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     2.3 @@ -30,7 +30,7 @@
     2.4  The /fact/ that this has proved possible, and the main technical 
     2.5  ideas involved, are assumed already known to the reader;
     2.6  and we are not concerned here with repeating standard material
     2.7 -already available in a dozen other textbooks . However 
     2.8 +already available in a dozen other textbooks. However 
     2.9  thermodynamics, in spite of its great successes, firmly established
    2.10  for over a century, has also produced a great deal of confusion
    2.11  and a long list of \ldquo{}paradoxes\rdquo{} centering mostly 
    2.12 @@ -571,7 +571,7 @@
    2.13  only by our patience in constructing the apparatus needed to control
    2.14  or measure all these quantities.
    2.15  
    2.16 -We leave it as an exercise for the reader (Problem 1) to find the most
    2.17 +We leave it as an exercise for the reader (Problem 1.1) to find the most
    2.18  general condition on the variables \(\{v_1, q_1, v_2, q_2, \ldots
    2.19  v_n,q_n\}\) which will ensure that a definite equation of state
    2.20  $f(P,V,t)=0$ is observed in spite of all these new degrees of
    2.21 @@ -1945,8 +1945,8 @@
    2.22  
    2.23  Equation (1-66) is recognized as an approximate form of the Vapor
    2.24  pressure formula
    2.25 -We note that AQ, AV, which appeared first as integration
    2.26 -constants for the entropy with no parti cular physical meaning,
    2.27 +We note that $A_\ell$, $A_v$, which appeared first as integration
    2.28 +constants for the entropy with no particular physical meaning,
    2.29  now play a role in determining the vapor pressure.
    2.30  
    2.31  ** The Second Law: Discussion
    2.32 @@ -2049,7 +2049,7 @@
    2.33  chemical activation energy hills, etc.
    2.34  
    2.35  
    2.36 -It was Maxwell himself who first ([[../sources/Maxwell-Heat.pdf][1871]])[fn::See also, the [[http://openlibrary.org/books/OL7243600M/Theory_of_heat][Open Library
    2.37 +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
    2.38  page]], where you can read and download Maxwell's book in a variety of formats.] suggested such 
    2.39  possibilities, in his invention of the \ldquo{}Maxwell Demon\rdquo{}, 
    2.40  an imaginary being (or mechanism) which can regulate valves so as to allow
    2.41 @@ -2075,7 +2075,6 @@
    2.42  fools of theorists who were too quick to assert that something
    2.43  cannot be done.
    2.44  
    2.45 -
    2.46  A recent example worth recalling concerns the Overhauser
    2.47  effect in magnetic resonance (enhancement of the polarization
    2.48  of one set of spins by irradiation of another set coupled to them).
    2.49 @@ -2087,7 +2086,7 @@
    2.50  the second law, or how to apply it in new situations.
    2.51  
    2.52  In this connection, there is a fascinating little gadget
    2.53 -known as the Hilsch tube or Vortex tube, in which a jet of
    2.54 +known as the [[http://en.wikipedia.org/wiki/Vortex_tube][Hilsch tube]] or Vortex tube, in which a jet of
    2.55  compressed air is injected into a pipe at right angles to its
    2.56  axis, but off center so that it sets up a rapid rotational
    2.57  motion of the gas. In some manner, this causes a separation of
    2.58 @@ -2097,11 +2096,12 @@
    2.59  air to escape, the other side is left open so that the hot air
    2.60  can escape. The result is that when compressed air at room
    2.61  temperature is injected, one can obtain air from the hot side
    2.62 -at +100^\circ F from the cold side at -70^\circ F, in sufficient quantities
    2.63 +at $+100^\circ$ F from the cold side at $-70^\circ$ F, in sufficient quantities
    2.64  to be used for quick-freezing small objects, or for cooling
    2.65 -photomultiplier tubes [for construction drawings and experi
    2.66 -mental data, see Stong (1960); for a partial thermodynamic
    2.67 -analysis, see Hilsch (19-47)].
    2.68 +photomultiplier tubes [for construction drawings and 
    2.69 +experimental data, see [[http://books.google.com/books?id=yOUWAAAAIAAJ][Stong (1960)]]; for a partial thermodynamic
    2.70 +analysis, see Hilsch (1947)[fn::Edit: Hilsch's paper is entitled /The use of the expansion of gases in
    2.71 +a centrifugal field as a cooling process./]].
    2.72  
    2.73  Of course, the air could also be cooled by adiabatic expansion 
    2.74  (i.e., by doing work against a piston); and it appears that
    2.75 @@ -2191,6 +2191,810 @@
    2.76  question whether such machines can or cannot be made.
    2.77  
    2.78  
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   2.103 +
   2.104 +
   2.105 +* COMMENT Use of Jacobians in Thermodynamics
   2.106 +
   2.107 +Many students find that thermodynamics, although mathematically almost
   2.108 +trivial, is nevertheless one of the most difficult subjects in their program.
   2.109 +A large part of the blame for this lies in the extremely cumbersome partial
   2.110 +derivative notation. In this chapter we develop a different mathematical
   2.111 +scheme, with which thermodynamic derivations can be carried out more easily,
   2.112 +and which gives a better physical insight into the meaning of thermodynamic
   2.113 +relations.
   2.114 +
   2.115 +***  COMMENT Editor's addendum
   2.116 +#+begin_quote
   2.117 +In order to help readers with the Jacobian material that follows, I
   2.118 +have included this section of supplementary material. --- Dylan
   2.119 +#+end_quote}
   2.120 +
   2.121 +Suppose your experimental parameters consist of three variables
   2.122 +$X,Y,Z$---say, volume, pressure, and temperature. Then the
   2.123 +physically allowed combinations $\langle x,y,z\rangle$ of $X,Y,Z$
   2.124 +comprise the /(equilibrium) state space/
   2.125 +of your thermodynamic system; the set of these combinations forms a
   2.126 +subset $\Omega$ of $\mathbb{R}^3$. (If there were four experimental
   2.127 +parameters, the state space would be a subset of $\mathbb{R}^4$, and
   2.128 +so on).
