Modern measurements give this numerical value as: 1

 calorie = 4.184 \times 10^7 ergs = 4.184 joules.

 # capitalize Joules? I think the convention is to spell them out in lowercase.

 

 The circumstances of this important work are worth noting. Joule was

 in frail health as a child, and was educated by private tutors,

 including the chemist, John Dalton, who had formulated the atomic

 hypothesis in the early nineteenth century. In 1839, when Joule was

 nineteen, his father (a wealthy brewer) built a private laboratory for

 him in Manchester, England; and the good use he made of it is shown by

 the fact that, within a few months of the opening of this laboratory

 (1840), he had completed his first important piece of work, at the

 age of twenty. This was his establishment of the law of \ldquo{}Joule

 heating,\rdquo{} $P=I^2 R$, due to the electric current in a

 resistor. He then used this effect to determine the universality and

 numerical value of the mechanical equivalent of heat, reported

 in 1843. His mechanical stirring experiments reported in 1849 yielded

 the value 1 calorie = 4.154 \times 10^7 ergs, amount 0.7% too low;

 this determination was not improved upon for several decades.

 

 The first law of thermodynamics may then be stated mathematically as

 follows:  

 

 #+begin_quote

 There exists a state function (i.e., a definite function of the

 thermodynamic state) $U$, representing the total energy of any system,

 such that in any process in which we change from one equilibrium to

 another, the net change in $U$ is given by the difference of the heat

 $Q$ supplied to the system, and the mechanical work $W$ done by the

 system.

 #+end_quote

 On an infinitesimal change of state, this becomes

 

 \begin{equation}

 dU = dQ - dW.

 \end{equation}

 

 For a system of two degrees of freedom, defined by pressure $P$,

 volume $V$, and temperature $t$, we have $dW = PdV$. Then if we regard

 $U$ as a function $U(V,t)$ of volume and temperature, the fact that

 $U$ is a state function means that $dU$ must be an exact differential;

 i.e., the integral

 

 \begin{equation}

 \int_1^2 dU = U(V_2,t_2) - U(V_1,t_1)

 \end{equation}

 between any two thermodynamic states must be independent of the

 path. Equivalently, the integral $\oint dU$ over any closed cyclic

 path  (for example, integrate from state 1 to state 2 along path A,

 then back to state 1 by a different path B) must be zero. From (1-15),

 this gives for any cyclic integral,

 

 \begin{equation}

 \oint dQ = \oint P dV

 \end{equation}

 

 another form of the first law, which states that in any process in

 which the system ends in the same thermodynamic state as the initial

 one, the total heat absorbed by the system must be equal to the total

 work done.

 

 Although the equations (1-15)-(1-17) are rather trivial

 mathematically, it is important to avoid later conclusions that we

 understand their exact meaning. In the first place, we have to

 understand that we are now measuring heat energy and mechanical energy

 in the same units; i.e. if we measured $Q$ in calories and $W$ in

 ergs, then (1-15)  would of course not be correct. It does

 not matter whether we apply Joule's mechanical equivalent of heat

 to express $Q$ in ergs, or whether we apply it in the opposite way

 to express $U$ and $W$ in calories; each procedure will be useful in

 various problems. We can develop the general equations of

 thermodynamics 

 without committing ourselves to any particular units,

 but of course all terms in a given equation must be expressed

 in the same units.

 

 Secondly, we have already stressed that the theory being

 developed must, strictly speaking, be a theory only of 

 equilibrium states, since otherwise we have no operational definition

 of temperature When we integrate over any \ldquo{}path\rdquo{} in the $(V-t)$

 plane, therefore, it must be understood that the path of 

 integration is, strictly speaking, just a /locus of equilibrium

 states/; nonequilibrium states cannot be represented by points

 in the $(V-t)$ plane.

 

 But then, what is the relation between path of equilibrium

 states appearing in our equations, and the sequence of conditions

 produced experimentally when we change the state of a system in

 the laboratory? With any change of state (heating, compression,

 etc.) proceeding at a finite rate we do not have equilibrium in

 termediate states; and so there is no corresponding \ldquo{}path\rdquo{} in

 the $(V-t)$ plane ; only the initial and final equilibrium states

 correspond to definite points. But if we carry out the change

 of state more and more slowly, the physical states produced are

 nearer and nearer to equilibrium state. Therefore, we interpret

 a path of integration in the $(V-t)$ plane, not as representing

 the intermediate states of any real experiment carried out at

 a finite rate, but as the /limit/ of this sequence of states, in

 the limit where the change of state takes place arbitrarily

 slowly.

 

 An arbitrarily slow process, so that we remain arbitrarily

 near to equilibrium at all times, has another important property.

 If heat is flowing at an arbitrarily small rate, the temperature

 difference producing it must be arbitrarily small, and therefore

 an arbitrarily small temperature change would be able to reverse

 the direction of heat flow. If the Volume is changing very

 slowly, the pressure difference responsible for it must be very

 small; so a small change in pressure would be able to reverse

 the direction of motion. In other words, a process carried out

 arbitrarily slowly is /reversible/; if a system is arbitrarily

 close to equilibrium, then an arbitrarily small change in its

 environment can reverse the direction of the process.

 Recognizing this, we can then say that the paths of integra

 tion in our equations are to be interpreted physically as

 /reversible paths/ In practice, some systems (such as gases)

 come to equilibrium so rapidly that rather fast changes of

 state (on the time scale of our own perceptions) may be quite

 good approximations to reversible changes; thus the change of

 state of water vapor in a steam engine may be considered 

 reversible to a useful engineering approximation.

 

 

 ** Intensive and Extensive Parameters

 

 The literature of thermodynamics has long recognized a distinction between two

 kinds of quantities that may be used to define the thermodynamic

 state. If we imagine a given system as composed of smaller 

 subsystems, we usually find that some of the thermodynamic variables

 have the same values in each subsystem, while others are additive,

 the total amount being the sum of the values of each subsystem.

 These are called /intensive/ and /extensive/ variables, respectively.

 According to this definition, evidently, the mass of a system is

 always an extensive quantity, and at equilibrium the temperature

 is an intensive ‘quantity. Likewise, the energy will be extensive

 provided that the interaction energy between the subsystems can

 be neglected.

 

 It is important to note, however, that in general the terms

 \ldquo{}intensive\rdquo{} and \ldquo{}extensive\rdquo{} 

 so defined cannot be regarded as

 establishing a real physical distinction between the variables.

 This distinction is, like the notion of number of degrees of

 freedom, in part an anthropomorphic one, because it may depend

 on the particular kind of subdivision we choose to imagine. For

 example, a volume of air may be imagined to consist of a number

 of smaller contiguous volume elements. With this subdivision,

 the pressure is the same in all subsystems, and is therefore in

 tensive; while the volume is additive and therefore extensive

 But we may equally well regard the volume of air as composed of

 its constituent nitrogen and oxygen subsystems (or we could re

 gard pure hydrogen as composed of two subsystems, in which the

 molecules have odd and even rotational quantum numbers 

 respectively, etc.) With this kind of subdivision the volume is the

 same in all subsystems, while the pressure is the sum of the

 partial pressures of its constituents; and it appears that the

 roles of \ldquo{}intensive\rdquo{} and \ldquo{}extensive\rdquo{}

  have been interchanged. Note that this ambiguity cannot be removed by requiring

 that we consider only spatial subdivisions, such that each sub

 system has the same local composi tion For, consider a s tressed

 elastic solid, such as a stretched rubber band. If we imagine

 the rubber band as divided, conceptually, into small subsystems

 by passing planes through it normal to its axis, then the tension

 is the same in all subsystems, while the elongation is additive.

 But if the dividing planes are parallel to the axis, the elonga

 tion is the same in all subsystems, while the tension is 

 additive; once again, the roles of \ldquo{}extensive\rdquo{} and

 \ldquo{}intensive\rdquo{} are

 interchanged merely by imagining a different kind of subdivision.

 In spite of the fundamental ambiguity of the usual definitions, 

 the notions of extensive and intensive variables are useful, 

 and in practice we seem to have no difficulty in deciding

 which quantities should be considered intensive. Perhaps the

 distinction is better characterized, not by considering 

 subdivisions at all, but by adopting a different definition, in which

 we recognize that some quantities have the nature of a \ldquo{}force\rdquo{}

 or \ldquo{}potential\rdquo{}, or some other local physical property, and are

 therefore called intensive, while others have the nature of a

 \ldquo{}displacement\rdquo{} or a \ldquo{}quantity\rdquo{} of 

 something (i.e. are proportional to the size of the system), 

 and are therefore called extensive. Admittedly, this definition is somewhat vague, in a

 way that can also lead to ambiguities ; in any event, let us agree

 to class pressure, stress tensor, mass density, energy density,

 particle density, temperature, chemical potential, angular

 velocity, as intensive, while volume, mass, energy, particle

 numbers, strain, entropy, angular momentum, will be considered

 extensive.

 

 ** The Kelvin Temperature Scale

 The form of the first law,

 $dU = dQ - dW$, expresses the net energy increment of a system as

 the heat energy supplied to it, minus the work done by it. In

 the simplest systems of two degrees of freedom, defined by 

 pressure and volume as the thermodynamic variables, the work done

 in an infinitesimal reversible change of state can be separated

 into a product $dW = PdV$ of an intensive and an extensive quantity.

 Furthermore, we know that the pressure $P$ is not only the 

 intensive factor of the work; it is also the \ldquo{}potential\rdquo{} 

 which governs mechanical equilibrium (in this case, equilibrium with respect

 to exchange of volume) between two systems; ie., if they are

 separated by a flexible but impermeable membrane, the two systems

 will exchange volume $dV_1 = -dV_2$ in a direction determined by the

 pressure difference, until the pressures are equalized. The

 energy exchanged in this way between the systems is a product

 of the form

 #+begin_quote

 (/intensity/ of something) \times (/quantity/ of something exchanged)

 #+end_quote

 

 Now if heat is merely a particular form of energy that can

 also be exchanged between systems, the question arises whether

 the quantity of heat energy $dQ$ exchanged in an infinitesimal 

 reversible change of state can also be written as a product of one

 factor which measures the \ldquo{}intensity\rdquo{} of the heat, 

 times another that represents the \ldquo{}quantity\rdquo{}

  of something exchanged between

 the systems, such that the intensity factor governs the 

 conditions of thermal equilibrium and the direction of heat exchange,

 in the same way that pressure does for volume exchange.

 

 

 But we already know that the /temperature/ is the quantity

 that governs the heat flow (i.e., heat flows from the hotter to

 the cooler body until the temperatures are equalized) So the

 intensive factor in $dQ$ must be essentially the temperature. But

 our temperature scale is at present still arbitrary, and we can

 hardly expect that such a factorization will be possible for all

 calibrations of our thermometers.

 

 The same thing is evidently true of pressure; if instead of

 the pressure $P$ as ordinarily defined, we worked with any mono

 tonic increasing function $P_1 = P_1 (P)$ we would find that $P_1$ is

 just as good as $P$ for determining the direction of volume 

 exchange and the condition of mechanical equilibrium; but the work

 done would not be given by $PdV$; in general, it could not even

 be expressed in the form $P_1 \cdot dF(V)$, where $F(V)$ is some function

 of V.

