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).
Dynamics
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).
Tutorial
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