   2.129 +
   2.130 +You can represent the flux of some physical quantities (such as
   2.131 +heat, entropy, or number of moles) as a vector field spread throughout
   2.132 +$\Omega$, i.e., a function $F:\Omega\rightarrow \mathbb{R}^n$ sending
   2.133 +each state to the value of the vector at that state. 
   2.134 +When you trace out different paths through the state space 
   2.135 +$\gamma:[a,b]\rightarrow \Omega$, you can measure the net quantity
   2.136 +exchanged by
   2.137 +
   2.138 +\begin{equation}
   2.139 +\text{net exchange} = \int_a^b (F\circ \gamma)\cdot \gamma^\prime.
   2.140 +\end{equation}
   2.141 +
   2.142 +Some quantities are conservative.
   2.143 +
   2.144 +- If the vector field $F$ (representing the flux of a physical
   2.145 +  quantity) is in fact the gradient of some function
   2.146 +  $\varphi:\Omega\rightarrow \mathbb{R}$, then $F$ is conservative and
   2.147 +  $\varphi$ represents the value of the conserved quantity at each state.
   2.148 +- In this case, the value of $\varphi$ is completely determined by
   2.149 +  specifying the values of the experimental parameters $X, Y, Z$. In
   2.150 +  particular, it doesn't matter by which path the state was reached.
   2.151 +
   2.152 +
   2.153 +Some physical quantities (such as entropy or number of moles) are
   2.154 +completely determined by your experimental parameters $X, Y, Z$. Others (such as
   2.155 +heat) are not. For those quantities that are,
   2.156 +you have functions $\phi:\Omega\rightarrow \mathbb{R}$ sending each state
   2.157 +to the value of the quantity at that state.
   2.158 +
   2.159 +
   2.160 +
   2.161 + and measure the change in physical
   2.162 +quantities (like entropy or number of moles)
   2.163 +
   2.164 +
   2.165 +Given your experimental parameters $X,Y,Z$, there may be other
   2.166 +physical quantities (such as entropy or number of moles) which are uniquely
   2.167 +defined by each combination of $\langle x,y,z\rangle$. Stated
   2.168 +mathematically, there is a function $f:\Omega\rightarrow \mathbb{R}$
   2.169 +sending each state to the value of the quantity at that state. 
   2.170 +
   2.171 +
   2.172 +
   2.173 +Now, sometimes you would like to use a different coordinate system to
   2.174 +describe the same physical situation.
   2.175 +A /change of variables/ is an
   2.176 +invertible differentiable transformation $g:\mathbb{R}^n\rightarrow
   2.177 +\mathbb{R}^n$---a function with $n$ input components (the $n$ old
   2.178 +variables) and $n$ output components (the $n$ new variables), where
   2.179 +each output component can depend on any number of the input components. For
   2.180 +example, in two dimensions you can freely switch between Cartesian
   2.181 +coordinates and polar coordinates; the familiar transformation is 
   2.182 +
   2.183 +\(g\langle x, y\rangle \mapsto \langle \sqrt{x^2+y^2}, \arctan{(y/x)}\rangle\)
   2.184 +
   2.185 +
   2.186 +
   2.187 +
   2.188 +
   2.189 +** Statement of the Problem
   2.190 +In fields other than thermodynamics , one usually starts out by stating
   2.191 +explicitly what variables shall be considered the independent ones, and then
   2.192 +uses partial derivatives without subscripts, the understanding being that all
   2.193 +independent variables other than the ones explicitly present are held constant
   2.194 +in the differentiation. This convention is used in most of mathematics and
   2.195 +physics without serious misunderstandings. But in thermodynamics, one never
   2.196 +seems to be able to maintain a fixed set of independent variables throughout
   2.197 +a derivation, and it becomes necessary to add one or more subscripts to every
   2.198 +derivative to indicate what is being held constant. The often-needed 
   2.199 +transformation from one constant quantity to another involves the
   2.200 +relation
   2.201 +
   2.202 +\begin{equation}
   2.203 +\left(\frac{\partial A}{\partial B}\right)_C = \left(\frac{\partial
   2.204 +A}{\partial B}\right)_D + \left(\frac{\partial A}{\partial D}\right)_B \left(\frac{\partial D}{\partial B}\right)_C 
   2.205 +\end{equation}
   2.206 +
   2.207 +which, although it expresses a fact that is mathematically trivial, assumes
   2.208 +such a complicated form in the usual notation that few people can remember it
   2.209 +long enough to write it down after the book is closed.
   2.210 +
   2.211 +As a further comment on notation, we note that in thermodynamics as well
   2.212 +as in mechanics and electrodynamics, our equations are made cumbersome if we
   2.213 +are forced to refer at all times to some particular coordinate system (i.e.,
   2.214 +set of independent variables). In the latter subjects this needless 
   2.215 +complication has long since been removed by the use of vector
   2.216 +notation, 
   2.217 +which enables us to describe physical relationships without reference to any particular
   2.218 +coordinate system. A similar house-cleaning can be effected for thermodynamics
   2.219 +by use of jacobians, which enable us to express physical relationships without
   2.220 +committing ourselves to any particular set of independent variables.
   2.221 +We have here an interesting example of retrograde progress in science:
   2.222 +for the historical fact is that use of jacobians was the original mathematical
   2.223 +method of thermodynamics. They were used extensively by the founder of modern
   2.224 +thermodynamics, Rudolph Clausius, in his work dating from about 1850. He used
   2.225 +the notation
   2.226 +
   2.227 +\begin{equation}
   2.228 +D_{xy} \equiv \frac{\partial^2 Q}{\partial x\partial y} -
   2.229 +\frac{\partial^2 Q}{\partial y \partial x}
   2.230 +\end{equation}
   2.231 +
   2.232 +
   2.233 +where $Q$ stands, as always, for heat, and $x$, $y$ are any 
   2.234 +two thermodynamic quantities. Since $dQ$ is not an exact differential, 
   2.235 +$D_{xy}$ is not identically zero. It is understandable that this notation, used in his published works, involved
   2.236 +Clausius in many controversies, which in retrospect appear highly amusing. An
   2.237 +account of some of them may be found in his book (Clausius, 1875). On the
   2.238 +other hand, it is unfortunate that this occurred, because it is probably for
   2.239 +this reason that the quantities $D_{xy}$ went out of general use for many years,
   2.240 +with only few exceptions (See comments at the end of this chapter). 