 

 

 Therefore we ask: out of all the monotonic functions $t_1(t)$ 

 corresponding to different empirical temperature scales, is

 there one (which we denote as $T(t)$) which forms a \ldquo{}natural\rdquo{}

 intensity factor for heat, such that in a reversible change

 $dQ = TdS$, where $S(U,V)$ is a new function of the thermodynamic

 state? If so, then the temperature scale $T$ will have a great

 theoretical advantage, in that the laws of thermodynamics will

 take an especially simple form in terms of this particular scale,

 and the new quantity $S$, which we call the /entropy/, will be a

 kind of \ldquo{}volume\rdquo{} factor for heat.

 

 We recall that $dQ = dU + PdV$ is not an exact differential;

 i.e., on a change from one equilibrium state to another the

 integral

 

 \[\int_1^2 dQ\]

 

 cannot be set equal to the difference $Q_2 - Q_1$ of values of any

 state function $Q(U,V)$, since the integral has different values

 for different paths connecting the same initial and final states.

 Thus there is no \ldquo{}heat function\rdquo{} $Q(U,V)$, and the notion of

 \ldquo{}amount of heat\rdquo{} $Q$ stored in a body has no meaning 

 (nor does the \ldquo{}amount of work\rdquo{} $W$; 

 only the total energy is a well-defined quantity). 

 But we want the entropy $S(U,V)$ to be a definite quantity,

 like the energy or volume, and so $dS$ must be an exact differential.

 On an infinitesimal reversible change from one equilibrium state

 to another, the first law requires that it satisfy[fn:: Edit: The first

 equality comes from our requirement that $dQ = T\,dS$. The second

 equality comes from the fact that $dU = dQ - dW$ (the first law) and

 that $dW = PdV$ in the case where the state has two degrees of

 freedom, pressure and volume.]

 

 \begin{equation}

 dS(U,V) = \frac{dQ}{T} = \frac{dU}{T} + \frac{P}{T}dV

 \end{equation}

 

 Thus $(1/T)$ must be an /integrating factor/ which converts $dQ$ into

 an exact differential [[fn::Edit: A differential $M(x,y)dx +

 N(x,y)dy$ is called /exact/ if there is a scalar function

 $\Phi(x,y)$ such that $M = \frac{\partial \Phi}{\partial x}$ and

 $N=\frac{\partial \Phi}{\partial y}$. If there is, \Phi is called the

 /potential function/ of the differential, Conceptually, this means

 that M(x,y)dx + N(x,y) dy is the derivative of a scalar potential and

 so consequently corresponds to a conservative field.

  

 Even if there is no such potential function

 \Phi for the given differential, it is possible to coerce an

 inexact differential into an exact one by multiplying by an unknown

 function $\mu(x,y)$ (called an /integrating factor/) and requiring the

 resulting differential $\mu M\, dx + \mu N\, dy$ to be exact.

 

 To complete the analogy, here we have the differential $dQ =

 dU + PdV$ (by the first law) which is not exact---conceptually, there

 is no scalar potential nor conserved quantity corresponding to

 $dQ$. We have introduced a new differential $dS = \frac{1}{T}dQ$, and we

 are searching for the temperature scale $T(U,V)$ which makes $dS$

 exact (i.e. which makes $S$ correspond to a conserved quantity). This means

 that $\frac{1}{T}$ is playing the role of the integrating factor

 \ldquo{}\mu\rdquo{} for the differential $dQ$.]]

 

 Now the question of the existence and properties of 

 integrating factors is a purely mathematical one, which can be 

 investigated independently of the properties of any particular

 substance. Let us denote this integrating factor for the moment

 by $w(U,V) = T^{-1}$; then the first law becomes

 

 \begin{equation}

 dS(U,V) = w dU + w P dV

 \end{equation}

 

 from which the derivatives are

 

 \begin{equation}

 \left(\frac{\partial S}{\partial U}\right)_V = w, \qquad

 \left(\frac{\partial S}{\partial V}\right)_U = wP. 

 \end{equation}

 

 The condition that $dS$ be exact is that the cross-derivatives be

 equal, as in (1-4):

 

 \begin{equation}

 \frac{\partial^2 S}{\partial U \partial V} = \frac{\partial^2

 S}{\partial V \partial U},

 \end{equation}

 

 or 

 

 \begin{equation}

 \left(\frac{\partial w}{\partial V}\right)_U = \left(\frac{\partial

 P}{\partial U}\right)_V + P\cdot \left(\frac{\partial w}{\partial U}\right)_V.

 \end{equation}

 

 Any function $w(U,V)$ satisfying this differential equation is an

 integrating factor for $dQ$.

 

 But if $w(U,V)$ is one such integrating factor, which leads

 to the new state function $S(U,V)$, it is evident that

 $w_1(U,V) \equiv w \cdot f(S)$ is an equally good integrating factor, where

 $f(S)$ is an arbitrary function. Use of $w_1$ will lead to a 

 different state function 

 

 #what's with the variable collision?

 \begin{equation}

 S_1(U,V) = \int^S f(S) dS

 \end{equation}

 

 The mere conversion of into an exact differential is, therefore, 

 not enough to determine any unique entropy function $S(U,V)$.

 However, the derivative

 

 \begin{equation}

 \left(\frac{dU}{dV}\right)_S = -P

 \end{equation}

 

 is evidently uniquely determined; so also, therefore, is the

 family of lines of constant entropy, called /adiabats/, in the

 $(U-V)$ plane. But, as (1-24) shows, the numerical value of $S$ on

 each adiabat is still completely undetermined.

 

 In order to fix the relative values of $S$ on different 

 adiabats we need to add the condition, not yet put into the equations, 

 that the integrating factor $w(U,V) = T^{-1}$ is to define a new

 temperature scale In other words, we now ask: out of the

 infinite number of different integrating factors allowed by

 the differential equation (1-23), is it possible to find one

 which is a function only of the empirical temperature $t$? If

 $w=w(t)$, we can write

 

 \begin{equation}

 \left(\frac{\partial w}{\partial V}\right)_U = \frac{dw}{dt}\left(\frac{\partial

 t}{\partial V}\right)_U

 \end{equation}

 \begin{equation}

 \left(\frac{\partial w}{\partial U}\right)_V = \frac{dw}{dt}\left(\frac{\partial

 t}{\partial U}\right)_V

 \end{equation}

 

 

 and (1-23) becomes

 \begin{equation}

 \frac{d}{dt}\log{w} = \frac{\left(\frac{\partial P}{\partial

 U}\right)_V}{\left(\frac{\partial t}{\partial V}\right)_U-P\left(\frac{\partial t}{\partial U}\right)_V}

 \end{equation}

 

 

 which shows that $w$ will be determined to within a multiplicative

 factor.

 

 Is the temperature scale thus defined independent of the

 empirical scale from which we started? To answer this, let

 $t_1 = t_1(t)$ be any monotonic function which defines a different

 empirical temperature scale. In place of (1-28), we then have

 

 \begin{equation}

 \frac{d}{dt_1}\log{w} \quad=\quad \frac{\left(\frac{\partial P}{\partial

 U}\right)_V}{\left(\frac{\partial t_1}{\partial V}\right)_U-P\left(\frac{\partial t_1}{\partial U}\right)_V}

 \quad = \quad 

  \frac{\left(\frac{\partial P}{\partial

 U}\right)_V}{\frac{dt_1}{dt}\left[ \left(\frac{\partial t}{\partial

 V}\right)_U-P\left(\frac{\partial t}{\partial U}\right)_V\right]},

 \end{equation}

 or

 \begin{equation}

 \frac{d}{dt_1}\log{w_1} = \frac{dt}{dt_1}\frac{d}{dt}\log{w}

 \end{equation}

 

 which reduces to $d \log{w_1} = d \log{w}$, or

 \begin{equation}

 w_1 = C\cdot w

 \end{equation}

 

 Therefore, integrating factors derived from whatever empirical

 temperature scale can differ among themselves only by a 

 multiplicative factor. For any given substance, therefore, except

 for this factor (which corresponds just to our freedom to choose

 the size of the units in which we measure temperature), there is

 only /one/ temperature scale $T(t) = 1/w$ with the property that

 $dS = dQ/T$ is an exact differential.

 

 To find a feasible way of realizing this temperature scale

 experimentally, multiply numerator and denominator of the right

 hand side of (1-28) by the heat capacity at constant volume,

 $C_V^\prime = (\partial U/\partial t) V$, the prime denoting that 

 it is in terms of the empirical temperature scale $t$. 

 Integrating between any two states denoted 1 and 2, we have

 

 \begin{equation}

 \frac{T_1}{T_2} = \exp\left\{\int_{t_1}^{t_2}

 \frac{\left(\frac{\partial P}{\partial t}\right)_V dt}{P - C_V^\prime

 \left(\frac{\partial t}{\partial V}\right)_U} \right\}

 \end{equation}

 

 If the quantities on the right-hand side have been determined

 experimentally, then a numerical integration yields the ratio

 of Kelvin temperatures of the two states.

 

 This process is particularly simple if we choose for our

 system a volume of gas with the property found in Joule's famous

 expansion experiment; when the gas expands freely into a vacuum

 (i.e., without doing work, or $U = \text{const.}$), there is no change in

 temperature. Real gases when sufficiently far from their condensation 

 points are found to obey this rule very accurately.

 But then 

 

 \begin{equation}

 \left(\frac{dt}{dV}\right)_U = 0

 \end{equation}

 

 and on a change of state in which we heat this gas at constant

 volume, (1-31) collapses to

 

 \begin{equation}

 \frac{T_1}{T_2} = \exp\left\{\int_{t_1}^{t_2}

 \frac{1}{P}\left(\frac{\partial P}{\partial t}\right)_V dt\right\} = \frac{P_2}{P_1}.

 \end{equation}

 

 Therefore, with a constant-volume ideal gas thermometer, (or more

 generally, a thermometer using any substance obeying (1-32) and

 held at constant volume), the measured pressure is directly 

 proportional to the Kelvin temperature.

 

 For an imperfect gas, if we have measured  $(\partial t /\partial

 V)_U$ and $C_V^\prime$, Eq. (1-31) determines the necessary

 corrections to (1-33). However, an alternative form of (1-31), in

 which the roles of pressure and volume are interchanged, proves to be

 more convenient for experimental determinations. To derive it, introduce the

 enthalpy function

 

 \begin{equation}H = U + PV\end{equation}

 

 with the property

 

 \begin{equation}

 dH = dQ + VdP

 \end{equation}

 

 Equation (1-19) then becomes

 

 \begin{equation}

 dS = \frac{dH}{T} - \frac{V}{T}dP.

 \end{equation}

 

 Repeating the steps (1-20) to (1-31) of the above derivation

 starting from (1-36) instead of from (1-19), we arrive at

 

 \begin{equation}

 \frac{T_2}{T_1} = \exp\left\{\int_{t_1}^{t_2}

 \frac{\left(\frac{dV}{dt}\right)_P dt}{V + C_P^\prime

 \left(\frac{\partial t}{\partial P}\right)_H}\right\}

 \end{equation}

 

 or

 

 \begin{equation}

 \frac{T_2}{T_1} = \exp\left\{\frac{\alpha^\prime

 dt}{1+\left(C_P^\prime \cdot \mu^\prime / V\right)}\right\}

 \end{equation}

 

 where

 \begin{equation}

 \alpha^\prime \equiv \frac{1}{V}\left(\frac{\partial V}{\partial t}\right)_P

 \end{equation}

 is the thermal expansion coefficient,

 \begin{equation}

 C_P^\prime \equiv \left(\frac{\partial H}{\partial t}\right)_P

 \end{equation}

 is the heat capacity at constant pressure, and

 \begin{equation}

 \mu^\prime \equiv \left(\frac{dt}{dP}\right)_H

 \end{equation}

 

 is the coefficient measured in the Joule-Thompson porous plug

 experiment, the primes denoting again that all are to be measured

 in terms of the empirical temperature scale $t$.

