J W Gibbs, Gibbs Models: computer-visualized thermodynamic surfaces

My interest in the visual models shown in this collection and, more generally, in the postulatory approach to thermodynamics arose from the first editions of the textbook by Modell and Reid, Thermodynamics and Its Applications, Prentice-Hall, 1974, 1983. I believe that the attention paid in that book to the Legendre transform and to the crucial role played by stability theory in any thorough understanding of classical thermodynamics makes it an essential resource for the serious student.

While on leave at the University of California at Berkeley in the early 1980s, working in Professor John Prausnitz' research group, my attention was first called to Gibbs' writings by a reviewer for The Journal of Chemical Education, Dr. Henry A. Bent, former Professor of Chemistry at North Carolina State University and at the University of Pittsburgh. While I was far from the first person to appreciate the visualizability of Gibbs' equations, I believe I was the first to try to create those drawings using computer graphics. I was assisted in the early stages of the work by Iowa State University graduate student Michael C. Schmitz ("Visualizing thermodynamic concepts through high-performance computer graphics," M.S. in ChE, ISU, 1991). Using a Silicon Graphics IRIS 3030 workstation, Michael created the first computer-drawn fundamental-equation surface in 1988 — the Helmholtz energy of a pure fluid.

Daniel C. Coy was my graduate student from 1988 through 1993 and is wholly responsible for the collection of drawings shown here. He developed the generating equations (for multicomponent systems, using extensive variables), wrote the convergence schemes to solve those equations, and created a transformation matrix that enabled him to plot any combination of thermodynamic variables attainable through the Legendre transformation. Dan also developed schemes for tessellating the surfaces so as to yield geometry files compatible with MOVIE.BYU graphics software. Dr. Coy's dissertation was an exemplary piece of scientific research and is mentioned elsewhere in this website.

I did not read Herbert Callen's highly regarded thermodynamics textbook until after I had begun work on these drawings, but his lineage and published writings placed him squarely on the road I was trying to follow. Professor Callen had been a graduate student of MIT physicist Laszlo Tisza, who had himself authored a book that discussed the postulatory approach, Generalized Thermodynamics, MIT Press, 1966. Dr. Callen was also a contemporary of Martin J. Klein, another Tisza graduate student who is now Professor Emeritus of The History of Science at Yale University and an eminent Gibbs historian.

In May 1989 I showed our first set of pure-fluid models at the Gibbs Symposium, a meeting organized by Professor Klein at Yale in honor of J. Willard Gibbs' 150th birthday. My presentation, entitled "Gibbs and the Art of Thermodynamics," may be found in the Proceedings of that Symposium. Soon after the meeting I wrote to Dr. Callen (then retired from the Physics Department at the University of Pennsylvania) and offered him a replica of the display. I received two letters from him that summer, and I would like to share excerpts from those notes with the viewers of this website:

June 16, 1989

Dear Prof. Jolls:

One of the pleasures resulting from writing my thermo book has certainly been the expressed appreciation of established scientists who succeed in conceptualizing the geometrical bases of the subject in new and sometimes ingenious ways. Personally, I think primarily in terms of the "Legendre potentials," usually quite abstractly, but sometimes visualized in terms of the family of surfaces of constant values of those potentials. The trick then is to somehow picture the manner in which those surfaces are transformed in shape by an additional Legendre transformation. I take it that you have succeeded in illustrating such transformations in an intuitively appealing way.... It also would be wonderful eventually to include some such illustrations in a thermodynamics text — perhaps in the form of a diskette that a student could exhibit in his own computer, and ponder over at his leisure! (I am certain that students will soon be able to generate and see such Legendre transforms for arbitrary functions in three dimensions on their home computers....)

Thank you for your kind letter, and I hope that your efforts to make the Gibbs geometrical formulation more intuitively clear and appealing will be enthusiastically appreciated.

Sincerely,

July 25, 1989

Dear Professor Jolls:

...

I did receive your elegant pictures, and I was awed by them. Firstly, for reasons which are not really central, but nevertheless real – that is, by admiration for the sheer computer mastery required to conceive of and to make them. I, of course, was reared in the pre-computer age.... My next reaction was a realization that students of thermodynamics in the future are going to have a more concrete view of a subject which has always been hampered by its abstract nature.

I suspect that the real impact of your computer techniques might well turn out to be in dynamics. How do fluctuations find a path for a system to emerge from a local minimum, and what time scales are involved? Here conventional theory is essentially powerless, whereas Monte Carlo methods grafted on to your representation of the thermodynamic potentials may well lead to real predictions of rate constants. (The scientific community is much more interested in new results than it is in pedagogical techniques, despite its lip-service to the latter!)....

I do wish that I had access to a color computer capable of displaying your elegant constructions. I hope that they develop further and become recognized. Good luck, and thanks!

Sincerely,

Both letters were signed:
Herbert Callen
Professor of Physics Emeritus

These notes are among the most valued mementos of my career. They have helped sustain me in the face of exactly the kind of "lip-service" to which Professor Callen alludes. While I never met him personally, his gracious response to a complete stranger who was obviously "seeking approval from the master" told me everything I needed to know about the man and the scholar. I therefore wish to dedicate this work and the benefits that it may create for "students of thermodynamics in the future" to his memory. An obituary outlining the details of Dr. Callen's career appeared in Physics Today, August 1994, page 74.