   2.241 +In a footnote in Chapter II of Planck's famous treatise (Planck, 1897), he explains
   2.242 +that he avoids using $dQ$ to represent an infinitesimal quantity of heat, because
   2.243 +that would imply that it is the differential of some quantity $Q$. This in turn
   2.244 +leads to the possibility of many fallacious arguments, all of which amount to
   2.245 +setting $D_{xy}=0$. However, a reading of Clausius‘ works makes it clear that
   2.246 +the quantities $D_{xy}$, when properly used, form the natural medium for discussion
   2.247 +of thermodynamics. They enabled him to carry out certain derivations with a
   2.248 +facility and directness which is conspicuously missing in most recent 
   2.249 +expositions. We leave it as an exercise for the reader to prove that $D_{xy}$ is a
   2.250 +jacobian (Problem 2.1).
   2.251 +
   2.252 +We now develop a condensed notation in which the algebra of jacobians
   2.253 +may be surveyed as a whole, in a form easy to remember since the abstract
   2.254 +relations are just the ones with which we are familiar in connection with the
   2.255 +properties of commutators in quantum mechanics.
   2.256 +
   2.257 +** 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). ]
   2.258 +Consider first a system with only two degrees of freedom. We define
   2.259 +
   2.260 +\begin{equation}
   2.261 +[A,B] \equiv \frac{\partial(A,B)}{\partial(x,y)} =
   2.262 +\left|\begin{matrix}\frac{\partial A}{\partial x}& \frac{\partial
   2.263 +A}{\partial y} \\
   2.264 +\frac{\partial B}{\partial x} & \frac{\partial B}{\partial y} \end{matrix}\right|
   2.265 +\end{equation}
   2.266 +where $x$, $y$ are any variables adequate to determine the state of the system.
   2.267 +
   2.268 +Since for any change of variables, $x,y \mapsto x^\prime, y^\prime$ we
   2.269 +have
   2.270 +
   2.271 +\begin{equation}
   2.272 +\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)}
   2.273 +\end{equation}
   2.274 +
   2.275 +or, in an easily understandable condensed notation,
   2.276 +
   2.277 +\begin{equation}
   2.278 +[A,B]^\prime = [A,B][x,y]^\prime
   2.279 +\end{equation}
   2.280 +
   2.281 +It follows that any equations that are homogeneous in the jacobians are in
   2.282 +variant in form under "coordinate transformations“, so that we can suppress
   2.283 +the independent variables x, y and carry out derivations without committing
   2.284 +ourselves to any particular set.
   2.285 +The algebra of these symbols is characterized by the following identities
   2.286 +(the comma may be omitted if A, B are single letters). The properties of
   2.287 +antisymmetry, linearity, and composition have the familiar form
   2.288 +In addition we have three cyclic identities, easily proved:
   2.289 +These relations are not all independent; for example, (2—ll) follows from
   2.290 +(2-9) and (2-13).
   2.291 +Putting dC = O in (2-9) , we obtain the rule
   2.292 +by means of which equations are translated from one language to the other.
   2.293 +
   2.294 +
   2.295 +From it one sees that the transformation law (2-l) now appears as a special
   2.296 +case of the identity (2-11) . Writing for the enthalpy, free energy, and Gibbs
   2.297 +function respectively ,
   2.298 +where U is the internal energy with the property dU = t :35 — P (N, we have as
   2.299 +consequences of (2-13) the relations
   2.300 +The advantages of this notation is shown particularly when we consider the
   2.301 +four Maxwe ll equati ons
   2.302 +Applying (2-14) , we see that each reduces to the single identity
   2.303 +
   2.304 +
   2.305 +Thus, all of the Maxwell equations are expressions in different "coordinate
   2.306 +systems" of the same basic fact (2-18) , which will receive a physical inter
   2.307 +pretation in Sec. 2.4. In a derivation, such as that of Eq. (1-49) , every
   2.308 +thing that can be gained by using any of the equations (2-17) is already
   2.309 +accomplished by application of the single relation (2-18).
   2.310 +Jacobians which involve the entropy in combinations other than are
   2.311 +related to various specific heats. The heat capacity at constant X is
   2.312 +and, using (2-14) we obtain the identity
   2.313 +C
   2.314 +In the simplest derivations, application of (2-18) or (2—20) is the essential
   2.315 +step.
   2.316 +In his well-known textbook, Zemansky (1943) shows that many of the ele
   2.317 +mentary derivations in thermodynamics may be reduced to application of the
   2.318 +In the above notation these equations are far from obvious and not easy to
   2.319 +remember. Note, however, that the T :38 equations are special cases of the
   2.320 +cyclic identity (2-9) for the sets of variables {TVS}, respectively,
   2.321 +while the energy equation is a consequence of (2-13) and the Maxwell relation:
   2.322 +
   2.323 +
   2.324 +From (2~l4) we see that this is the energy equation in jacobian notation.
   2.325 +2 .3 Elementary Examples
   2.326 +In a large class of problems, the objective is to express some quantity
   2.327 +of interest, or some condition of interest, in terms of experimentally mea
   2.328 +surable quantities. Therefore, the “sense of direction“ in derivations is
   2.329 +provided by the principle that we want to get rid of any explicit appearance
   2.330 +of the entropy and the various energies U, H, F, G. Thus, if the entropy
   2.331 +appears in the combination [TS], we use the Maxwell relation to replace it
   2.332 +with . If it appears in some other combination , we can use the
   2.333 +identity (2-20) .