 Since $\alpha^\prime$, $C_P^\prime$, $\mu^\prime$ are all 

 easily measured in the laboratory, Eq. (1-38) provides a 

 feasible way of realizing the Kelvin temperature scale experimentally, 

 taking account of the imperfections of real gases. 

 For an account of the work of Roebuck and others based on this

 relation, see [[http://books.google.com/books?id=KKJKAAAAMAAJ][Zemansky (1943)]]; pp. 252-255.

 

 Note that if $\mu^\prime = O$ and we heat the gas at constant

 pressure, (1-38) reduces to

 

 \begin{equation}

 \frac{T_2}{T_1} = \exp\left\{ \int_{t_1}^{t_2}

 \frac{1}{V}\left(\frac{\partial V}{\partial t}\right)_P dt \right\} = \frac{V_2}{V_1}

 \end{equation}

 

 so that, with a constant-pressure gas thermometer using a gas for

 which the Joule-Thomson coefficient is zero, the Kelvin temperature is

 proportional to the measured volume.

 

 Now consider another empirical fact, [[http://en.wikipedia.org/wiki/Boyle%27s_law][Boyle's law]]. For gases

 sufficiently far from their condensation points---which is also

 the condition under which (1-32) is satisfied---Boyle found that

 the product $PV$ is a constant at any fixed temperature. This

 product is, of course proportional to the number of moles $n$

 present, and so Boyle's equation of state takes the form

 

 \begin{equation}PV = n \cdot f(t)\end{equation}

 

 where f(t) is a function that depends on the particular empirical

 temperature scale used. But from (1-33) we must then have 

 $f(t) = RT$, where $R$ is a constant, the universal gas constant whose

 numerical value (1.986 calories per mole per degree K), depends

 on the size of the units in which we choose to measure the Kelvin

 temperature $T$. In terms of the Kelvin temperature, the ideal gas

 equation of state is therefore simply

 

 \begin{equation}

 PV = nRT

 \end{equation}

 

 

 The relations (1-32) and (1-44) were found empirically, but

 with the development of thermodynamics one could show that they

 are not logically independent. In fact, all the material needed

 for this demonstration is now at hand, and we leave it as an

 exercise for the reader to prove that Joule‘s relation (1-32) is

 a logical consequence of Boyle's equation of state (1-44) and the

 first law.

 

 

 Historically, the advantages of the gas thermometer were

 discovered empirically before the Kelvin temperature scale was

 defined; and the temperature scale \theta defined by

 

 \begin{equation}

 \theta = \lim_{P\rightarrow 0}\left(\frac{PV}{nR}\right)

 \end{equation}

 

 was found to be convenient, easily reproducible, and independent

 of the properties of any particular gas. It was called the

 /absolute/ temperature scale; and from the foregoing it is clear

 that with the same choice of the numerical constant $R$, the 

 absolute and Kelvin scales are identical.

 

 

 For many years the unit of our temperature scale was the

 Centigrade degree, so defined that the difference $T_b - T_f$ of

 boiling and freezing points of water was exactly 100 degrees.

 However, improvements in experimental techniques have made another

 method more reproducible; and the degree was redefined by the

 Tenth General Conference of Weights and Measures in 1954, by

 the condition that the triple point of water is at 273.l6^\circ K,

 this number being exact by definition. The freezing point, 0^\circ C,

 is then 273.15^\circ K. This new degree is called the Celsius degree.

 For further details, see the U.S. National Bureau of Standards

 Technical News Bulletin, October l963.

 

 

 The appearance of such a strange and arbitrary-looking

 number as 273.16 in the /definition/ of a unit is the result of

 the historical development, and is the means by which much

 greater confusion is avoided. Whenever improved techniques make

 possible a new and more precise (i.e., more reproducible) 

 definition of a physical unit, its numerical value is of course chosen

 so as to be well inside the limits of error with which the old

 unit could be defined. Thus the old Centigrade and new Celsius

 scales are the same, within the accuracy with which the 

 Centigrade scale could be realized; so the same notation, ^\circ C, is used

 for both Only in this way can old measurements retain their

 value and accuracy, without need of corrections every time a

 unit is redefined.

 

 #capitalize Joules?

 Exactly the same thing has happened in the definition of

 the calorie; for a century, beginning with the work of Joule,

 more and more precise experiments were performed to determine

 the mechanical equivalent of heat more and more accurately But

 eventually mechanical and electrical measurements of energy be

 came far more reproducible than calorimetric measurements; so

 recently the calorie was redefined to be 4.1840 Joules, this

 number now being exact by definition. Further details are given

 in the aforementioned Bureau of Standards Bulletin.

 

 

 The derivations of this section have shown that, for any

 particular substance, there is (except for choice of units) only

 one temperature scale $T$ with the property that $dQ = TdS$ where

 $dS$ is the exact differential of some state function $S$. But this

 in itself provides no reason to suppose that the /same/ Kelvin

 scale will result for all substances; i.e., if we determine a

 \ldquo{}helium Kelvin temperature\rdquo{} and a 

 \ldquo{}carbon dioxide Kelvin temperature\rdquo{} by the measurements 

 indicated in (1-38), and choose the units so that they agree numerically at one point, will they then

 agree at other points? Thus far we have given no reason to 

 expect that the Kelvin scale is /universal/, other than the empirical

 fact that the limit (1-45) is found to be the same for all gases.

 In section 2.0 we will see that this universality is a conse

 quence of the second law of thermodynamics (i.e., if we ever

 find two substances for which the Kelvin scale as defined above

 is different, then we can take advantage of this to make a 

 perpetual motion machine of the second kind).

 

 

 Usually, the second law is introduced before discussing

 entropy or the Kelvin temperature scale. We have chosen this

 unusual order so as to demonstrate that the concepts of entropy

 and Kelvin temperature are logically independent of the second

 law; they can be defined theoretically, and the experimental

 procedures for their measurement can be developed, without any

 appeal to the second law. From the standpoint of logic, there

 fore, the second law serves /only/ to establish that the Kelvin

 temperature scale is the same for all substances.

 

 

 ** Entropy of an Ideal Boltzmann Gas

 

 At the present stage we are far from understanding the physical 

 meaning of the function $S$ defined by (1-19); but we can investigate

 its  mathematical form and numerical values. Let us do this for a

 system 

 consisting of $n$ moles of a substance which obeys the ideal gas

 equation of state

 

 \begin{equation}PV = nRT\end{equation}

 

 and for which the heat capacity at constant volume 

 $C_V$ is a constant. The difference in entropy between any two states (1)

 and (2) is from (1-19),

 

 \begin{equation}

 S_2 - S_1 = \int_1^2 \frac{dQ}{T} = \int_1^2

 \left[\left(\frac{\partial S}{\partial V}\right)+\left(\frac{\partial S}{\partial T}\right)_V dT\right]

 \end{equation}

 

 where we integrate over any reversible path connecting the two

 states. From the manner in which $S$ was defined, this integral

 must be the same whatever path we choose. Consider, then, a

 path consisting of a reversible expansion at constant 

 temperature to a state 3 which has the initial temperature $T_1$, and the

 the final volume $V_2$; followed by heating at constant volume to the

 final temperature $T_2$. 

 Then (1-47) becomes

 

 \begin{equation}

 S_2 - S_1 = \int_1^3 \left(\frac{\partial S}{\partial V}\right)_T dV +

 \int_3^2 \left(\frac{\partial S}{\partial T}\right)_V dT

 \end{equation}

 

 To evaluate the integral over $(1\rightarrow 3)$, note that since $dU

 = TdS - PdV$, the Helmholtz free energy function $F \equiv U -TS$ has

 the property $dF = -SdT - PdV$; and of course $dF$ is an exact

 differential since $F$ is a definite state function. The condition

 that $dF$ be exact is, analogous to (1-22),

 

 \begin{equation}

 \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial

 P}{\partial T}\right)_V

 \end{equation}

 

 which is one of the Maxwell relations, discussed further in

 Chapter 2. But [the value of this expression] is determined by the equation of state

 (1-46):

 

 \begin{equation}

 \left(\frac{\partial S}{\partial V}\right)_T = \frac{nR}{V}

 \end{equation}

 

 Likewise, along the path $(3\rightarrow 2)$, we have

 

 \begin{equation}

 \left(\frac{\partial S}{\partial T}\right)_V = \frac{n C_V}{T}

 \end{equation}

 

 where $C_V$ is the molar heat capacity at constant volume. 

 Collecting these results, we have

 

 \begin{equation}

 S_2 - S_1 = \int_1^3 \frac{nR}{V} dV + \int_2^3 \frac{n C_V}{T} dT =

 nR\log{(V_2/V_1)} + nC_V \log{(T_2/T_1)}

 \end{equation}

 

 since $C_V$ was assumed independent of $T$. Thus the entropy function

 must have the form

 

 \begin{equation}

 S(n,V,T) = nR \log{V} + n C_V \log{T} + (\text{const.})

 \end{equation}

 

 From the derivation, the additive constant must be independent

 of V and T; but it can still depend on n. We indicate this by

 writing

 

 \begin{equation}

 S(n,V,T) = n\left[R \log{V} + C_V \log{T}\right] + f(n)

 \end{equation}

 

 where $f(n)$ is a function not determined by the definition (1-47).

 The form of $f(n)$ is, however, restricted by the condition that

 the entropy be an extensive quantity; i.e., two identical systems

 placed together should have twice the entropy of a single system; or

 more generally,

 

 \begin{equation}

 S(qn, qV, T) = q\cdot S(n,v,T),\qquad 0<q<\infty

 \end{equation}

 

 Substituting (1-54) into (1-55), we find that $f(n)$ must satisfy

 the functional equation

 

 \begin{equation}

 f(q\cdot n) = q\cdot f(n) - R\cdot n\cdot q\log{q}\end{equation}

 

 

 To solve this, one can differentiate with respect to $q$ and set

 $q = 1$; we then obtain the differential equation

 

 \begin{equation}

 n\cdot f^\prime(n) - f(n) + R\cdot n = 0

 \end{equation}

 # xy' - y + rx = 0

 which is readily solved; alternatively, just set $n = 1$ in (1-56)

 and replace $q$ by $n$ By either procedure we find

 

 \begin{equation}

 f(n) = n\cdot f(1) - R\cdot n \log{n} (1-58)

 \end{equation}

 

 As a check, it is easily verified that this is the solution of (1-56)

 and (1-57). We then have finally,

 

 \begin{equation}

 S(n,V,t) = n\left[C_v\cdot\log{t} + R\cdot \log{\left(\frac{V}{n}\right)} +

 A\right]

 \end{equation}

 

 where $A\equiv f(1)$ is still an arbitrary constant, not determined

 by the definition (1-19), or by the condition (1-55) that $S$ be

 extensive. However, $A$ is not without physical meaning; we will

 see in the next Section that the vapor pressure of this 

 substance (and more generally, its chemical potential) depends on

 $A$. Later, it will appear that the numerical value of $A$ involves

 Planck's constant, and its theoretical determination therefore

 requires quantum statistics.

