# HG changeset patch
# User Dylan Holmes <ocsenave@gmail.com>
# Date 1335659570 18000
# Node ID 26acdaf2e8c753921917294f912b0362db09f4ec

beginit begins.

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+#+TITLE: Statistical Mechanics
+#+AUTHOR:    E.T. Jaynes; edited by  Dylan Holmes
+#+EMAIL:     rlm@mit.edu
+#+KEYWORDS: statistical mechanics, thermostatics, thermodynamics, temperature, paradoxes, Jaynes
+#+SETUPFILE: ../../aurellem/org/setup.org
+#+INCLUDE:   ../../aurellem/org/level-0.org
+#+MATHJAX: align:"left" mathml:t path:"http://www.aurellem.org/MathJax/MathJax.js"
+
+# "extensions/eqn-number.js"
+
+#+begin_quote
+*Note:* The following is a typeset version of
+ [[../sources/stat.mech.1.pdf][this unpublished book draft]], written by [[http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes][E.T. Jaynes]]. I have only made
+ minor changes, e.g. to correct typographical errors, add references, or format equations. The
+ content itself is intact. --- Dylan
+#+end_quote
+
+* Development of Thermodynamics
+Our first intuitive, or \ldquo{}subjective\rdquo{} notions of temperature
+arise from the sensations of warmth and cold associated with our
+sense of touch . Yet science has been able to convert this qualitative 
+sensation into an accurately defined quantitative notion,
+which can be applied far beyond the range of our direct experience. 
+Today an experimentalist will report confidently that his
+spin system was at a temperature of 2.51 degrees Kelvin; and a
+theoretician will report with almost as much confidence that the 
+temperature at the center of the sun is about \(2 \times 10^7\) degrees
+Kelvin.
+
+The /fact/ that this has proved possible, and the main technical 
+ideas involved, are assumed already known to the reader;
+and we are not concerned here with repeating standard material
+already available in a dozen other textbooks . However 
+thermodynamics, in spite of its great successes, firmly established
+for over a century, has also produced a great deal of confusion
+and a long list of \ldquo{}paradoxes\rdquo{} centering mostly 
+around the second law and the nature of irreversibility. 
+For this reason and others noted below, we want to dwell here at 
+some length on the /logic/ underlying the development of 
+thermodynamics . Our aim is to emphasize certain points which, 
+in the writer's opinion, are essential for clearing up the 
+confusion and resolving the paradoxes; but which are not 
+sufficiently ernphasized---and indeed in many cases are 
+totally ignored---in other textbooks.
+
+This attention to logic 
+would not be particularly needed if we regarded classical 
+thermodynamics (or, as it is becoming called increasingly, 
+/thermostatics/) as a closed subject, in which the fundamentals 
+are already completely established, and there is
+nothing more to be learned about them. A person who believes
+this will probably prefer a pure axiomatic approach, in which
+the basic laws are simply stated as arbitrary axioms, without
+any attempt to present the evidence for them; and one proceeds
+directly to working out their consequences.
+However, we take the attitude here that thermostatics, for
+all its venerable age, is very far from being a closed subject,
+we still have a great deal to learn about such matters as the
+most general definitions of equilibrium and reversibility, the
+exact range of validity of various statements of the second and
+third laws, the necessary and sufficient conditions for 
+applicability of thermodynamics to special cases such as 
+spin systems, and how thermodynamics can be applied to such 
+systems as putty or polyethylene, which deform under force, 
+but retain a \ldquo{}memory\rdquo{} of their past deformations. 
+Is it possible to apply thermodynamics to a system such as a vibrating quartz crystal? We can by
+no means rule out the possibility that still more laws of 
+thermodynamics exist, as yet undiscovered, which would be 
+useful in such applications.
+
+
+It is only by careful examination of the logic by which
+present thermodynamics was created, asking exactly how much of
+it is mathematical theorems, how much is deducible from the laws
+of mechanics and electrodynamics, and how much rests only on
+empirical evidence, how compelling is present evidence for the
+accuracy and range of validity of its laws; in other words,
+exactly where are the boundaries of present knowledge, that we
+can hope to uncover new things. Clearly, much research is still
+needed in this field, and we shall be able to accomplish only a
+small part of this program in the present review.
+
+
+It will develop that there is an astonishingly close analogy
+with the logic underlying statistical theory in general, where
+again a qualitative feeling that we all have (for the degrees of
+plausibility of various unproved and undisproved assertions) must
+be convertefi into a precisely defined quantitative concept 
+(probability). Our later development of probability theory in 
+Chapter 6,7 will be, to a considerable degree, a paraphrase 
+of our present review of the logic underlying classical 
+thermodynamics.
+
+** The Primitive Thermometer. 