   2.334 +Similarly, if combinations such as or [UX] appear, we can use (2-16)
   2.335 +and replace them with
   2.336 +it cannot be eliminated in this way. However, since in phenomenological
   2.337 +thermodynamics the absolute value of the entropy has no meaning, this situa
   2.338 +tion cannot arise in any expression representing a definite physical quantity.
   2.339 +For problems of this simplest type, the jacobian formalism works like a
   2.340 +well-oiled machine, as the following examples show. We denote the isothermal
   2.341 +compressibility, thermal expansion coefficient, and ratio of specific heats
   2.342 +bY K1 5: Y, réspectively:
   2.343 +
   2.344 +
   2.345 +and note that from (2-27) and (2-28) we have
   2.346 +(2-30)
   2.347 +Several derivatives, chosen at random, are now evaluated in terms of these
   2.348 +quantities:
   2.349 +A more difficult type of problem is the following: We have given a num
   2.350 +ber of quantities and wish to find the general relation, if any, connecting
   2.351 +them. In one sense, the question whether relations exist can be answered
   2.352 +
   2.353 +
   2.354 +immediately; for any two quantities A, B a necessary and sufficient condition
   2.355 +for the existence of a functional relation A f(B) in a region R is:
   2.356 += O in R}. In a system of two degrees of freedom it is clear that between
   2.357 +any three quantities A, B, C there is necessarily at least one functional
   2.358 +relation f(A,B,C) = O, as is implied by the identity (2-9) [Problem 2.2] . An
   2.359 +example is the equation of state f(PVT) = O. This , however, is not the type
   2.360 +of relation one usually has in mind. For each choice of A, B, C and each
   2.361 +particular system of two degrees of freedom, some functional relationship
   2.362 +must exist, but in general it will depend on the physical nature of the system
   2.363 +and can be obtained only when one has sufficient information, obtained from
   2.364 +measurement or theory, about the system.
   2.365 +The problem is rather to find those relations between various quantities
   2.366 +which hold generally, regardless of the nature of the particular system.
   2.367 +Mathematically, all such relations are trivial in the sense that they must be
   2.368 +special cases of the basic identities already given. Their physical meaning
   2.369 +may, however, be far from trivial and they may be difficult to find. Note,
   2.370 +for example, that the derivative computed in (2-35) is just the Joule—Thomson
   2.371 +coefficient 11. Suppose the problem had been stated as: "Given the five
   2.372 +quantities V, Cp, 8, determine whether there is a general relation
   2.373 +between them and if so find it." Now, although a repetition of the argument
   2.374 +of (2-35) would be successful in this case, this success must be viewed as a
   2.375 +lucky accident from ‘the standpoint of the problem just formulated. It is not
   2.376 +a general rule for attacking this type of problem because there is no way of
   2.377 +ensuring that the answer will come out in terms of the desired quantities.
   2.378 +To illustrate a general rule of procedure, consider the problem of find
   2.379 +ing a relationship, if any, between iCp, CV, V, T, B, K}. First we write
   2.380 +these quantities in terms of jacobians.
   2.381 +
   2.382 +
   2.383 +At this point we make a definite choice of some coordinate system. Since
   2.384 +[TP] occurs more often than any other jacobian, we adopt x = T, y = P as the
   2.385 +The variables in jacobians are P, V, T, S, for which (2-11) gives
   2.386 +[PV][TS] + [VT] [PS] + = 0 (2-40)
   2.387 +or, in this case
   2.388 +Substituting the expressions (2-39) into this we obtain
   2.389 +or, rearranging, we have the well—known law
   2.390 +which is now seen as a special case of (2-11).
   2.391 +There are several points to notice in this derivation: (1) no use has
   2.392 +been made of the fact that the quantities T, V were given explicitly; the
   2.393 +
   2.394 +
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   2.432 +
   2.433 +
   2.434 +* Gibbs Formalism \mdash{} Physical Derivation
   2.435 +
   2.436 +
   2.437 +In this Chapter we present physical arguments by which the Gibbs 
   2.438 +formalism can be derived and justified, deliberately avoiding all use
   2.439 +of probability theory. This will serve to convince us of the /validity/ of Gibbs’ formalism
   2.440 +for the particular applications given by Gibbs, and will give us an intuitive
   2.441 +physical understanding of the second law, as well as the physical meaning of
   2.442 +the Kelvin temperature.
   2.443 +
   2.444 +Later on (Chapter 9) we will present an entirely different derivation in
   2.445 +terms of a general problem of statistical estimation, deliberately avoiding
   2.446 +all use of physical ideas, and show that the identical mathematical formalism
   2.447 +emerges. This will serve to convince us of the /generality/ of the
   2.448 +Gibbs methods, and show that their applicability is in no way restricted to equilibrium
   2.449 +problems; or indeed, to physics.
   2.450 +
   2.451 +
   2.452 +It is interesting to note that most of Gibbs‘ important results were
   2.453 +found independently and almost simultaneously by Einstein (1902); but it is
   2.454 +to Gibbs that we owe the elegant mathematical formulation of the theory. In
   2.455 +the following we show how, from mechanical considerations involving
   2.456 +the microscopic state of a system, the Gibbs rules emerge as a
   2.457 +description of  equilibrium macroscopic properties. Having this, we can then reason
   2.458 +backwards, and draw inferences about microscopic conditions from macroscopic experimental
   2.459 +data. We will consider only classical mechanics here; however, none of this
   2.460 +classical theory will have to be unlearned later, because the Gibbs formalism
   2.461 +lost none of its validity through the development of quantum theory. Indeed,
   2.462 +the full power of Gibbs‘ methods has been realized only through their 
   2.463 +successful application to quantum theory.