 

 #edit: "is constant"

 We conclude from this that, in any region where experimentally 

 $C_V$ is constant, and the ideal gas equation of state is

 obeyed, the entropy must have the form (1-59) The fact that

 classical statistical mechanics does not lead to this result,

 the term $n\cdot R \cdot \log{(1/n)}$ being missing (Gibbs paradox), 

 was historically one of the earliest clues indicating the need for the

 quantum theory.

 

 In the case of a liquid, the volume does not change 

 appreciably on heating, and so $dS = n\cdot C_V\cdot dT/T$, and if

 $C_V$ is independent of temperature, we would have in place of (1-59),

 

 \begin{equation}

 S = n\left[C_V\ln{T}+A_\ell\right]

 \end{equation}

 

 where $A_\ell$ is an integration constant, which also has physical

 meaning in connection with conditions of equilibrium between

 two different phases.

 

 ** The Second Law: Definition

 

 Probably no proposition in physics has been the subject of more deep

 and sustained confusion

 than the second law of thermodynamics It is not in the province

 of macroscopic thermodynamics to explain the underlying reason

 for the second law; but at this stage we should at least be able

 to /state/ this law in clear and experimentally meaningful terms.

 However, examination of some current textbooks reveals that,

 after more than a century, different authors still disagree as

 to the proper statement of the second law, its physical meaning,

 and its exact range of validity.

 

 Later on in this book it will be one of our major objectives

 to show, from several different viewpoints, how much clearer and

 simpler these problems now appear in the light of recent develop

 ments in statistical mechanics For the present, however, our

 aim is only to prepare the way for this by pointing out exactly

 what it is that is to be proved later. As a start on this at

 tempt, we note that the second law conveys a certain piece of

 informations about the /direction/ in which processes take place.

 In application it enables us to predict such things as the final

 equilibrium state of a system, in situations where the first law

 alone is insufficient to do this.

 

 

 A concrete example will be helpful. We have a vessel

 equipped with a piston, containing $N$ moles of carbon dioxide.

 

 #changed V_f to V_1

 The system is initially at thermal equilibrium at temperature $T_0$,

 volume $V_0$ and pressure $P_O$; and under these conditions it contains

 $n$ moles of CO_2 in the vapor phase and $N-n$ moles in the liquid

 phase The system is now thermally insulated from its 

 surroundings, and the piston is moved rapidly (i.e., so that $n$ does not

 change appreciably during the motion) so that the system has a

 new volume $V_1$; and immediately after the motion, a new pressure

 $P_1$ The piston is now held fixed in its new position, and the

 system allowed to come once more to equilibrium. During this

 process, will the CO_2 tend to evaporate further, or condense further? 

 What will be the final equilibrium temperature $T_{eq}$, 

 the final pressure $P_eq$, and final value of $n_{eq}$?

 

 It is clear that the first law alone is incapable of answering

 these questions; for if the only requirement is conservation of

 energy, then the CO_2 might condense, giving up its heat of 

 vaporization and raising the temperature of the system; or it might

 evaporate further, lowering the temperature. Indeed, all values

 of $n_{eq}$ in $O \leq n_{eq} \leq N$ would be possible without any 

 violation of the first law. In practice, however, this process will be found

 to go in only one direction and the system will reach a definite

 final equilibrium state with a temperature, pressure, and vapor

 density predictable from the second law.

 

 

 Now there are dozens of possible verbal statements of the

 second law; and from one standpoint, any statement which conveys

 the same information has equal right to be called \ldquo{}the second

 law.\rdquo{} However, not all of them are equally direct statements of

 experimental fact, or equally convenient for applications, or

 equally general; and it is on these grounds that we ought to

 choose among them.

 

 Some of the mos t popular statements of the second law 

 belong to the class of the well-known \ldquo{}impossibility\rdquo{} 

 assertions; i.e., it is impossible to transfer heat from a lower to a higher

 temperature without leaving compensating changes in the rest of

 the universe, it is impossible to convert heat into useful work

 without leaving compensating changes, it is impossible to make

 a perpetual motion machine of the second kind, etc.

 

 Suoh formulations have one clear logical merit; they are

 stated in such a way that, if the assertion should be false, a

 single experiment would suffice to demonstrate that fact 

 conclusively. It is good to have our principles stated in such a

 clear, unequivocal way.

 

 However, impossibility statements also have some 

 disadvantages In the first place, /they are not, and by their very

 nature cannot be, statements of eiperimental fact/. Indeed, we

 can put it more strongly; we have no record of anyone having

 seriously tried to do any of the various things which have been

 asserted to be impossible, except for one case which actually

 succeeded. In the experimental realization of negative spin

 temperatures, one can transfer heat from a lower to a higher

 temperature without external changes; and so one of the common

 impossibility statements is now known to be false [for a clear

 discussion of this, see the [[../sources/Ramsey.pdf][article of N. F. Ramsey (1956)]];

 experimental details of calorimetry with negative temperature

 spin systems are given by Abragam and Proctor (1958)]

 

 

 Finally, impossibility statements are of very little use in

 /applications/ of thermodynamics; the assertion that a certain kind

 of machine cannot be built, or that a certain laboratory feat

 cannot be performed, does not tell me very directly whether my

 carbon dioxide will condense or evaporate. For applications,

 such assertions must first be converted into a more explicit

 mathematical form.

 

 

 For these reasons, it appears that a different kind of

 statement of the second law will be, not necessarily more

 \ldquo{}correct\rdquo{}, but more useful in practice. Now both Clausius (1875)

 and Planck (1897) have laid great stress on their conclusion

 that the most general statement, and also the most immediately

 useful in applications, is simply the existence of a state

 function, called the entropy, which tends to increase. More

 precisely: in an adiabatic change of state, the entropy of

 a system may increase or may remain constant, but does not

 decrease. In a process involving heat flow to or from the

 system, the total entropy of all bodies involved may increase

 or may remain constant; but does not decrease; let us call this

 the \ldquo{}weak form\rdquo{} of the second law.

 

 The weak form of the second law is capable of answering the

 first question posed above; thus the carbon dioxide will 

 evaporate further if, and only if, this leads to an increase in the

 total entropy of the system This alone, however, is not enough

 to answer the second question; to predict the exact final 

 equilibrium state, we need one more fact.

 

 The strong form of the second law is obtained by adding the

 further assertion that the entropy not only \ldquo{}tends\rdquo{} to increase;

 in fact it /will/ increase, /to the maximum value permitted by the

 constraints imposed[fn::Note, however, that the second law has

 nothing to say about how rapidly this approach to equilibrium takes place.]/. In the case of the carbon dioxide, these

 constraints are: fixed total energy (first law), fixed total

 amount of carbon dioxide, and fixed position of the piston. The

 final equilibrium state is the one which has the maximum entropy

 compatible with these constraints, and it can be predicted 

 quantitatively from the strong form of the second law if we know,

 from experiment or theory, the thermodynamic properties of carbon

 dioxide (ie, heat capacity, equation of state, heat of vapor

 ization)

 

 To illustrate this, we set up the problem in a crude 

 approximation which supposes that (l) in the range of conditions

 of interest, the molar heat capacity $C_v$ of the vapor, and $C_\ell$ of

 the liquid, and the molar heat of vaporization $L$, are all con

 stants, and the heat capacities of cylinder and piston are 

 negligible; (2) the liquid volume is always a small fraction of the

 total $V$, so that changes in vapor volume may be neglected; (3) the

 vapor obeys the ideal gas equation of state $PV = nRT$. The 

 internal energy functions of liquid and vapor then have the form

 

 \begin{equation}

 U_\ell = (N-n)\left[C_\ell\cdot T + A\right]

 \end{equation}

 \begin{equation}

 U_v = n\left[C_v\cdot T + A + L\right]

 \end{equation}

 

 where $A$ is a constant which plays no role in the problem. The

 appearance of $L$ in (1-62) recognizes that the zero from which we

 measure energy of the vapor is higher than that of the liquid by

 the energy $L$ necessary to form the vapor. On evaporation of $dn$

 moles of liquid, the total energy increment is $dU = dU_\ell + dU_v =

 0$; or

 

 \begin{equation}

 \left[n\cdot C_v + (N-n)C_\ell\right] dT + \left[(C_v-C_\ell)T + L\right]dn = 0

 \end{equation}

 

 which is the constraint imposed by the first law. As we found

 previously (1-59), (1-60) the entropies of vapor and liquid are

 given by

 

 \begin{equation}

 S_v = n\left[C_v\cdot\ln{T} + R\cdot \ln{\left(V/n\right)} + A_v\right]

 \end{equation}

 \begin{equation}

 S_\ell = (N-n)\left[C_\ell\cdot \ln{T}+A_\ell\right]

 \end{equation}

 

 where $A_v$, $A_\ell$ are the constants of integration discussed in the

 last Section.

 

 

 We leave it as an exercise for the reader to complete the

 derivation from this point, and show that the total entropy

 $S = S_\ell + S_v$ is maximized subject to the constraint (1-63), when

 the values $n_{eq}$, $T_{eq}$ are related by

 

 \begin{equation}

 \frac{n_{eq}}{V}= B\cdot T_{eq}^a\cdot \exp{\left(-\frac{L}{RT_{eq}}\right)}

 \end{equation}

 

 where $B\equiv \exp{(-1-a-\frac{A_\ell-A_v}{R})}$ and $a\equiv

 (C_v-C_\ell)/R$ are constants.

 

 

 Equation (1-66) is recognized as an approximate form of the Vapor

 pressure formula

 We note that $A_\ell$, $A_v$, which appeared first as integration

 constants for the entropy with no particular physical meaning,

 now play a role in determining the vapor pressure.

 

 ** The Second Law: Discussion

 

 We have emphasized the distinction between the weak and strong forms 

 of the second law

 because (with the exception of Boltzmann's original unsuccessful

 argument based on the H-theorem), most attempts to deduce the

 second law from statistical mechanics have considered only the

 weak form; whereas it is evidently the strong form that leads

 to definite quantitative predictions, and is therefore needed

 for most applications. As we will see later, a demonstration of

 the weak form is today almost trivial---given the Hamiltonian form

 of the equations of motion, the weak form is a necessary 

 condition for any experiment to be reproducible. But demonstration

 of the strong form is decidedly nontrivial; and we recognize from

 the start that the job of statistical mechanics is not complete

 until that demonstration is accomplished.

 

 

 As we have noted, there are many different forms of the

 seoond law, that have been favored by various authors. With

 regard to the entropy statement of the second law, we note the

 following. In the first place, it is a direct statement of 

 experimental fact, verified in many thousands of quantitative 

 measurements, /which have actually been performed/. This is worth a

 great deal in an age when theoretical physics tends to draw

 sweeping conclusions from the assumed outcomes of 

 \ldquo{}thought-experiments.\rdqquo{} Secondly, it has stood the test 

 of time; it is the entropy statement which remained valid in the case

 of negative spin temperatures, where some others failed. Thirdly, it

 is very easy to apply in practice, the weak form leading 

 immediately to useful predictions as to which processes will go and

 which will not; the strong form giving quantitative predictions

 of the equilibrium state. At the present time, therefore, we

 cannot understand what motivates the unceasing attempts of many

 textbook authors to state the second law in new and more 

 complicated ways.

