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Some Early Ideas on the Metric Geometry of Thermodynamics

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Abstract

It is a pleasure to write for this 90th anniversary volume of Journal of Low Temperature Physics dedicated to Horst Meyer at Duke University. I was a PhD student with Horst in the period 1975–1980, working in experimental low temperature physics. While in Horst’s group, I also did a theoretical physics project on the side. This project in the metric geometry of thermodynamics was motivated by my work in Horst’s lab, and helped me to understand the theory of critical phenomena, very much in play in Horst’s lab. In this paper, I explain the essence of my theory project and give a few accounts of its future development, focussing on topics where I interacted with Horst. I pay particular attention to the pure fluid critical point.

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Notes

  1. Robert Richardson, one of Horst’s former PhD students, shared a Nobel Prize for this discovery.

  2. But this language is very powerful. If we must understand macroscopic physics by building up from the level of the molecules, there is simply no rational alternative. If I had been much more sophisticated theoretically, and had access to a book such as the one by Altland and Simon [4], my future direction in theory might have been quite different.

  3. In 1971, I took Physics I as a freshman with Bill, and he promised us students gravity wave data when we were sophomores. This timetable turned out to be a bit too optimistic (finally, LIGO 2016!), but I remember being very impressed by the spirit of the thing.

  4. Michael Ryschkewitsch, a graduate student in Horst’s lab, showed me this paper when it was published. It was natural that this paper would be shown to me, since my lugging Gravitation around the lab marked me as having a bit of a “mathematical bent,” as Horst put it.

  5. Thermodynamic fluctuations are usually presented in the books as an add-on to thermodynamics, and not really necessary. But I remember being very impressed at the time by an old paper by G. N. Lewis arguing that, in fact, fluctuation theory was logically necessary for thermodynamics [8].

  6. I use the intuitive word “surface” rather than the mathematically more accurate “manifold.” “Surface” implies being embedded in a 3D flat space, not intended here. But for the purposes of visualization, such technicalities will not concern us in this paper.

  7. When I first got this idea for probability distance, I thought that it was a novel find since I had seen it nowhere in the statistical mechanics literature. However, probability distance is an element of information theory in the form of the Fisher information metric, which dates back well before my efforts. My contribution was to help bring the idea of probability distance to thermodynamics, and then, in particular, to pay serious attention to the induced thermodynamic curvature R.

  8. I always found Widom’s papers to be relatively nontechnical, free of heavy math and jargon, clear to read, and filled with good physical ideas.

  9. I add that Eq. (12) leads to the hyperscaling critical exponent relation.

  10. I first learned about path integrals from reading Tisza’s book [15] about Markov processes. Also very interesting to me was Feynman’s book on quantum mechanics and path integrals [16], which Moses Chan gave me.

  11. I got interested in R for black hole thermodynamics when Åman et al. [30] worked it out for several simple examples, though not necessarily with my physical interpretation in mind.

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Acknowledgments

I thank Horst Meyer for his continual support. While a lot of good work would have been done in geometry of thermodynamics without him on the scene, it probably would not have been done by me. I also thank my many friends and collaborators.

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  1. Division of Natural Sciences, New College of Florida, 5800 Bay Shore Road, Sarasota, FL, 34243-2109, USA

    George Ruppeiner

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Correspondence to George Ruppeiner.

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Ruppeiner, G. Some Early Ideas on the Metric Geometry of Thermodynamics. J Low Temp Phys 185, 246–261 (2016). https://doi.org/10.1007/s10909-016-1605-x

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