   2.464 +
   2.465 +** COMMENT Review of Classical Mechanics (SICM)
   2.466 +In [[http://mitpress.mit.edu/sicm/][classical mechanics]] a complete description of the state of a system is
   2.467 +given by specifying $n$ coordinates $q_1\ldots q_n$, and the corresponding velocities
   2.468 +$D{q}_1\ldots Dq_n$. The equations of motion are then determined by a Lagrangian function
   2.469 +which in simple mechanical problems is
   2.470 +
   2.471 +\begin{equation}
   2.472 +L(t,q(t),Dq(t)) = T - V
   2.473 +\end{equation}
   2.474 +
   2.475 +
   2.476 +where $T$ and $V$ are the kinetic and potential energies. In problems involving
   2.477 +coupling of particles to an electromagnetic field, the Lagrangian function
   2.478 +takes a more general form, as we will see later. In either case, the 
   2.479 +equations of motion are
   2.480 +
   2.481 +\begin{equation}
   2.482 +D(\partial_2 L \circ \Gamma[q]) - \partial_1 L \circ \Gamma[q] = 0
   2.483 +\end{equation}
   2.484 +
   2.485 +where $\Gamma[q]$ is the function $t\mapsto \langle
   2.486 +t,q(t),Dq(t)\rangle$, and $\partial_i$ denotes the derivative with
   2.487 +respect to the \(i\)th argument ($i=0,1,2,\ldots$).
   2.488 +
   2.489 +The advantage of the Lagrangian form (5-2) over the original Newtonian form
   2.490 +(to which it is completely equivalent in simple mechanical problems)
   2.491 +
   2.492 +\begin{equation}
   2.493 +D^2 (m\cdot x(t)) = -\partial_1 V \circ \Gamma[x]
   2.494 +\end{equation}
   2.495 +
   2.496 +is that (5-2) holds for arbitrary choices of the coordinates $q_i$;
   2.497 +they can include angles, or any other parameters which serve to locate a particle in
   2.498 +space. The Newtonian equations (5-3), on the other hand, hold only when the
   2.499 +$x_i$ are rectangular (cartesian) coordinates of a particle.
   2.500 +Still more convenient for our purposes is the Hamiltonian form of the
   2.501 +equations of motion. Define the momentum \ldquo{}canonically
   2.502 +conjugate\rdquo{} to the 
   2.503 +coordinate $q$ by
   2.504 +
   2.505 +\begin{equation}
   2.506 +p(t) \equiv \partial_1 L \circ \Gamma[q]
   2.507 +\end{equation}
   2.508 +
   2.509 +let $\mathscr{V}(t,q,p) = Dq$, and define a Hamiltonian function $H$ by
   2.510 +
   2.511 +\begin{equation}
   2.512 +H(t,q,p) = p \cdot V(t,q,p) - L(t,q, V(t,q,p)
   2.513 +\end{equation}
   2.514 +
   2.515 +the notation indicating that after forming the right-hand side of (5-5) the
   2.516 +velocities $\dot{q}_i$ are eliminated mathematically, so that the
   2.517 +Hamiltonian is 
   2.518 +expressed as a function of the coordinates and momenta only.
   2.519 +
   2.520 +#+begin_quote
   2.521 +------
   2.522 +*Problem (5.1).* A particle of mass $m$ is located by specifying
   2.523 +$(q_1,q_2,q_3)=(r,\theta,z)$ respectively, where $r$, $\theta$, $z$
   2.524 +are a cylindrical coordinate system 
   2.525 +related to the cartesian $x, y, z$ by $x + iy = re^{i\theta}$, $z=z$. The
   2.526 +particle moves in a potential $V(q_1,q_2,q_3)$. Show that the
   2.527 +Hamiltonian in this coordinate system is
   2.528 +
   2.529 +\begin{equation}
   2.530 +H = \frac{p_1^2}{2m} + \frac{p_2^2}{2m}+\frac{p_3^2}{2m} + V(q_1,q_2,q_3)
   2.531 +\end{equation}
   2.532 +
   2.533 +and discuss the physical meaning of $p_1$, $p_2$, $p_3$.
   2.534 +------
   2.535 +
   2.536 +
   2.537 +
   2.538 +*Problem (5.2).* Find the Hamiltonian for the same particle, in the spherical
   2.539 +coordinate system $(q_1,q_2,q_3) = (r,\theta,\phi)$ related to the
   2.540 +Cartesian by 
   2.541 +$x + iy = r\,\sin{\theta}\,e^{i\phi}$, $z=r\,\cos{\theta}$., and again
   2.542 +discuss the physical meaning of $p_1$, $p_2$, $p_3$ .
   2.543 +------
   2.544 +#+end_quote
   2.545 +
   2.546 +
   2.547 +
   2.548 +
   2.549 +
   2.550 +
   2.551 +
   2.552 +
   2.553 +
   2.554 +
   2.555 +** Review of Classical Mechanics
   2.556 +In [[http://mitpress.mit.edu/sicm/][classical mechanics]] a complete description of the state of a system is
   2.557 +given by specifying $n$ coordinates $q_1\ldots q_n$, and the corresponding velocities
   2.558 +$\dot{q}_1\ldots \dot{q}_n$. The equations of motion are then determined by a Lagrangian function
   2.559 +which in simple mechanical problems is
   2.560 +
   2.561 +\begin{equation}
   2.562 +L(q_i,\dot{q}_i) = T - V
   2.563 +\end{equation}
   2.564 +
   2.565 +
   2.566 +where $T$ and $V$ are the kinetic and potential energies. In problems involving
   2.567 +coupling of particles to an electromagnetic field, the Lagrangian function
   2.568 +takes a more general form, as we will see later. In either case, the 
   2.569 +equations of motion are
   2.570 +
   2.571 +\begin{equation}
   2.572 +\frac{\partial L}{\partial q_i} - \frac{d}{dt}\frac{\partial
   2.573 +L}{\partial \dot{q}_i} = 0.