 

 One of the most persistent of these attempts involves the

 use of [[http://en.wikipedia.org/wiki/Second_law_of_thermodynamics#Principle_of_Carath.C3.A9odory][Carath\eacute{}odory's principle]]. This states that, in the 

 neighborhood of any thermodynamic state there are other states which

 cannot be reached by an adiabatic process. After some mathematical

 analysis 

 [Margenau and Murphy (1943), pp. 26-31; or Wannier (1966),

 pp. 126-132] 

 one infers the existence of a state function (entropy) which tends 

 to increase; or at least, cannot decrease. From a /mathematical/ 

 standpoint there can be no objection at all to this; the analysis 

 is quite rigorous. But from a /physical/ standpoint it is subject 

 to the same objection that its premise is an impossibility 

 statement, and therefore not an experimental fact. 

 Indeed, the conclusion of Carath\eacute{}odory's

 argument is a far more direct statement of observed fact than its

 premise; and so it would seem more logical to use the argument

 backwards. Thus, from the experimental fact that the entropy

 tends to increase, we would infer that there must exist 

 neighboring states inaccessible in an adiabatic process; but the

 result is then trivial. In a similar way, other impossibility

 statements follow trivially from the entropy statement of the

 second law.

 

 

 Finally, we note that all statements of the second law are

 subject to a very important qualification, not always sufficiently

 emphasized. As we stress repeatedly, conventional thermodynamics

 is a theory only of states of thermal equilibrium; such concepts

 as temperature and entropy are not even defined for others. 

 Therefore, all the above statements of the second law must be under

 stood as describing only the /net result/ of processes /which begin

 and end in states of complete thermal equilibrium/. Classical

 thermodynamics has nothing to say about processes that do not

 meet this condition, or about intermediate states of processes

 that do. Again, it is nuclear magnetic resonance (NMR) 

 experiments which provide the most striking evidence showing how 

 essential this qualification is; the spin-echo experiment 

 (Hahn, 1950) is, as we will see in detail later, a gross violation of

 any statement of the second law that fails to include it.

 

 

 This situation has some interesting consequences, in that

 impossibility statements may be misleading if we try to read too

 much into them. From classical thermodynamics alone, we cannot

 logically infer the impossibility of a \ldquo{}perpetual motion machine\rdquo{}

 of the second kind (i.e., a machine which converts heat energy

 into useful work without requiring any low temperature heat sink,

 as does the Carnot engine); we can infer only that such a machine

 cannot operate between equilibrium states. More specifically, if

 the machine operates by carrying out some cyclic process, then

 the states of (machine + environment) at the beginning and end

 of a cycle cannot be states of complete thermal equilibrium, as

 in the reversible Carnot engine. But no real machine operates

 between equilibrium states anyway. Without some further analysis

 involving statistical mechanics, we cannot be at all certain that

 a sufficiently clever inventor could not find a way to convert

 heat energy into useful work on a commercially profitable scale;

 the energy is there, and the only question is whether we could

 persuade it to \ldquo{}organize\rdquo{} itself enough to perform useful work

 against pistons, magnets, gravitational or electric fields,

 chemical activation energy hills, etc.

 

 

 It was Maxwell himself who first ([[../sources/Maxwell-Heat.pdf][1871]])[fn::Edit: See also, the [[http://openlibrary.org/books/OL7243600M/Theory_of_heat][Open Library

 page]], where you can read and download Maxwell's book in a variety of formats.] suggested such 

 possibilities, in his invention of the \ldquo{}Maxwell Demon\rdquo{}, 

 an imaginary being (or mechanism) which can regulate valves so as to allow

 fast molecules to pass through a partition in one direction only,

 thus heating up one side at the expense of the other. We could

 then allow the heat to flow back from the hot side to the cold

 through a conventional Carnot engine, generating useful work; and

 the whole arrangement would constitute a perpetual motion machine

 of the second kind.

 

 #http://naca.larc.nasa.gov/search.jsp?R=19760010893&qs=Ns%3DLoaded-Date|1%26N%3D4294709597

 

 Maxwell did not regard such a device as impossible in principle; 

 only very difficult technically. Later authors ([[../sources/Szilard.pdf][Szilard, 1929]];

 Brillouin, 1951, 1956) 

 have argued, on the basis of quantum

 theory or connections between entropy and information, that it

 fundamentally impossible. However, all these arguments seem

 to contain just enough in the way of questionable assumptions or

 loopholes in the logic, as to leave the critical reader not quite

 convinced. This is particularly so when we recall the lessons

 of history; clever experimenters have, over and over again, made

 fools of theorists who were too quick to assert that something

 cannot be done.

 

 A recent example worth recalling concerns the Overhauser

 effect in magnetic resonance (enhancement of the polarization

 of one set of spins by irradiation of another set coupled to them).

 When this effect was first proposed, several well-known 

 authorities on thermodynamics and statistical mechanics ridiculed the

 suggestion and asserted that the effect could not possibly exist,

 because it violated the second law of thermodynamics. This 

 incident is a valuable reminder of how little we really understand

 the second law, or how to apply it in new situations.

 

 In this connection, there is a fascinating little gadget

 known as the [[http://en.wikipedia.org/wiki/Vortex_tube][Hilsch tube]] or Vortex tube, in which a jet of

 compressed air is injected into a pipe at right angles to its

 axis, but off center so that it sets up a rapid rotational

 motion of the gas. In some manner, this causes a separation of

 the fast and slow molecules, cold air collecting along the axis

 of the tube, and hot air at the walls. On one side of the jet,

 a diaphragm with a small hole at the center allows only the cold

 air to escape, the other side is left open so that the hot air

 can escape. The result is that when compressed air at room

 temperature is injected, one can obtain air from the hot side

 at $+100^\circ$ F from the cold side at $-70^\circ$ F, in sufficient quantities

 to be used for quick-freezing small objects, or for cooling

 photomultiplier tubes [for construction drawings and 

 experimental data, see [[http://books.google.com/books?id=yOUWAAAAIAAJ][Stong (1960)]]; for a partial thermodynamic

 analysis, see Hilsch (1947)[fn::Edit: Hilsch's paper is entitled /The use of the expansion of gases in

 a centrifugal field as a cooling process./]].

 

 Of course, the air could also be cooled by adiabatic expansion 

 (i.e., by doing work against a piston); and it appears that

 the amount of cooling achieved in vortex tubes is comparable to,

 but somewhat less than, what could be obtained this way for the

 same pressure drop. However, the operation of the vortex tube

 is manifestly not simple adiabatic since no work is

 done; rather, part of the gas is heated up, at the cost of cooling

 the rest; i.e., fast and slow molecules are separated spatially.

 There is, apparently, no violation of the laws of thermodynamics,

 since work must be supplied to compress the air; nevertheless,

 the device resembles the Maxwell Demon so much as to make one

 uncomfortable.. This is so particularly because of our 

 embarrassing inability to explain in detail (i.e., in molecular terms)

 how such asimple device works. If we did understand it, would

 we be able to see still more exciting possibilities? No one

 knows.

 

 

 It is interesting to note in passing that such considerations 

 were very much in Planck's mind also; in his [[http://books.google.com/books?id=kOjy3FQqXPQC&printsec=frontcover][/Treatise on Thermodynamics/]] (Planck, 1897; 116), he begins his discussion

 of the second law in these words (translation of A. Ogg): 

 #+begin_quote 

 \ldquo{}We

 $\ldots$ put forward the following proposition $\ldots$ : 

 /it is impossible to construct an engine which will work a complete cycle,

 and produce no effect except the raising of a weight and the cooling of a heat-reservoir./ Such an engine could be used simultaneously

 as a motor and a refrigerator without any waste of energy or

 material, and would in any case be the most profitable engine

 ever made. It would, it is true, not be equivalent to perpetual

 motion, for it does not produce work from nothing, but from the

 heat which it draws from the reservoir. It would not, therefore,

 like perpetual motion, contradict the principle of energy, but

 would nevertheless possess for man the essential advantage of

 perpetual motion, the supply of work without cost; for the in

 exhaustible supply of heat in the earth, in the atmosphere, and

 in the sea, would, like the oxygen of the atmosphere, be at

 everybody ‘s immediate disposal. For this reason we take the

 above proposition as our starting point. Since we are to deduce

 the second law from it, we expect, at the same time, to make a

 most serviceable application of any natural phenomenon which may

 be discovered to deviate from the second law.\rdquo{}

 #+end_quote

 The ammonia maser ([[../sources/Townes-Maser.pdf][Townes, 1954]]) is another example of an

 experimental device which, at first glance, violates the second

 law by providing \ldquo{}useful work\rdquo{} in the form of coherent microwave

 radiation at the expense of thermal energy. The ammonia molecule

 has two energy levels separated by 24.8 GHz, with a large electric

 dipole moment matrix element connecting them. We cannot obtain

 radiation from ordinary ammonia gas because the lower state

 populations are slightly greater than the upper, as given by

 the usual Boltzmann factors. However, if we release ammonia gas

 slowly from a tank into a vacuum so that a well-collimated jet

 of gas is produced, we can separate the upper state molecules

 from the lower. In an electric field, there is a quadratic

 Stark effect, the levels \ldquo{}repelling\rdquo{} each other according to

 the well-known rule of second-order perturbation theory. Thus,

 the thermally excited upper-state molecules have their energy

 raised further by a strong field; and vice versa for the lower

 state molecules. If the field is inhomogeneous, the result is

 that upper-state molecules experience a force drawing them into

 regions of weak field; and lower-state molecules are deflected

 toward strong field regions. The effect is so large that, in a

 path length of about 15 cm, one can achieve an almost complete

 spatial separation. The upper-state molecules then pass through

 a small hole into a microwave cavity, where they give up their

 energy in the form of coherent radiation.

 

 

 Again, we have something very similar to a Maxwell Demon;

 for without performing any work (since no current flows to the

 electrodes producing the deflecting field) we have separated

 the high-energy molecules from the low-energy ones, and obtained

 useful work from the former. This, too, was held to be 

 impossible by some theorists before the experiment succeeded!

 

 Later in this course, when we have learned how to formulate

 a general theory of irreversible processes, we will see that the

 second law can be extended to a new principle that tells us which

 nonequilibrium states can be reached, reproducibly, from others;

 and this will of course have a direct bearing on the question of

 perpetual motion machines of the second kind. However, the full

 implications of this generalized second law have not yet been

 worked out; our understanding has advanced just to the point

 where confident, dogmatic statements on either side now seem

 imprudent. For the present, therefore, we leave it as an open

 question whether such machines can or cannot be made.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 * COMMENT Use of Jacobians in Thermodynamics

 

 Many students find that thermodynamics, although mathematically almost

 trivial, is nevertheless one of the most difficult subjects in their program.

 A large part of the blame for this lies in the extremely cumbersome partial

 derivative notation. In this chapter we develop a different mathematical

 scheme, with which thermodynamic derivations can be carried out more easily,

 and which gives a better physical insight into the meaning of thermodynamic

 relations.

