1.1 --- a/org/stat-mech.org	Sat Apr 28 19:32:50 2012 -0500
     1.2 +++ b/org/stat-mech.org	Sat Apr 28 22:03:39 2012 -0500
     1.3 @@ -298,7 +298,7 @@
     1.4  Suppose now that two bodies have the same temperature; i.e.,
     1.5  a given thermometer reads the same steady value when in contact
     1.6  with either. In order that the statement, \ldquo{}two bodies have the
     1.7 -same temperature\rdquo{} shall describe a physi cal property of the bodies,
     1.8 +same temperature\rdquo{} shall describe a physical property of the bodies,
     1.9  and not merely an accidental circumstance due to our having used
    1.10  a particular kind of thermometer, it is necessary that /all/ 
    1.11  thermometers agree in assigning equal temperatures to them if 
    1.12 @@ -390,7 +390,7 @@
    1.13  by specifying any two of the variables arbitrarily, whereupon the
    1.14  third, and all others we may introduce, are determined. 
    1.15  Mathematically, this is expressed by the existence of a functional
    1.16 -relationship of the form[fn::Edit: The set of solutions to an equation
    1.17 +relationship of the form[fn:: /Edit./: The set of solutions to an equation
    1.18  like /f(X,x,t)=/ const. is called a /level set/. Here, Jaynes is
    1.19  saying that the quantities /X/, /x/, and /t/ follow a \ldquo{}functional
    1.20  rule\rdquo{}, so the set of physically allowed combinations of /X/,
    1.21 @@ -403,8 +403,8 @@
    1.22  the level set /f(X,x,t)=0/ is a sphere, and specifying /x=1/, /t=1/
    1.23  leaves you with two potential possibilities for /X/ =\pm 1.)
    1.24  
    1.25 -A function like /f/ has to possess one more propery in order to
    1.26 -express a constraint relationship: it must be monotonic in
    1.27 +A function like /f/ has to possess one more propery in order for its
    1.28 +level set to express a constraint relationship: it must be monotonic in
    1.29  each of its variables /X/, /x/, and /t/.
    1.30  #the partial derivatives of /f/ exist for every allowed combination of
    1.31  #inputs /x/, /X/, and /t/. 
    1.32 @@ -423,7 +423,7 @@
    1.33  where $X$ is a generalized force (pressure, tension, electric or
    1.34  magnetic field, etc.), $x$ is the corresponding generalized 
    1.35  displacement (volume, elongation, electric or magnetic polarization,
    1.36 -etc.), and $t$ is the empirical temperature. Equation (1) is
    1.37 +etc.), and $t$ is the empirical temperature. Equation (1-1) is
    1.38  called /the equation of state/.
    1.39  
    1.40  At the risk of belaboring it, we emphasize once again that
    1.41 @@ -465,7 +465,7 @@
    1.42  plane through which the present values were approached; or, as we 
    1.43  usually say it, \(dV\) is an /exact differential/.
    1.44  
    1.45 -Therefore, although at first glance the relation (2) appears
    1.46 +Therefore, although at first glance the relation (1-2) appears
    1.47  nontrivial and far from obvious, a trivial mathematical analysis
    1.48  convinces us that it must hold regardless of our particular
    1.49  temperature scale, and that it is true not only of oxygen; it must
    1.50 @@ -473,7 +473,7 @@
    1.51  definite, reproducible equation of state \(f(P,V,t)=0\).
    1.52  
    1.53  But this understanding also enables us to predict situations in which
    1.54 -(2) will /not/ hold. Equation (2), as we have just learned, expresses
    1.55 +(1-2) will /not/ hold. Equation (1-2), as we have just learned, expresses
    1.56  the fact that an equation of state exists involving only the three
    1.57  variables \((P,V,t)\). Now suppose we try to apply it to a liquid such
    1.58  as nitrobenzene. The nitrobenzene molecule has a large electric dipole
    1.59 @@ -557,4 +557,83 @@
    1.60  physical system has, for all practical purposes, an /arbitrarily
    1.61  large/ number of  degrees of freedom. In the case of nitrobenzene, for
    1.62  example, we may impose any variety of nonuniform electric fields on
    1.63 -our sample. Suppose we place $(n+1)$   
    1.64 +our sample. Suppose we place $(n+1)$ different electrodes, labelled
    1.65 +\(\{e_0,e_1, e_2 \ldots e_n\}\) in contact with the liquid in various
    1.66 +positions. Regarding \(e_0\) as the \ldquo{}ground\rdquo{}, maintained
    1.67 +at zero potential, we can then impose $n$ different potentials
    1.68 +\(\{v_1, \ldots, v_n\}\) on the other electrodes independently, and we
    1.69 +can also measure the $n$ different conjugate displacements, as the
    1.70 +charges \(\{q_1,\ldots, q_n\}\) accumulated on electrodes 
    1.71 +\(\{e_1,\ldots e_n\}\). Together with the pressure (understood as the
    1.72 +pressure measured at one given position), volume, and temperature, our
    1.73 +sample of nitrobenzene is now a thermodynamic system of $(n+1)$
    1.74 +degrees of freedom. This number may be as large as we please, limited
    1.75 +only by our patience in constructing the apparatus needed to control
    1.76 +or measure all these quantities.