   2.574 +\end{equation}
   2.575 +
   2.576 +The advantage of the Lagrangian form (5-2) over the original Newtonian form
   2.577 +(to which it is completely equivalent in simple mechanical problems)
   2.578 +
   2.579 +\begin{equation}
   2.580 +m\ddot{x}_i = -\frac{\partial V}{\partial x_i}
   2.581 +\end{equation}
   2.582 +
   2.583 +is that (5-2) holds for arbitrary choices of the coordinates $q_i$;
   2.584 +they can include angles, or any other parameters which serve to locate a particle in
   2.585 +space. The Newtonian equations (5-3), on the other hand, hold only when the
   2.586 +$x_i$ are rectangular (cartesian) coordinates of a particle.
   2.587 +Still more convenient for our purposes is the Hamiltonian form of the
   2.588 +equations of motion. Define the momentum \ldquo{}canonically
   2.589 +conjugate\rdquo{} to the 
   2.590 +coordinate $q_i$ by
   2.591 +
   2.592 +\begin{equation}
   2.593 +p_i \equiv \frac{\partial L}{\partial q_i}
   2.594 +\end{equation}
   2.595 +
   2.596 +and a Hamiltonian function $H$ by
   2.597 +
   2.598 +\begin{equation}
   2.599 +H(q_1,p_1;\cdots ; q_n,p_n) \equiv \sum_{i=1}^n p\cdot \dot{q}_i -
   2.600 +L(q_1,\ldots, q_n). 
   2.601 +\end{equation}
   2.602 +
   2.603 +the notation indicating that after forming the right-hand side of (5-5) the
   2.604 +velocities $\dot{q}_i$ are eliminated mathematically, so that the Hamiltonian is ex
   2.605 +pressed as a function of the coordinates and momenta only.
   2.606 +
   2.607 +#+begin_quote
   2.608 +------
   2.609 +*Problem (5.1).* A particle of mass $m$ is located by specifying
   2.610 +$(q_1,q_2,q_3)=(r,\theta,z)$ respectively, where $r$, $\theta$, $z$
   2.611 +are a cylindrical coordinate system 
   2.612 +related to the cartesian $x, y, z$ by $x + iy = re^{i\theta}$, $z=z$. The
   2.613 +particle moves in a potential $V(q_1,q_2,q_3)$. Show that the
   2.614 +Hamiltonian in this coordinate system is
   2.615 +
   2.616 +\begin{equation}
   2.617 +H = \frac{p_1^2}{2m} + \frac{p_2^2}{2m}+\frac{p_3^2}{2m} + V(q_1,q_2,q_3)
   2.618 +\end{equation}
   2.619 +
   2.620 +and discuss the physical meaning of $p_1$, $p_2$, $p_3$.
   2.621 +------
   2.622 +
   2.623 +
   2.624 +
   2.625 +*Problem (5.2).* Find the Hamiltonian for the same particle, in the spherical
   2.626 +coordinate system $(q_1,q_2,q_3) = (r,\theta,\phi)$ related to the
   2.627 +Cartesian by 
   2.628 +$x + iy = r\,\sin{\theta}\,e^{i\phi}$, $z=r\,\cos{\theta}$., and again
   2.629 +discuss the physical meaning of $p_1$, $p_2$, $p_3$ .
   2.630 +------
   2.631 +#+end_quote
   2.632 +
   2.633 +In terms of the Hamiltonian, the equations of motion assume a more
   2.634 +symmetrical form:
   2.635 +
   2.636 +\begin{equation}
   2.637 +\cdot{q}_i = \frac{\partial H}{\partial p_i}\qquad \dot{p}_i =
   2.638 +-\frac{\partial H}{\partial q_i}
   2.639 +\end{equation}
   2.640 +
   2.641 +of which the first follows from the definition (5-5) , while the second is
   2.642 +equivalent to (5-2).
   2.643 +
   2.644 +The above formulation of mechanics holds only when all forces are 
   2.645 +conservative; i.e. derivable from a potential energy function
   2.646 +$V(q_1,\ldots q_n)$ , and
   2.647 +in this case the Hamiltonian is numerically equal to the total energy $(T + V)$.
   2.648 +Often, in addition to the conservative forces we have non-conservative ones
   2.649 +which depend on the velocities as well as the coordinates. The Lagrangian
   2.650 +and Hamiltonian form of the equations of motion can be preserved if there
   2.651 +exists a new potential function $M(q_i,\dot{q}_i)$ such that the non-conservative force
   2.652 +acting on coordinate $q_i$ is
   2.653 +
   2.654 +\begin{equation}
   2.655 +F_i = \frac{d}{dt}\frac{\partial M}{\partial \dot{q}_i} -
   2.656 +\frac{\partial M}{\partial q_i}
   2.657 +\end{equation}
   2.658 +
   2.659 +We then define the Lagrangian as $L \equiv T - V - M$.
   2.660 +
   2.661 +#+begin_quote
   2.662 +------
   2.663 +*Problem (5.3).* Show that the Lagrangian equations of motion (5-2)
   2.664 +are correct with this modified Lagrangian. Find the new momenta and
   2.665 +Hamiltonian. Carry this through explicitly for the case of a charged particle moving in a
   2.666 +time-varying electromagnetic field $\vec{E}(x,y,z,t),
   2.667 +\vec{H}(x,y,z,t)$, for which the 
   2.668 +non-conservative force is given by the Lorentz force law,
   2.669 +
   2.670 +\(\vec{F} = e\left(\vec{E} + \frac{1}{c}\vec{v} \times \vec{B}\right)\)
   2.671 +
   2.672 +# Jaynes wrote \dot{A}. typo?
   2.673 +/Hint:/ Express the potential $M$ in terms of the vector and scalar
   2.674 +potentials of the field \(\vec{A},\phi,\) defined by
   2.675 +\(\vec{B}=\vec{\nabla}\times\vec{A},
   2.676 +\vec{E}=-\vec{\nabla}{\phi}-\frac{1}{c}\vec{A}\).