 

 ***  COMMENT Editor's addendum

 #+begin_quote

 In order to help readers with the Jacobian material that follows, I

 have included this section of supplementary material. --- Dylan

 #+end_quote}

 

 Suppose your experimental parameters consist of three variables

 $X,Y,Z$---say, volume, pressure, and temperature. Then the

 physically allowed combinations $\langle x,y,z\rangle$ of $X,Y,Z$

 comprise the /(equilibrium) state space/

 of your thermodynamic system; the set of these combinations forms a

 subset $\Omega$ of $\mathbb{R}^3$. (If there were four experimental

 parameters, the state space would be a subset of $\mathbb{R}^4$, and

 so on).

 

 You can represent the flux of some physical quantities (such as

 heat, entropy, or number of moles) as a vector field spread throughout

 $\Omega$, i.e., a function $F:\Omega\rightarrow \mathbb{R}^n$ sending

 each state to the value of the vector at that state. 

 When you trace out different paths through the state space 

 $\gamma:[a,b]\rightarrow \Omega$, you can measure the net quantity

 exchanged by

 

 \begin{equation}

 \text{net exchange} = \int_a^b (F\circ \gamma)\cdot \gamma^\prime.

 \end{equation}

 

 Some quantities are conservative.

 

 - If the vector field $F$ (representing the flux of a physical

   quantity) is in fact the gradient of some function

   $\varphi:\Omega\rightarrow \mathbb{R}$, then $F$ is conservative and

   $\varphi$ represents the value of the conserved quantity at each state.

 - In this case, the value of $\varphi$ is completely determined by

   specifying the values of the experimental parameters $X, Y, Z$. In

   particular, it doesn't matter by which path the state was reached.

 

 

 Some physical quantities (such as entropy or number of moles) are

 completely determined by your experimental parameters $X, Y, Z$. Others (such as

 heat) are not. For those quantities that are,

 you have functions $\phi:\Omega\rightarrow \mathbb{R}$ sending each state

 to the value of the quantity at that state.

 

 

 

  and measure the change in physical

 quantities (like entropy or number of moles)

 

 

 Given your experimental parameters $X,Y,Z$, there may be other

 physical quantities (such as entropy or number of moles) which are uniquely

 defined by each combination of $\langle x,y,z\rangle$. Stated

 mathematically, there is a function $f:\Omega\rightarrow \mathbb{R}$

 sending each state to the value of the quantity at that state. 

 

 

 

 Now, sometimes you would like to use a different coordinate system to

 describe the same physical situation.

 A /change of variables/ is an

 invertible differentiable transformation $g:\mathbb{R}^n\rightarrow

 \mathbb{R}^n$---a function with $n$ input components (the $n$ old

 variables) and $n$ output components (the $n$ new variables), where

 each output component can depend on any number of the input components. For

 example, in two dimensions you can freely switch between Cartesian

 coordinates and polar coordinates; the familiar transformation is 

 

 \(g\langle x, y\rangle \mapsto \langle \sqrt{x^2+y^2}, \arctan{(y/x)}\rangle\)

 

 

 

 

 

 ** Statement of the Problem

 In fields other than thermodynamics , one usually starts out by stating

 explicitly what variables shall be considered the independent ones, and then

 uses partial derivatives without subscripts, the understanding being that all

 independent variables other than the ones explicitly present are held constant

 in the differentiation. This convention is used in most of mathematics and

 physics without serious misunderstandings. But in thermodynamics, one never

 seems to be able to maintain a fixed set of independent variables throughout

 a derivation, and it becomes necessary to add one or more subscripts to every

 derivative to indicate what is being held constant. The often-needed 

 transformation from one constant quantity to another involves the

 relation

 

 \begin{equation}

 \left(\frac{\partial A}{\partial B}\right)_C = \left(\frac{\partial

 A}{\partial B}\right)_D + \left(\frac{\partial A}{\partial D}\right)_B \left(\frac{\partial D}{\partial B}\right)_C 

 \end{equation}

 

 which, although it expresses a fact that is mathematically trivial, assumes

 such a complicated form in the usual notation that few people can remember it

 long enough to write it down after the book is closed.

 

 As a further comment on notation, we note that in thermodynamics as well

 as in mechanics and electrodynamics, our equations are made cumbersome if we

 are forced to refer at all times to some particular coordinate system (i.e.,

 set of independent variables). In the latter subjects this needless 

 complication has long since been removed by the use of vector

 notation, 

 which enables us to describe physical relationships without reference to any particular

 coordinate system. A similar house-cleaning can be effected for thermodynamics

 by use of jacobians, which enable us to express physical relationships without

 committing ourselves to any particular set of independent variables.

 We have here an interesting example of retrograde progress in science:

 for the historical fact is that use of jacobians was the original mathematical

 method of thermodynamics. They were used extensively by the founder of modern

 thermodynamics, Rudolph Clausius, in his work dating from about 1850. He used

 the notation

 

 \begin{equation}

 D_{xy} \equiv \frac{\partial^2 Q}{\partial x\partial y} -

 \frac{\partial^2 Q}{\partial y \partial x}

 \end{equation}

 

 

 where $Q$ stands, as always, for heat, and $x$, $y$ are any 

 two thermodynamic quantities. Since $dQ$ is not an exact differential, 

 $D_{xy}$ is not identically zero. It is understandable that this notation, used in his published works, involved

 Clausius in many controversies, which in retrospect appear highly amusing. An

 account of some of them may be found in his book (Clausius, 1875). On the

 other hand, it is unfortunate that this occurred, because it is probably for

 this reason that the quantities $D_{xy}$ went out of general use for many years,

 with only few exceptions (See comments at the end of this chapter). 

 In a footnote in Chapter II of Planck's famous treatise (Planck, 1897), he explains

 that he avoids using $dQ$ to represent an infinitesimal quantity of heat, because

 that would imply that it is the differential of some quantity $Q$. This in turn

 leads to the possibility of many fallacious arguments, all of which amount to

 setting $D_{xy}=0$. However, a reading of Clausius‘ works makes it clear that

 the quantities $D_{xy}$, when properly used, form the natural medium for discussion

 of thermodynamics. They enabled him to carry out certain derivations with a

 facility and directness which is conspicuously missing in most recent 

 expositions. We leave it as an exercise for the reader to prove that $D_{xy}$ is a

 jacobian (Problem 2.1).

 

 We now develop a condensed notation in which the algebra of jacobians

 may be surveyed as a whole, in a form easy to remember since the abstract

 relations are just the ones with which we are familiar in connection with the

 properties of commutators in quantum mechanics.

 

 ** Formal Properties of Jacobians[fn::For any function $F:\mathbb{R}^n\rightarrow \mathbb{R}^n$, $F:\langle x_1,\ldots, x_n\rangle \mapsto \langle F_1(x), F_2(x),\ldots F_n(x)\rangle$ we can define the Jacobian matrix of $F$ to be \(JF = \begin{bmatrix}\partial_1{F_1}&\ldots& \partial_n{F_n}\\\vdots&\ddots&\vdots\\\partial_1 F_n & \ldots & \partial_n F_n\\\end{bmatrix}\), and the Jacobian (determinant) of $f$ to be the determinant of this matrix (provided all partial derivatives exist). ]

 Consider first a system with only two degrees of freedom. We define

 

 \begin{equation}

 [A,B] \equiv \frac{\partial(A,B)}{\partial(x,y)} =

 \left|\begin{matrix}\frac{\partial A}{\partial x}& \frac{\partial

 A}{\partial y} \\

 \frac{\partial B}{\partial x} & \frac{\partial B}{\partial y} \end{matrix}\right|

 \end{equation}

 where $x$, $y$ are any variables adequate to determine the state of the system.

 

 Since for any change of variables, $x,y \mapsto x^\prime, y^\prime$ we

 have

 

 \begin{equation}

 \frac{\partial(A,B)}{\partial(x^\prime,y^\prime)} = \frac{\partial(A,B)}{\partial(x,y)}\frac{\partial(x,y)}{\partial(x^\prime,y^\prime)}

 \end{equation}

 

 or, in an easily understandable condensed notation,

 

 \begin{equation}

 [A,B]^\prime = [A,B][x,y]^\prime

 \end{equation}

 

 It follows that any equations that are homogeneous in the jacobians are in

 variant in form under "coordinate transformations“, so that we can suppress

 the independent variables x, y and carry out derivations without committing

 ourselves to any particular set.

 The algebra of these symbols is characterized by the following identities

 (the comma may be omitted if A, B are single letters). The properties of

 antisymmetry, linearity, and composition have the familiar form

 In addition we have three cyclic identities, easily proved:

 These relations are not all independent; for example, (2—ll) follows from

 (2-9) and (2-13).

 Putting dC = O in (2-9) , we obtain the rule

 by means of which equations are translated from one language to the other.

 

 

 From it one sees that the transformation law (2-l) now appears as a special

 case of the identity (2-11) . Writing for the enthalpy, free energy, and Gibbs

 function respectively ,

 where U is the internal energy with the property dU = t :35 — P (N, we have as

 consequences of (2-13) the relations

 The advantages of this notation is shown particularly when we consider the

 four Maxwe ll equati ons

 Applying (2-14) , we see that each reduces to the single identity

 

 

 Thus, all of the Maxwell equations are expressions in different "coordinate

 systems" of the same basic fact (2-18) , which will receive a physical inter

 pretation in Sec. 2.4. In a derivation, such as that of Eq. (1-49) , every

 thing that can be gained by using any of the equations (2-17) is already

 accomplished by application of the single relation (2-18).

 Jacobians which involve the entropy in combinations other than are

 related to various specific heats. The heat capacity at constant X is

 and, using (2-14) we obtain the identity

 C

 In the simplest derivations, application of (2-18) or (2—20) is the essential

 step.

 In his well-known textbook, Zemansky (1943) shows that many of the ele

 mentary derivations in thermodynamics may be reduced to application of the

 In the above notation these equations are far from obvious and not easy to

 remember. Note, however, that the T :38 equations are special cases of the

 cyclic identity (2-9) for the sets of variables {TVS}, respectively,

 while the energy equation is a consequence of (2-13) and the Maxwell relation:

 

 

 From (2~l4) we see that this is the energy equation in jacobian notation.

 2 .3 Elementary Examples

 In a large class of problems, the objective is to express some quantity

 of interest, or some condition of interest, in terms of experimentally mea

 surable quantities. Therefore, the “sense of direction“ in derivations is

 provided by the principle that we want to get rid of any explicit appearance

 of the entropy and the various energies U, H, F, G. Thus, if the entropy

 appears in the combination [TS], we use the Maxwell relation to replace it

 with . If it appears in some other combination , we can use the

 identity (2-20) .

 Similarly, if combinations such as or [UX] appear, we can use (2-16)

 and replace them with

 it cannot be eliminated in this way. However, since in phenomenological

 thermodynamics the absolute value of the entropy has no meaning, this situa

 tion cannot arise in any expression representing a definite physical quantity.

