# HG changeset patch # User Dylan Holmes # Date 1335668619 18000 # Node ID 4da2176e489001485f21c312036d83400995e486 # Parent 26acdaf2e8c753921917294f912b0362db09f4ec Transcribed up to section 1.5: heat diff -r 26acdaf2e8c7 -r 4da2176e4890 org/stat-mech.org --- a/org/stat-mech.org Sat Apr 28 19:32:50 2012 -0500 +++ b/org/stat-mech.org Sat Apr 28 22:03:39 2012 -0500 @@ -298,7 +298,7 @@ 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 @@ -390,7 +390,7 @@ 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/, @@ -403,8 +403,8 @@ 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 +A function like /f/ has to possess one more propery in order for its +level set to express a constraint relationship: it must be monotonic in each of its variables /X/, /x/, and /t/. #the partial derivatives of /f/ exist for every allowed combination of #inputs /x/, /X/, and /t/. @@ -423,7 +423,7 @@ 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 @@ -465,7 +465,7 @@ 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 @@ -473,7 +473,7 @@ 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 @@ -557,4 +557,83 @@ 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 |