   2.677 +Notice that, since the potentials are not uniquely determined by $E$, $H$, there is no longer any
   2.678 +unique connection between momentum and velocity; or between the Hamiltonian
   2.679 +and the energy. Nevertheless, the Lagrangian and Hamiltonian equations of
   2.680 +motion still describe the correct physical laws.
   2.681 +-----
   2.682 +#+end_quote
   2.683 +** Liouville's Theorem
   2.684 +The Hamiltonian form (5-7) is of particular value because of the following
   2.685 +property. Let the coordinates and momenta $(q_1,p_1;\ldots;q_n,p_n)$
   2.686 +be regarded as coordinates of a single point in a $2n$-dimensional /phase space/. This point moves,
   2.687 +by virtue of the equations of motion, with a velocity $v$ whose
   2.688 +components are $\langle \dot{q}_1, \dot{p}_1; \ldots; \dot{q}_n,\dot{p}_n\rangle$.
   2.689 +At each point of phase space there is specified in this way a
   2.690 +particular velocity, and the equations of motion thus generate a continuous
   2.691 +flow pattern in phase space, much like the flow pattern of a fluid in ordinary
   2.692 +space. The divergence of the velocity of this flow pattern is
   2.693 +
   2.694 +\begin{eqnarray}
   2.695 +\vec{\nabla}\cdot {v}&=&\sum_{i=1}^n \left[\frac{\partial \dot{q}_i}{\partial q_i} +
   2.696 +\frac{\partial \dot{p}_i}{\partial p_i}\right]\\
   2.697 +&=& \sum_{i=1}^n \left[\frac{\partial^2 H}{\partial q_i \partial
   2.698 +p_i}-\frac{\partial^2 H}{\partial p_i \partial q_i}\right]\\
   2.699 +&=& 0
   2.700 +\end{eqnarray}
   2.701 +
   2.702 +# note: this is a sort of Jacobian determinant/commutator|((d_q q_p)(d_q d_p))|
   2.703 +
   2.704 +so that the flow in phase space corresponds to that of an [[http://en.wikipedia.org/wiki/Incompressible_flow][incompressible fluid]].
   2.705 +In an incompressible flow, the volume occupied by any given mass of the
   2.706 +fluid remains constant as time goes on and the mass of fluid is carried into
   2.707 +various regions. An exactly analogous property holds in phase space by virtue
   2.708 +of (5-9). Consider at time $t = 0$ any $2n$-dimensional region
   2.709 +$\Gamma_0$ consisting of some possible range of initial conditions
   2.710 +$q_i(O), p_i(O)$  for a mechanical system, as shown in Fig. (5.1). This region has a total phase volume
   2.711 +
   2.712 +\begin{equation}
   2.713 +\Omega(0) = \int_{\Gamma_{0}} dq_1\ldots dp_n
   2.714 +\end{equation}
   2.715 +
   2.716 +In time t, each point $\langle q_1(O) \ldots p_n(O)\rangle$ of
   2.717 +$\Gamma_0$ is carried, by the equations of
   2.718 +motion, into a new point $\langle q_1(t),\ldots,p_n(t)\rangle$. The totality of all points which
   2.719 +were originally in $\Gamma_0$ now defines a new region $\Gamma_t$ with phase volume
   2.720 +
   2.721 +\(\Omega(t) = \int_{\Gamma_{t}} dq_1\ldots dp_n\)
   2.722 +
   2.723 +and from (5-9) it can be shown that
   2.724 +
   2.725 +\begin{equation}
   2.726 +\Omega(t) = \Omega(0)
   2.727 +\end{equation}
   2.728 +
   2.729 +#+caption: Figure 5.1: Volume-conserving flow in phase space.
   2.730 +[[../images/volume-conserved.jpg]]
   2.731 +
   2.732 +
   2.733 +An equivalent statement is that the Jacobian determinant of the
   2.734 +transformation \( \langle q_1(0), \ldots, p_n(0)\rangle \mapsto
   2.735 +\langle q_1(t), \ldots , p_n(t)\rangle \) is identically equal to
   2.736 +unity:
   2.737 +
   2.738 +\begin{equation}
   2.739 +\frac{\partial(q_{1t},\ldots p_{nt})}{\partial(q_{10}\ldots q_{n0})} =
   2.740 +\left|
   2.741 +\begin{matrix}
   2.742 +\frac{\partial q_{1t}}{\partial q_{10}}&\cdots &
   2.743 +\frac{\partial p_{nt}}{\partial q_{10}}\\ 
   2.744 +\vdots&\ddots&\vdots\\
   2.745 +\frac{\partial q_{1t}}{\partial p_{n0}}&\cdots &
   2.746 +\frac{\partial p_{nt}}{\partial p_{n0}}\\ 
   2.747 +\end{matrix}\right| = 1
   2.748 +\end{equation}
   2.749 +
   2.750 +#+begin_quote
   2.751 +------
   2.752 +*Problem (5.4).* Prove that (5-9), (5-11), and (5-12) are equivalent statements.
   2.753 +(/Hint:/ See A. I. Khinchin, /Mathematical Foundations of Statistical
   2.754 +Mechanics/, Chapter II.)
   2.755 +------
   2.756 +#+end_quote
   2.757 +
   2.758 +This result was termed by Gibbs the \ldquo{}Principle of conservation
   2.759 +of extension-in—phase\rdquo{}, and is usually referred to nowadays as /Liouville's theorem/.
   2.760 +An important advantage of considering the motion of a system referred to phase
   2.761 +space (coordinates and momenta) instead of the coordinate—velocity space of
   2.762 +the Lagrangian is that in general no such conservation law holds in the latter
   2.763 +space (although they amount to the same thing in the special case where all
   2.764 +the $q_i$ are cartesian coordinates of particles and all forces are conservative
   2.765 +in the sense of Problem 5.3).