+
+The earliest stages of our
+story are necessarily speculative, since they took place long
+before the beginnings of recorded history. But we can hardly
+doubt that primitive man learned quickly that objects exposed
+to the sun‘s rays or placed near a fire felt different from
+those in the shade away from fires; and the same difference was
+noted between animal bodies and inanimate objects.
+
+
+As soon as it was noted that changes in this feeling of
+warmth were correlated with other observable changes in the
+behavior of objects, such as the boiling and freezing of water, 
+cooking of meat, melting of fat and wax, etc., the notion of
+warmth took its first step away from the purely subjective 
+toward an objective, physical notion capable of being studied
+scientifically.
+
+One of the most striking manifestations of warmth (but far
+from the earliest discovered) is the almost universal expansion
+of gases, liquids, and solids when heated . This property has
+proved to be a convenient one with which to reduce the notion
+of warmth to something entirely objective. The invention of the
+/thermometer/, in which expansion of a mercury column, or a gas,
+or the bending of a bimetallic strip, etc. is read off on a
+suitable scale, thereby giving us a /number/ with which to work,
+was a necessary prelude to even the crudest study of the physical
+nature of heat. To the best of our knowledge, although the
+necessary technology to do this had been available for at least
+3,000 years, the first person to carry it out in practice was
+Galileo, in 1592.
+
+Later on we will give more precise definitions of the term
+\ldquo{}thermometer.\rdquo{} But at the present stage we 
+are not in a position to do so (as Galileo was not), because 
+the very concepts needed have not yet been developed; 
+more precise definitions can be
+given only after our study has revealed the need for them. In
+deed, our final definition can be given only after the full
+mathematical formalism of statistical mechanics is at hand.
+
+Once a thermometer has been constructed, and the scale
+marked off in a quite arbitrary way (although we will suppose
+that the scale is at least monotonic: i.e., greater warmth always
+corresponds to a greater number), we are ready to begin scien
+tific experiments in thermodynamics. The number read eff from
+any such instrument is called the /empirical temperature/, and we
+denote it by \(t\). Since the exact calibration of the thermometer
+is not specified), any monotonic increasing function 
+\(t‘ = f(t)\) provides an equally good temperature scale for the
+present.
+
+
+** Thermodynamic Systems.
+
+The \ldquo{}thermodynamic systems\rdquo{} which
+are the objects of our study may be, physically, almost any
+collections of objects. The traditional simplest system with
+which to begin a study of thermodynamics is a volume of gas.
+We shall, however, be concerned from the start also with such
+things as a stretched wire or membrane, an electric cell, a
+polarized dielectric, a paramagnetic body in a magnetic field, etc.
+
+The /thermodynamic state/ of such a system is determined by
+specifying (i.e., measuring) certain macrcoscopic physical 
+properties. Now, any real physical system has many millions of such
+preperties; in order to have a usable theory we cannot require
+that /all/ of them be specified. We see, therefore, that there
+must be a clear distinction between the notions of 
+\ldquo{}thermodynamic system\rdquo{} and \ldquo{}physical
+system.\rdquo{}
+A given /physical/ system may correspond to many different
+/thermodynamic systems/, depending
+on which variables we choose to measure or control; and which
+we decide to leave unmeasured and/or uncontrolled.
+
+
+For example, our physical system might consist of a crystal
+of sodium chloride. For one set of experiments we work with
+temperature, volume, and pressure; and ignore its electrical
+properties. For another set of experiments we work with 
+temperature, electric field, and electric polarization; and 
+ignore the varying stress and strain. The /physical/ system, 
+therefore, corresponds to two entirely different /thermodynamic/ 
+systems. Exactly how much freedom, then, do we have in choosing 
+the variables which shall define the thermodynamic state of our
+system? How many must we choose? What [criteria] determine when
+we have made an adequate choice? These questions cannot be
+answered until we say a little more about what we are trying to
+accomplish by a thermodynamic theory. A mere collection of
+recorded data about our system, as in the [[http://en.wikipedia.org/wiki/CRC_Handbook_of_Chemistry_and_Physics][/Handbook of Physics and
+Chemistry/]], is a very useful thing, but it hardly constitutes
+a theory. In order to construct anything deserving of such a
+name, the primary requirement is that we can recognize some kind
+of reproducible connection between the different properties con
+sidered, so that information about some of them will enable us
+to predict others. And of course, in order that our theory can
+be called thermodynamics (and not some other area of physics),
+it is necessary that the temperature be one of the quantities
+involved in a nontrivial way.
+
+The gist of these remarks is that the notion of 
+\ldquo{}thermodynamic system\rdquo{} is in part 
+an anthropomorphic one; it is for us to
+say which set of variables shall be used. If two different
+choices both lead to useful reproducible connections, it is quite
+meaningless to say that one choice is any more \ldquo{}correct\rdquo{} 
+than the other. Recognition of this fact will prove crucial later in
+avoiding certain ancient paradoxes.
