The Thermodynamics Program

Thermodynamics and his Big Brother Statistical Mecanics seem to have an inexorable hold on physicists. My interest in this field was stimulated by an introductory article in Physics Today: F. Weinhold, Geometry and Thermodynamics, Physics Today 29(3), 23-30 (1976). Weinhold's insight was to use the concavity property (when viewed from the right direction) of the equation of state surface in a geometric fashion. To be explicit, the set of second partial derivatives provide a set of matrix elements for a matrix that is positive semi-definite. This set of coefficients constitutes the ``second fundamental (curvature) form'' of classical diferential geometry. This is distinct from the first fundamental (distance) form of differential geometry that is used to measure distances between points on a surface. Here are the original contributions of Weinhold:

F. Weinhold,
Metric geometry of equilibrium thermodynamics,
J. Chem. Phys. 63(6), 2479-2483 (1975).

F. Weinhold,
Metric geometry of equilibrium thermodynamics.
II. Scaling, homogeneity, and generalized Gibbs-Duhem relations,
J. Chem. Phys. 63(6), 2484-2487 (1975).

F. Weinhold,
Metric geometry of equilibrium thermodynamics.
III. Elementary formal structure of a vector-algebraic representation of equilibrium thermodynamics,
J. Chem. Phys. 63(6), 2488-2495 (1975).

F. Weinhold,
Metric geometry of equilibrium thermodynamics.
IV. Vector-algebraic evaluation of thermodynamic derivatives,
J. Chem. Phys. 63(6), 2496-2501 (1975).

F. Weinhold, Metric geometry of equilibrium thermodynamics.
V. Aspects of heterogeneous equilibrium,
J. Chem. Phys. 65(2), 559-564 (1976).

Understanding the implications of this form preasented a challenge that was too good to pass up.

Thermodynamic Partial Derivatives
The first response to this challenge was to understand the role that differentials play on the equation of state surface. This is a simple problem of Linear Algebra. This understanding quickly leads to a simple, straightforward algorithm for resolving the bete-noire of all students of classical thermodynamics: that of computing thermodynamic partial derivatives. This problem was resolved in R. Gilmore, Thermodynamic Partial Derivatives, J. Chem. Phys. 75(12), 5964-5966 (1981).
This algorithm is implemented by a Maple Worksheet. An algorithm for computing higher thermodynamic partial derivatives has also been developed: R. Gilmore, Higher Thermodynamic Partial Derivatives, J. Chem. Phys. 77(11), 5853-5855 (1982).

Uncertainty Relations
Probability distributions occupy a central position in Statistical Mechanics. Wherever there is a probability distribution there is an uncertainty relation relating the probabilistic and statistical aspects of the theory. The uncertainty relations of statistical Mechanics are immediate consequences of inequalities derived from Weinhold's metric. They were derived here for the first time. R. Gilmore, Uncertainty relations of statistical mechanics, Phys. Rev. A 31(5), 3144-3146 (1985).

Discussion about the Length and Curvature Forms
Physicists who haven't had the opportunity to get their fingers into Gaussian geometry often have not seen the difference between the First Fundamental Form that describes distances and the Second Fundamental Form that describes curvature. The failure to understand this distinction lead to misunderstandings in the literature. The first attempt to correct this misperception is presented here. R. Gilmore, Length and curvature in the geometry of thermodynamics, Phys. Rev. A 30(4), 1994-1997 (1984). The message wasn't heard, so a second attempt was made. R. Gilmore, Reply to ``Comment of `Length and curvature in the geometry of thermodynamics' '', Phys. Rev. A 32(5), 3144-3146 (1985). This message also met with a lot of resistance. A last attempt to set the ideas straight was made here. B. Andresen, R. S. Berry, R. Gilmore, E. Ihrig, and P. Salamon, Thermodynamic geometry and the metrics of Weinhold and Gilmore, Phys. Rev. A 37(3), 845-848 (1988).

LeChatelier Relations
LeChatelier's Relations are often poorly understood and presented in an even more incomprehensible way. The short and simple idea is that when an equilibrium is defined by the minimum of a potential, displacements from the original minimum ultimately relax to the lowest available minimum: the old unless constraints prevent this. Determination of the new minimum is (not surprisingly) a simple problem of Linear Algebra. What is surprising is that the LeChatelier Relations exhibit some elegant symmetries, previously unnoticed. The announcement of these symmetries is presented here. R. Gilmore, LeChatelier Reciprocal Relations, J. Chem. Phys. 76(11), 5551-5553 (1982). These symmetries are illustrated in both a simple chemical context and also a mechanical context here: R. Gilmore, LeChatelier Reciprocal Relations and the Mechanical Analog, American Journal of Physics 51(8), 733-743 (1983). This Principle was extended to the case where several widely-separated relaxation time scales exist: R. Gilmore and R. D. Levine, LeChatelier's Principle with multiple relaxation channels, Phys. Rev. A 33(5), 3328-3332 (1986).

It is always a surprise to students that Thermodynamics says nothing about dynamics. Dynamics is described in the context of Nonequilibrium Thermodynamics. In this field the kinetic coefficients play the role that the standard thermodynamic susceptibilities play in classical thermodynamics. One might expect that the relaxation of a thermodynamic system to an equilibrium should involve both sets of coefficients. It was a delightful surprise to find (at long last) that the relaxation time scales and processes are defined by a generalized eigenvalue problem involving both sets of coefficients in a very natural way. This result was provoked by an invitation to offer a contribution to a Birthday Celebration for my colleague Prof. Raphael D. Levine: R. Gilmore, LeChatelier Dynamics, Molecular Physics 106(8), 991-997 (2008).

Thermodynamics is presented in very many books. Few approach this subject from the geometric point of view. I find this approach exceedingly convenient, if only for the bookkeeping simplifications that it allows. Here is a short summary of Classical Thermodynamics that I offer to my students. It is a (relatively) quick read that allows an understanding of the structure of classical thermodynamics, given in the straitjacket of geometry (the geometry of the second fundamental (curvature) form). R. Gilmore, The Structure of Thermodynamics, an internal document for my classes.

Saha Equation
It has been a frustrating experience to find a straightforward derivation/explanation of the Saha equation. In particular, this equation must be closely related to the partition functions of the appropriate component species. This problem was formulated in such a way that it was presentable to an undergraduate class in Statistical Mechanics: R. Gilmore, Saha's Equation and the Partition Function: an internal document for my classes.

Last Revision: February 23, 2012