jaynes: org/stat-mech.org comparison

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Transcribed up to section 1.5: heat

author	Dylan Holmes <ocsenave@gmail.com>
date	Sat, 28 Apr 2012 22:03:39 -0500
parents	26acdaf2e8c7
children	afbe1fe19b36

comparison

equal deleted inserted replaced

-:26acdaf2e8c7
+:4da2176e4890
 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,
+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
 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
+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.
 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
+A function like /f/ has to possess one more propery in order for its
-express a constraint relationship: it must be monotonic in
+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.]
 \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
+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
 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
+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
-(2) will /not/ hold. Equation (2), as we have just learned, expresses
+(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
 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 $(n+1)$
+our sample. Suppose we place $(n+1)$ different electrodes, labelled
+\(\{e_0,e_1, e_2 \ldots e_n\}\) in contact with the liquid in various
+positions. Regarding \(e_0\) as the \ldquo{}ground\rdquo{}, maintained
+at zero potential, we can then impose $n$ different potentials
+\(\{v_1, \ldots, v_n\}\) on the other electrodes independently, and we
+can also measure the $n$ different conjugate displacements, as the
+charges \(\{q_1,\ldots, q_n\}\) accumulated on electrodes
+\(\{e_1,\ldots e_n\}\). Together with the pressure (understood as the
+pressure measured at one given position), volume, and temperature, our
+sample of nitrobenzene is now a thermodynamic system of $(n+1)$
+degrees of freedom. This number may be as large as we please, limited
+only by our patience in constructing the apparatus needed to control
+or measure all these quantities.
+We leave it as an exercise for the reader (Problem 1) to find the most
+general condition on the variables \(\{v_1, q_1, v_2, q_2, \ldots
+v_n,q_n\}\) which will ensure that a definite equation of state
+$f(P,V,t)=0$ is observed in spite of all these new degrees of
+freedom. The simplest special case of this relation is, evidently, to
+ground all electrodes, thereby inposing the conditions $v_1 = v_2 =
+\ldots = v_n = 0$. Equally well (if we regard nitrobenzene as having
+negligible electrical conductivity) we may open-circuit all
+electrodes, thereby imposing the conditions \(q_i = \text{const.}\) In
+the latter case, in addition to an equation of state of the form
+\(f(P,V,t)=0\), which contains these constants as fixed parameters,
+there are \(n\) additional equations of state of the form $v_i =
+v_i(P,t)$. But if we choose to ignore these voltages, there will be no
+contradiction in considering our nitrobenzene to be a thermodynamic
+system of two degrees of freedom, involving only the variables
+\(P,V,t\).
+Similarly, if our system of interest is a crystal, we may impose on it
+a wide variety of nonuniform stress fields; each component of the
+stress tensor $T_{ij}$ may bary with position. We might expand each of
+these functions in a complete orthonormal set of functions
+\(\phi_k(x,y,z)\):
+\begin{equation}
+T_{ij}(x,y,z) = \sum_k a_{ijk} \phi_k(x,y,z)
+\end{equation}
+and with a sufficiently complicated system of levers which in various
+ways squeeze and twist the crystal, we might vary each of the first
+1,000 expansion coefficients $a_{ijk}$ independently, and measure the
+conjugate displacements $q_{ijk}$. Our crystal is then a thermodynamic
+system of over 1,000 degrees of freedom.
+The notion of \ldquo{}numbers of degrees of freedom\rdquo{} is
+therefore not a /physical property/ of any system; it is entirely
+anthropomorphic, since any physical system may be regarded as a
+thermodynamic system with any number of degrees of freedom we please.
+If new thermodynamic variables are always introduced in pairs,
+consisting of a \ldquo{}force\rdquo{} and conjugate
+\ldquo{}displacement\rdquo{}, then a thermodynamic system of $n$
+degrees of freedom must possess $(n-1)$ independent equations of
+state, so that specifying $n$ quantities suffices to determine all
+others.
+This raises an interesting question; whether the scheme of classifying
+thermodynamic variables in conjugate pairs is the most general
+one. Why, for example, is it not natural to introduce three related
+variables at a time? To the best of the writer's knowledge, this is an
+open question; there seems to be no fundamental reason why variables
+/must/ always be introduced in conjugate pairs, but there seems to be
+no known case in which a different scheme suggests itself as more
+appropriate.
+** Heat
+We are now in a position to consider the results and interpretation of
+a number of elementary experiments involving
+* Appendix
+| Generalized Force  | Generalized Displacement |
+|--------------------+--------------------------|
+| force              | displacement             |
+| pressure           | volume                   |
+| electric potential | charge                   |

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