 For problems of this simplest type, the jacobian formalism works like a

 well-oiled machine, as the following examples show. We denote the isothermal

 compressibility, thermal expansion coefficient, and ratio of specific heats

 bY K1 5: Y, réspectively:

 

 

 and note that from (2-27) and (2-28) we have

 (2-30)

 Several derivatives, chosen at random, are now evaluated in terms of these

 quantities:

 A more difficult type of problem is the following: We have given a num

 ber of quantities and wish to find the general relation, if any, connecting

 them. In one sense, the question whether relations exist can be answered

 

 

 immediately; for any two quantities A, B a necessary and sufficient condition

 for the existence of a functional relation A f(B) in a region R is:

 = O in R}. In a system of two degrees of freedom it is clear that between

 any three quantities A, B, C there is necessarily at least one functional

 relation f(A,B,C) = O, as is implied by the identity (2-9) [Problem 2.2] . An

 example is the equation of state f(PVT) = O. This , however, is not the type

 of relation one usually has in mind. For each choice of A, B, C and each

 particular system of two degrees of freedom, some functional relationship

 must exist, but in general it will depend on the physical nature of the system

 and can be obtained only when one has sufficient information, obtained from

 measurement or theory, about the system.

 The problem is rather to find those relations between various quantities

 which hold generally, regardless of the nature of the particular system.

 Mathematically, all such relations are trivial in the sense that they must be

 special cases of the basic identities already given. Their physical meaning

 may, however, be far from trivial and they may be difficult to find. Note,

 for example, that the derivative computed in (2-35) is just the Joule—Thomson

 coefficient 11. Suppose the problem had been stated as: "Given the five

 quantities V, Cp, 8, determine whether there is a general relation

 between them and if so find it." Now, although a repetition of the argument

 of (2-35) would be successful in this case, this success must be viewed as a

 lucky accident from ‘the standpoint of the problem just formulated. It is not

 a general rule for attacking this type of problem because there is no way of

 ensuring that the answer will come out in terms of the desired quantities.

 To illustrate a general rule of procedure, consider the problem of find

 ing a relationship, if any, between iCp, CV, V, T, B, K}. First we write

 these quantities in terms of jacobians.

 

 

 At this point we make a definite choice of some coordinate system. Since

 [TP] occurs more often than any other jacobian, we adopt x = T, y = P as the

 The variables in jacobians are P, V, T, S, for which (2-11) gives

 [PV][TS] + [VT] [PS] + = 0 (2-40)

 or, in this case

 Substituting the expressions (2-39) into this we obtain

 or, rearranging, we have the well—known law

 which is now seen as a special case of (2-11).

 There are several points to notice in this derivation: (1) no use has

 been made of the fact that the quantities T, V were given explicitly; the

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 * Gibbs Formalism \mdash{} Physical Derivation

 

 

 In this Chapter we present physical arguments by which the Gibbs 

 formalism can be derived and justified, deliberately avoiding all use

 of probability theory. This will serve to convince us of the /validity/ of Gibbs’ formalism

 for the particular applications given by Gibbs, and will give us an intuitive

 physical understanding of the second law, as well as the physical meaning of

 the Kelvin temperature.

 

 Later on (Chapter 9) we will present an entirely different derivation in

 terms of a general problem of statistical estimation, deliberately avoiding

 all use of physical ideas, and show that the identical mathematical formalism

 emerges. This will serve to convince us of the /generality/ of the

 Gibbs methods, and show that their applicability is in no way restricted to equilibrium

 problems; or indeed, to physics.

 

 

 It is interesting to note that most of Gibbs‘ important results were

 found independently and almost simultaneously by Einstein (1902); but it is

 to Gibbs that we owe the elegant mathematical formulation of the theory. In

 the following we show how, from mechanical considerations involving

 the microscopic state of a system, the Gibbs rules emerge as a

 description of  equilibrium macroscopic properties. Having this, we can then reason

 backwards, and draw inferences about microscopic conditions from macroscopic experimental

 data. We will consider only classical mechanics here; however, none of this

 classical theory will have to be unlearned later, because the Gibbs formalism

 lost none of its validity through the development of quantum theory. Indeed,

 the full power of Gibbs‘ methods has been realized only through their 

 successful application to quantum theory.

 

 ** COMMENT Review of Classical Mechanics (SICM)

 In [[http://mitpress.mit.edu/sicm/][classical mechanics]] a complete description of the state of a system is

 given by specifying $n$ coordinates $q_1\ldots q_n$, and the corresponding velocities

 $D{q}_1\ldots Dq_n$. The equations of motion are then determined by a Lagrangian function

 which in simple mechanical problems is

 

 \begin{equation}

 L(t,q(t),Dq(t)) = T - V

 \end{equation}

 

 

 where $T$ and $V$ are the kinetic and potential energies. In problems involving

 coupling of particles to an electromagnetic field, the Lagrangian function

 takes a more general form, as we will see later. In either case, the 

 equations of motion are

 

 \begin{equation}

 D(\partial_2 L \circ \Gamma[q]) - \partial_1 L \circ \Gamma[q] = 0

 \end{equation}

 

 where $\Gamma[q]$ is the function $t\mapsto \langle

 t,q(t),Dq(t)\rangle$, and $\partial_i$ denotes the derivative with

 respect to the \(i\)th argument ($i=0,1,2,\ldots$).

 

 The advantage of the Lagrangian form (5-2) over the original Newtonian form

 (to which it is completely equivalent in simple mechanical problems)

 

 \begin{equation}

 D^2 (m\cdot x(t)) = -\partial_1 V \circ \Gamma[x]

 \end{equation}

 

 is that (5-2) holds for arbitrary choices of the coordinates $q_i$;

 they can include angles, or any other parameters which serve to locate a particle in

 space. The Newtonian equations (5-3), on the other hand, hold only when the

 $x_i$ are rectangular (cartesian) coordinates of a particle.

 Still more convenient for our purposes is the Hamiltonian form of the

 equations of motion. Define the momentum \ldquo{}canonically

 conjugate\rdquo{} to the 

 coordinate $q$ by

 

 \begin{equation}

 p(t) \equiv \partial_1 L \circ \Gamma[q]

 \end{equation}

 

 let $\mathscr{V}(t,q,p) = Dq$, and define a Hamiltonian function $H$ by

 

 \begin{equation}

 H(t,q,p) = p \cdot V(t,q,p) - L(t,q, V(t,q,p)

 \end{equation}

 

 the notation indicating that after forming the right-hand side of (5-5) the

 velocities $\dot{q}_i$ are eliminated mathematically, so that the

 Hamiltonian is 

 expressed as a function of the coordinates and momenta only.

 

 #+begin_quote

 ------

 *Problem (5.1).* A particle of mass $m$ is located by specifying

 $(q_1,q_2,q_3)=(r,\theta,z)$ respectively, where $r$, $\theta$, $z$

 are a cylindrical coordinate system 

 related to the cartesian $x, y, z$ by $x + iy = re^{i\theta}$, $z=z$. The

 particle moves in a potential $V(q_1,q_2,q_3)$. Show that the

 Hamiltonian in this coordinate system is

 

 \begin{equation}

 H = \frac{p_1^2}{2m} + \frac{p_2^2}{2m}+\frac{p_3^2}{2m} + V(q_1,q_2,q_3)

 \end{equation}

 

 and discuss the physical meaning of $p_1$, $p_2$, $p_3$.

 ------

 

 

 

 *Problem (5.2).* Find the Hamiltonian for the same particle, in the spherical

 coordinate system $(q_1,q_2,q_3) = (r,\theta,\phi)$ related to the

 Cartesian by 

 $x + iy = r\,\sin{\theta}\,e^{i\phi}$, $z=r\,\cos{\theta}$., and again

 discuss the physical meaning of $p_1$, $p_2$, $p_3$ .

 ------

 #+end_quote

 

 

 

 

 

 

 

 

 

 

 ** Review of Classical Mechanics

 In [[http://mitpress.mit.edu/sicm/][classical mechanics]] a complete description of the state of a system is

 given by specifying $n$ coordinates $q_1\ldots q_n$, and the corresponding velocities

 $\dot{q}_1\ldots \dot{q}_n$. The equations of motion are then determined by a Lagrangian function

 which in simple mechanical problems is

 

 \begin{equation}

 L(q_i,\dot{q}_i) = T - V

 \end{equation}

 

 

 where $T$ and $V$ are the kinetic and potential energies. In problems involving

 coupling of particles to an electromagnetic field, the Lagrangian function

 takes a more general form, as we will see later. In either case, the 

 equations of motion are

 

 \begin{equation}

 \frac{\partial L}{\partial q_i} - \frac{d}{dt}\frac{\partial

 L}{\partial \dot{q}_i} = 0.

 \end{equation}

 

 The advantage of the Lagrangian form (5-2) over the original Newtonian form

 (to which it is completely equivalent in simple mechanical problems)

 

 \begin{equation}

 m\ddot{x}_i = -\frac{\partial V}{\partial x_i}

 \end{equation}

 

 is that (5-2) holds for arbitrary choices of the coordinates $q_i$;

 they can include angles, or any other parameters which serve to locate a particle in

 space. The Newtonian equations (5-3), on the other hand, hold only when the

 $x_i$ are rectangular (cartesian) coordinates of a particle.

 Still more convenient for our purposes is the Hamiltonian form of the

 equations of motion. Define the momentum \ldquo{}canonically

 conjugate\rdquo{} to the 

 coordinate $q_i$ by

 

 \begin{equation}

 p_i \equiv \frac{\partial L}{\partial q_i}

 \end{equation}

 

 and a Hamiltonian function $H$ by

 

 \begin{equation}

 H(q_1,p_1;\cdots ; q_n,p_n) \equiv \sum_{i=1}^n p\cdot \dot{q}_i -

 L(q_1,\ldots, q_n). 

 \end{equation}

 

 the notation indicating that after forming the right-hand side of (5-5) the

 velocities $\dot{q}_i$ are eliminated mathematically, so that the Hamiltonian is ex

 pressed as a function of the coordinates and momenta only.

 

 #+begin_quote

 ------

 *Problem (5.1).* A particle of mass $m$ is located by specifying

 $(q_1,q_2,q_3)=(r,\theta,z)$ respectively, where $r$, $\theta$, $z$

 are a cylindrical coordinate system 

 related to the cartesian $x, y, z$ by $x + iy = re^{i\theta}$, $z=z$. The

 particle moves in a potential $V(q_1,q_2,q_3)$. Show that the

 Hamiltonian in this coordinate system is

 

 \begin{equation}

 H = \frac{p_1^2}{2m} + \frac{p_2^2}{2m}+\frac{p_3^2}{2m} + V(q_1,q_2,q_3)

 \end{equation}

 

 and discuss the physical meaning of $p_1$, $p_2$, $p_3$.

 ------

 

 

 

 *Problem (5.2).* Find the Hamiltonian for the same particle, in the spherical

 coordinate system $(q_1,q_2,q_3) = (r,\theta,\phi)$ related to the

 Cartesian by 

 $x + iy = r\,\sin{\theta}\,e^{i\phi}$, $z=r\,\cos{\theta}$., and again

 discuss the physical meaning of $p_1$, $p_2$, $p_3$ .

 ------

 #+end_quote

 

 In terms of the Hamiltonian, the equations of motion assume a more

 symmetrical form:

 

 \begin{equation}

 \cdot{q}_i = \frac{\partial H}{\partial p_i}\qquad \dot{p}_i =

 -\frac{\partial H}{\partial q_i}

 \end{equation}

 

 of which the first follows from the definition (5-5) , while the second is

 equivalent to (5-2).