   2.766 +
   2.767 +#+begin_quote
   2.768 +------
   2.769 +*Problem (5.5).* Liouville's theorem holds only because of the special form of
   2.770 +the Hamiltonian equations of motion, which makes the divergence (5-9) 
   2.771 +identically zero. Generalize it to a mechanical system whose state is defined by a
   2.772 +set of variables $\{x_1,x_2,\ldots,x_n\}$ with equations of motion for
   2.773 +$x_i(t)$:
   2.774 +\begin{equation}
   2.775 +\dot{x}_i(t) = f_i(x_1,\ldots,x_n),\qquad i=1,2,\ldots,n
   2.776 +\end{equation}
   2.777 +
   2.778 +The jacobian (5-12) then corresponds to
   2.779 +
   2.780 +\begin{equation}
   2.781 +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)]}
   2.782 +\end{equation}
   2.783 +
   2.784 +Prove that in place of Liouville's theorem $J=1=\text{const.}$, we now
   2.785 +have
   2.786 +
   2.787 +\begin{equation}
   2.788 +$J(t) = J(0)\,\exp\left[\int_0^t \sum_{i=1}^n \frac{\partial
   2.789 +f[x_1(t),\ldots, x_n(t)]}{\partial x_i(t)}
   2.790 +dt\right].
   2.791 +\end{equation}
   2.792 +------
   2.793 +#+end_quote
   2.794 +
   2.795 +** The Structure Function
   2.796 +
   2.797 +One of the essential dynamical properties of a system, which controls its
   2.798 +thermodynamic properties, is the total phase volume compatible with various
   2.799 +experimentally observable conditions. In particular, for a system in which
   2.800 +the Hamiltonian and the energy are the same, the total phase volume below a
   2.801 +certain energy $E$ is
   2.802 +
   2.803 +\begin{equation}
   2.804 +\Omega(E) = \int \vartheta[E-H(q_i,p_i)] dq_i\ldots dp_n
   2.805 +\end{equation}
   2.806 +(When limits of integration are unspecified, we understand integration over
   2.807 +all possible values of qi, pi.) In (5-16) , $\vartheta(x)$ is the unit
   2.808 +step function
   2.809 +
   2.810 +\begin{equation}
   2.811 +\vartheta(x) \equiv \begin{cases}1,&x>0\\ 0,&x<0\end{cases}
   2.812 +\end{equation}
   2.813 +
   2.814 +The differential phase volume, called the /structure function/, is
   2.815 +given by
   2.816 +\begin{equation}
   2.817 +\rho(E) = \frac{d\Omega}{dE} = \int \delta[E-H(q_i,p_i)] dq_1\ldots dp_n
   2.818 +\end{equation}
   2.819 +
   2.820 +and it will appear presently that essentially all thermodynamic properties of
   2.821 +the system are known if $\rho(E)$ is known, in its dependence on such parameters
   2.822 +as volume and mole numbers.
   2.823 +
   2.824 +
   2.825 +Calculation of $\rho(E)$ directly from the definition (5-18) is generally
   2.826 +very difficult. It is much easier to calculate first its [[http://en.wikipedia.org/wiki/Laplace_transform][Laplace transform]],
   2.827 +known as the /partition function/:
   2.828 +
   2.829 +\begin{equation}
   2.830 +Z(\beta) = \int_0^\infty e^{-\beta E} \rho(E)\, dE
   2.831 +\end{equation}
   2.832 +
   2.833 +where we have assumed that all possible values of energy are positive; this
   2.834 +can always be accomplished for the systems of interest by
   2.835 +appropriately choosing the zero from which we measure energy. In addition, it will develop that
   2.836 +full thermodynamic information is easily extracted directly from the partition
   2.837 +function $Z(\beta)$ , so that calculation of the structure function
   2.838 +$\rho(E)$ is 
   2.839 +unnecessary for some purposes.
   2.840 +
   2.841 +* COMMENT 
   2.842 +Using (1-18) , the partition function can be written as
   2.843 +which is the form most useful for calculation. If the structure function p (E)
   2.844 +is needed, it is then found by the usual rule for inverting a Laplace trans
   2.845 +form:
   2.846 +the path of integration passing to the right of all singularities of Z(B) , as
   2.847 +in Fig. (5.2) -
   2.848 +
   2.849 +
   2.850 +Figure 5.2. Path of integration in Equation (5-21) .
   2.851 +To illustrate the above relations, we now compute the partition function
   2.852 +and structure function in two simple examples.
   2.853 +Example 1. Perfect monatomic gas. We have N atoms, located by cartesian co
   2.854 +ordinates ql...qN, and denote a particular component (direction in space) by
   2.855 +an index oz, 0: = l, 2, 3. Thus, qia denotes the component of the position
   2.856 +vector of the particle. Similarly, the vector momenta of the particles
   2.857 +are denoted by pl.. .pN, and the individual components by pig. The Hamiltonian
   2.858 +and the potential function u(q) defines the box of volume V containing the
   2.859 +
   2.860 +
   2.861 +otherwise
   2.862 +The arbitrary additive constant uo, representing the zero from which we
   2.863 +measure our energies, will prove convenient later. The partition function is
   2.864 +then
   2.865 +
   2.866 +If N is an even number, the integrand is analytic everywhere in the com
   2.867 +plex except for the pole of order 3N/2 at the origin. If E > Nuo,
   2.868 +the integrand tends to zero very rapidly as GO in the left half—plane
   2.869 +Re(;,%) 5 O. The path of integration may then be extended to a closed one by
   2.870 +addition of an infinite semicircle to the left, as in Fig. (5.3), the integral
   2.871 +over the semicircle vanishing. We can then apply the Cauchy residue theorem
   2.872 +where the closed contour C, illustrated in Fig. (5.4) , encloses the point
   2.873 +z = a once in a counter—clockwise direction, and f(z) is analytic everywhere
   2.874 +on and within C.
   2.875 +
   2.876 +
   2.877 +
   2.878 +
   2.879 +
   2.880 +
   2.881 +
   2.882 +
   2.883  * COMMENT  Appendix
   2.884  
   2.885  | Generalized Force  | Generalized Displacement |