+
+At this stage we can determine only empirically which other
+physical properties need to be introduced before reproducible
+connections appear. Once any such connection is established, we
+can analyze it with the hope of being able to (1) reduce it to a
+/logical/ connection rather than an empirical one; and (2) extend
+it to an hypothesis applying beyond the original data, which
+enables us to predict further connections capable of being
+tested by experiment. Examples of this will be given presently.
+
+
+There will remain, however, a few reproducible relations
+which to the best of present knowledge, are not reducible to
+logical relations within the context of classical thermodynamics
+(and. whose demonstration in the wider context of mechanics,
+electrodynamics, and quantum theory remains one of probability
+rather than logical proof); from the standpoint of thermodynamics
+these remain simply statements of empirical fact which must be
+accepted as such without any deeper basis, but without which the
+development of thermodynamics cannot proceed. Because of this
+special status, these relations have become known as the
+\ldquo{}laws\rdquo{}
+of thermodynamics . The most fundamental one is a qualitative
+rather than quantitative relation, the \ldquo{}zero'th law.\rdquo{}
+
+** Equilibrium; the \ldquo{}Zero‘th Law.\rdquo{}
+
+ It is a common experience
+that when objects are placed in contact with each other but
+isolated from their surroundings, they may undergo observable
+changes for a time as a result; one body may become warmer,
+another cooler, the pressure of a gas or volume of a liquid may
+change; stress or magnetization in a solid may change, etc. But
+after a sufficient time, the observable macroscopic properties
+settle down to a steady condition, after which no further changes
+are seen unless there is a new intervention from the outside.
+When this steady condition is reached, the experimentalist says
+that the objects have reached a state of /equilibrium/ with each
+other. Once again, more precise definitions of this term will
+be needed eventually, but they require concepts not yet developed.
+In any event, the criterion just stated is almost the only one
+used in actual laboratory practice to decide when equilibrium
+has been reached.
+
+
+A particular case of equilibrium is encountered when we
+place a thermometer in contact with another body. The reading
+\(t\) of the thermometer may vary at first, but eventually it reach es
+a steady value. Now the number \(t\) read by a thermometer is always.
+by definition, the empirical temperature /of the thermometer/ (more
+precisely, of the sensitive element of the thermometer). When
+this number is constant in time, we say that the thermometer is
+in /thermal equilibrium/ with its surroundings; and we then extend
+the notion of temperature, calling the steady value \(t\) also the
+/temperature of the surroundings/.
+
+We have repeated these elementary facts, well known to every
+child, in order to emphasize this point: Thermodynamics can be
+a theory /only/ of states of equilibrium, because the very 
+procedure by which the temperature of a system is defined by 
+operational means, already presupposes the attainment of 
+equilibrium. Strictly speaking, therefore, classical 
+thermodynamics does not even contain the concept of a 
+\ldquo{}time-varying temperature.\rdquo{}
+
+Of course, to recognize this limitation on conventional
+thermodynamics (best emphasized by calling it instead, 
+thermostatics) in no way rules out the possibility of 
+generalizing the notion of temperature to nonequilibrium states. 
+Indeed, it is clear that one could define any number of 
+time-dependent quantities all of which reduce, in the special 
+case of equilibrium, to the temperature as defined above. 
+Historically, attempts to do this even antedated the discovery 
+of the laws of thermodynamics, as is demonstrated by 
+\ldquo{}Newton's law of cooling.\rdquo{} Therefore, the
+question is not whether generalization is /possible/, but only
+whether it is in any way /useful/; i.e., does the temperature so
+generalized have any connection with other physical properties
+of our system, so that it could help us to predict other things?
+However, to raise such questions takes us far beyond the
+domain of thermostatics; and the general laws of nonequilibrium
+behavior are so much more complicated that it would be virtually
+hopeless to try to unravel them by empirical means alone. For
+example, even if two different kinds of thermometer are calibrated
+so that they agree with each other in equilibrium situations,
+they will not agree in general about the momentary value a
+\ldquo{}time-varying temperature.\rdquo{} To make any real 
+progress in this area, we have to supplement empirical observation by the guidance
+of a rather hiqhly-developed theory. The notion of a 
+time-dependent temperature is far from simple conceptually, and we
+will find that nothing very helpful can be said about this until
+the full mathematical apparatus of nonequilibrium statistical
+mechanics has been developed.
+
+Suppose now that two bodies have the same temperature; i.e.,
+a given thermometer reads the same steady value when in contact
+with either. In order that the statement, \ldquo{}two bodies have the
+same temperature\rdquo{} shall describe a physi cal 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 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) 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 (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
+(2) will /not/ hold. Equation (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 $(n+1)$   
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