 

 The above formulation of mechanics holds only when all forces are 

 conservative; i.e. derivable from a potential energy function

 $V(q_1,\ldots q_n)$ , and

 in this case the Hamiltonian is numerically equal to the total energy $(T + V)$.

 Often, in addition to the conservative forces we have non-conservative ones

 which depend on the velocities as well as the coordinates. The Lagrangian

 and Hamiltonian form of the equations of motion can be preserved if there

 exists a new potential function $M(q_i,\dot{q}_i)$ such that the non-conservative force

 acting on coordinate $q_i$ is

 

 \begin{equation}

 F_i = \frac{d}{dt}\frac{\partial M}{\partial \dot{q}_i} -

 \frac{\partial M}{\partial q_i}

 \end{equation}

 

 We then define the Lagrangian as $L \equiv T - V - M$.

 

 #+begin_quote

 ------

 *Problem (5.3).* Show that the Lagrangian equations of motion (5-2)

 are correct with this modified Lagrangian. Find the new momenta and

 Hamiltonian. Carry this through explicitly for the case of a charged particle moving in a

 time-varying electromagnetic field $\vec{E}(x,y,z,t),

 \vec{H}(x,y,z,t)$, for which the 

 non-conservative force is given by the Lorentz force law,

 

 \(\vec{F} = e\left(\vec{E} + \frac{1}{c}\vec{v} \times \vec{B}\right)\)

 

 # Jaynes wrote \dot{A}. typo?

 /Hint:/ Express the potential $M$ in terms of the vector and scalar

 potentials of the field \(\vec{A},\phi,\) defined by

 \(\vec{B}=\vec{\nabla}\times\vec{A},

 \vec{E}=-\vec{\nabla}{\phi}-\frac{1}{c}\vec{A}\).

 Notice that, since the potentials are not uniquely determined by $E$, $H$, there is no longer any

 unique connection between momentum and velocity; or between the Hamiltonian

 and the energy. Nevertheless, the Lagrangian and Hamiltonian equations of

 motion still describe the correct physical laws.

 -----

 #+end_quote

 ** Liouville's Theorem

 The Hamiltonian form (5-7) is of particular value because of the following

 property. Let the coordinates and momenta $(q_1,p_1;\ldots;q_n,p_n)$

 be regarded as coordinates of a single point in a $2n$-dimensional /phase space/. This point moves,

 by virtue of the equations of motion, with a velocity $v$ whose

 components are $\langle \dot{q}_1, \dot{p}_1; \ldots; \dot{q}_n,\dot{p}_n\rangle$.

 At each point of phase space there is specified in this way a

 particular velocity, and the equations of motion thus generate a continuous

 flow pattern in phase space, much like the flow pattern of a fluid in ordinary

 space. The divergence of the velocity of this flow pattern is

 

 \begin{eqnarray}

 \vec{\nabla}\cdot {v}&=&\sum_{i=1}^n \left[\frac{\partial \dot{q}_i}{\partial q_i} +

 \frac{\partial \dot{p}_i}{\partial p_i}\right]\\

 &=& \sum_{i=1}^n \left[\frac{\partial^2 H}{\partial q_i \partial

 p_i}-\frac{\partial^2 H}{\partial p_i \partial q_i}\right]\\

 &=& 0

 \end{eqnarray}

 

 # note: this is a sort of Jacobian determinant/commutator|((d_q q_p)(d_q d_p))|

 

 so that the flow in phase space corresponds to that of an [[http://en.wikipedia.org/wiki/Incompressible_flow][incompressible fluid]].

 In an incompressible flow, the volume occupied by any given mass of the

 fluid remains constant as time goes on and the mass of fluid is carried into

 various regions. An exactly analogous property holds in phase space by virtue

 of (5-9). Consider at time $t = 0$ any $2n$-dimensional region

 $\Gamma_0$ consisting of some possible range of initial conditions

 $q_i(O), p_i(O)$  for a mechanical system, as shown in Fig. (5.1). This region has a total phase volume

 

 \begin{equation}

 \Omega(0) = \int_{\Gamma_{0}} dq_1\ldots dp_n

 \end{equation}

 

 In time t, each point $\langle q_1(O) \ldots p_n(O)\rangle$ of

 $\Gamma_0$ is carried, by the equations of

 motion, into a new point $\langle q_1(t),\ldots,p_n(t)\rangle$. The totality of all points which

 were originally in $\Gamma_0$ now defines a new region $\Gamma_t$ with phase volume

 

 \(\Omega(t) = \int_{\Gamma_{t}} dq_1\ldots dp_n\)

 

 and from (5-9) it can be shown that

 

 \begin{equation}

 \Omega(t) = \Omega(0)

 \end{equation}

 

 #+caption: Figure 5.1: Volume-conserving flow in phase space.

 [[../images/volume-conserved.jpg]]

 

 

 An equivalent statement is that the Jacobian determinant of the

 transformation \( \langle q_1(0), \ldots, p_n(0)\rangle \mapsto

 \langle q_1(t), \ldots , p_n(t)\rangle \) is identically equal to

 unity:

 

 \begin{equation}

 \frac{\partial(q_{1t},\ldots p_{nt})}{\partial(q_{10}\ldots q_{n0})} =

 \left|

 \begin{matrix}

 \frac{\partial q_{1t}}{\partial q_{10}}&\cdots &

 \frac{\partial p_{nt}}{\partial q_{10}}\\ 

 \vdots&\ddots&\vdots\\

 \frac{\partial q_{1t}}{\partial p_{n0}}&\cdots &

 \frac{\partial p_{nt}}{\partial p_{n0}}\\ 

 \end{matrix}\right| = 1

 \end{equation}

 

 #+begin_quote

 ------

 *Problem (5.4).* Prove that (5-9), (5-11), and (5-12) are equivalent statements.

 (/Hint:/ See A. I. Khinchin, /Mathematical Foundations of Statistical

 Mechanics/, Chapter II.)

 ------

 #+end_quote

 

 This result was termed by Gibbs the \ldquo{}Principle of conservation

 of extension-in—phase\rdquo{}, and is usually referred to nowadays as /Liouville's theorem/.

 An important advantage of considering the motion of a system referred to phase

 space (coordinates and momenta) instead of the coordinate—velocity space of

 the Lagrangian is that in general no such conservation law holds in the latter

 space (although they amount to the same thing in the special case where all

 the $q_i$ are cartesian coordinates of particles and all forces are conservative

 in the sense of Problem 5.3).

 

 #+begin_quote

 ------

 *Problem (5.5).* Liouville's theorem holds only because of the special form of

 the Hamiltonian equations of motion, which makes the divergence (5-9) 

 identically zero. Generalize it to a mechanical system whose state is defined by a

 set of variables $\{x_1,x_2,\ldots,x_n\}$ with equations of motion for

 $x_i(t)$:

 \begin{equation}

 \dot{x}_i(t) = f_i(x_1,\ldots,x_n),\qquad i=1,2,\ldots,n

 \end{equation}

 

 The jacobian (5-12) then corresponds to

 

 \begin{equation}

 J(x_1(0),\ldots,x_n(0);t) \equiv \frac{\partial[x_1(t),\ldots, x_n(t)]}{\partial[x_1(0),\ldots,x_n(0)]}

 \end{equation}

 

 Prove that in place of Liouville's theorem $J=1=\text{const.}$, we now

 have

 

 \begin{equation}

 $J(t) = J(0)\,\exp\left[\int_0^t \sum_{i=1}^n \frac{\partial

 f[x_1(t),\ldots, x_n(t)]}{\partial x_i(t)}

 dt\right].

 \end{equation}

 ------

 #+end_quote

 

 ** The Structure Function

 

 One of the essential dynamical properties of a system, which controls its

 thermodynamic properties, is the total phase volume compatible with various

 experimentally observable conditions. In particular, for a system in which

 the Hamiltonian and the energy are the same, the total phase volume below a

 certain energy $E$ is

 

 \begin{equation}

 \Omega(E) = \int \vartheta[E-H(q_i,p_i)] dq_i\ldots dp_n

 \end{equation}

 (When limits of integration are unspecified, we understand integration over

 all possible values of qi, pi.) In (5-16) , $\vartheta(x)$ is the unit

 step function

 

 \begin{equation}

 \vartheta(x) \equiv \begin{cases}1,&x>0\\ 0,&x<0\end{cases}

 \end{equation}

 

 The differential phase volume, called the /structure function/, is

 given by

 \begin{equation}

 \rho(E) = \frac{d\Omega}{dE} = \int \delta[E-H(q_i,p_i)] dq_1\ldots dp_n

 \end{equation}

 

 and it will appear presently that essentially all thermodynamic properties of

 the system are known if $\rho(E)$ is known, in its dependence on such parameters

 as volume and mole numbers.

 

 

 Calculation of $\rho(E)$ directly from the definition (5-18) is generally

 very difficult. It is much easier to calculate first its [[http://en.wikipedia.org/wiki/Laplace_transform][Laplace transform]],

 known as the /partition function/:

 

 \begin{equation}

 Z(\beta) = \int_0^\infty e^{-\beta E} \rho(E)\, dE

 \end{equation}

 

 where we have assumed that all possible values of energy are positive; this

 can always be accomplished for the systems of interest by

 appropriately choosing the zero from which we measure energy. In addition, it will develop that

 full thermodynamic information is easily extracted directly from the partition

 function $Z(\beta)$ , so that calculation of the structure function

 $\rho(E)$ is 

 unnecessary for some purposes.

 

 * COMMENT 

 Using (1-18) , the partition function can be written as

 which is the form most useful for calculation. If the structure function p (E)

 is needed, it is then found by the usual rule for inverting a Laplace trans

 form:

 the path of integration passing to the right of all singularities of Z(B) , as

 in Fig. (5.2) -

 

 

 Figure 5.2. Path of integration in Equation (5-21) .

 To illustrate the above relations, we now compute the partition function

 and structure function in two simple examples.

 Example 1. Perfect monatomic gas. We have N atoms, located by cartesian co

 ordinates ql...qN, and denote a particular component (direction in space) by

 an index oz, 0: = l, 2, 3. Thus, qia denotes the component of the position

 vector of the particle. Similarly, the vector momenta of the particles

 are denoted by pl.. .pN, and the individual components by pig. The Hamiltonian

 and the potential function u(q) defines the box of volume V containing the

 

 

 otherwise

 The arbitrary additive constant uo, representing the zero from which we

 measure our energies, will prove convenient later. The partition function is

 then

 

 If N is an even number, the integrand is analytic everywhere in the com

 plex except for the pole of order 3N/2 at the origin. If E > Nuo,

 the integrand tends to zero very rapidly as GO in the left half—plane

 Re(;,%) 5 O. The path of integration may then be extended to a closed one by

 addition of an infinite semicircle to the left, as in Fig. (5.3), the integral

 over the semicircle vanishing. We can then apply the Cauchy residue theorem

 where the closed contour C, illustrated in Fig. (5.4) , encloses the point

 z = a once in a counter—clockwise direction, and f(z) is analytic everywhere

 on and within C.

 

 

 

 

 

 

 

 

 * COMMENT  Appendix

 

 | Generalized Force  | Generalized Displacement |

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

 | force              | displacement             |

 | pressure           | volume                   |

 | electric potential | charge                   |