    1.77 +
    1.78 +We leave it as an exercise for the reader (Problem 1) to find the most
    1.79 +general condition on the variables \(\{v_1, q_1, v_2, q_2, \ldots
    1.80 +v_n,q_n\}\) which will ensure that a definite equation of state
    1.81 +$f(P,V,t)=0$ is observed in spite of all these new degrees of
    1.82 +freedom. The simplest special case of this relation is, evidently, to
    1.83 +ground all electrodes, thereby inposing the conditions $v_1 = v_2 =
    1.84 +\ldots = v_n = 0$. Equally well (if we regard nitrobenzene as having
    1.85 +negligible electrical conductivity) we may open-circuit all
    1.86 +electrodes, thereby imposing the conditions \(q_i = \text{const.}\) In
    1.87 +the latter case, in addition to an equation of state of the form
    1.88 +\(f(P,V,t)=0\), which contains these constants as fixed parameters,
    1.89 +there are \(n\) additional equations of state of the form $v_i =
    1.90 +v_i(P,t)$. But if we choose to ignore these voltages, there will be no
    1.91 +contradiction in considering our nitrobenzene to be a thermodynamic
    1.92 +system of two degrees of freedom, involving only the variables
    1.93 +\(P,V,t\).
    1.94 +
    1.95 +Similarly, if our system of interest is a crystal, we may impose on it
    1.96 +a wide variety of nonuniform stress fields; each component of the
    1.97 +stress tensor $T_{ij}$ may bary with position. We might expand each of
    1.98 +these functions in a complete orthonormal set of functions
    1.99 +\(\phi_k(x,y,z)\):
   1.100 +
   1.101 +\begin{equation}
   1.102 +T_{ij}(x,y,z) = \sum_k a_{ijk} \phi_k(x,y,z)
   1.103 +\end{equation}
   1.104 +
   1.105 +and with a sufficiently complicated system of levers which in various
   1.106 +ways squeeze and twist the crystal, we might vary each of the first
   1.107 +1,000 expansion coefficients $a_{ijk}$ independently, and measure the
   1.108 +conjugate displacements $q_{ijk}$. Our crystal is then a thermodynamic
   1.109 +system of over 1,000 degrees of freedom. 
   1.110 +
   1.111 +The notion of \ldquo{}numbers of degrees of freedom\rdquo{} is
   1.112 +therefore not a /physical property/ of any system; it is entirely
   1.113 +anthropomorphic, since any physical system may be regarded as a
   1.114 +thermodynamic system with any number of degrees of freedom we please.
   1.115 +
   1.116 +If new thermodynamic variables are always introduced in pairs,
   1.117 +consisting of a \ldquo{}force\rdquo{} and conjugate
   1.118 +\ldquo{}displacement\rdquo{}, then a thermodynamic system of $n$
   1.119 +degrees of freedom must possess $(n-1)$ independent equations of
   1.120 +state, so that specifying $n$ quantities suffices to determine all
   1.121 +others.
   1.122 +
   1.123 +This raises an interesting question; whether the scheme of classifying
   1.124 +thermodynamic variables in conjugate pairs is the most general
   1.125 +one. Why, for example, is it not natural to introduce three related
   1.126 +variables at a time? To the best of the writer's knowledge, this is an
   1.127 +open question; there seems to be no fundamental reason why variables
   1.128 +/must/ always be introduced in conjugate pairs, but there seems to be
   1.129 +no known case in which a different scheme suggests itself as more
   1.130 +appropriate.
   1.131 +
   1.132 +** Heat
   1.133 +We are now in a position to consider the results and interpretation of
   1.134 +a number of elementary experiments involving
   1.135 +
   1.136 +
   1.137 +* Appendix
   1.138 +
   1.139 +| Generalized Force  | Generalized Displacement |
   1.140 +|--------------------+--------------------------|
   1.141 +| force              | displacement             |
   1.142 +| pressure           | volume                   |
   1.143 +| electric potential | charge                   |