References about rigorous thermodynamics

Question

Can you suggest some references for rigorous treatment of thermodynamics? I want things like reversibility, equilibrium to be clearly defined in terms of the basic assumptions of the framework.

Answer 1

1) The pioneer of the rigorous treatment of thermodynamics is Constantin Carathéodory. His aticle (Carathéodory, C., Untersuchung über die Grundlagen der Thermodynamik, Math. Annalen 67, 355-386) is cited everywhere in this context, but probably you want some newer and more modern things.

2) Buchdahl wrote a lot of papers about this subject in the 40's, 50's and 60's. He summarized these in the book: H.A. Buchdahl, The Concepts of Classical Thermodynamics (Cambridge Monographs on Physics), 1966.

3) There was a recent series of articles on this subject by Lieb and Yngavason which became famous. You can find the online version of these here, here, here and here :).

4) Finally, I have come across the book T. Matolcsi, "Ordinary Thermodynamics" (since a few friends of mine went to the class of the author), which treats thermodynamics in a mathematically very rigorous way.

I hope some of these will help you.

Greetings, Zoltan

Answer 2

As some people point out, it depends on what you mean by "rigorous". If you mean rigorous, in the sense of a mathematician the books by David Ruelle are very nice.

1. Statistical mechanics: rigorous results by David Ruelle. This is actually a very nice books, which takes the statistical mechanics point of view (instead of a phenomenological/geometric approach). Everything is defined very rigorously, the existence of the thermodynamic limit is proven in both classical/quantum and lattice/continuous systems (for well-behaved interactions) and the equivalence of different ensembles shown. Also phase transitions are covered where very general versions of the Lee-Yang and the Mermin-Wagner theorems are proven. And much more. The book can be rather dry and formal, so make sure to find the old papers by Michael Fisher which can be helpful.
2. Thermodynamic formalism: the mathematical structures of equilibrium statistical mechanics by David Ruelle. I don't know this book very well, but it is also a very rigorous (and more advanced than 1.) treatment of thermodynamics.

Hope you will find this useful.

Answer 3

First I'll recommend the book I usually promote. There are very nice description of thermodynamics from point of view of differential geometry.

Also you can start from this link for similar treatment.

Answer 4

My paper ''Phenomenological thermodynamics'' http://www.mat.univie.ac.at/~neum/ms/phenTherm.pdf rigorously derives the core of thermodynamics in 18 pages, starting from a few simple axioms. No physics background is required; the main mathematical took is convexity (and of course calculus).

The exposition is far simpler than Caratheodory or Lieb & Yngvason.

A slightly different version - mainly augmented by cross references - is also available as Chapter 7 of my online book http://lanl.arxiv.org/abs/0810.1019 , but can be read independent of the remainder of the book. Chapters 8-10 of the book rigorously deduce the phenomenological axioms of Chapter 7 from statistical mechanics.

Answer 5

David R Owen, A first course in the Mathematical foundations of thermodynamics.

C Truesdell , The origins of rational thermodynamics. Well written, Truesdell sets thermodynamics on a stronger foot than all the mumbo-jumbo usualy found in books written by people that do not master math. The intro is a must be read.

And of course, for a correct derivation of macroscopic thermodynamics from kinetic theory, Radu Ballescu's numerous books.

Answer 6

I think this book is what you are looking for: (although this question is already answered a long time ago)

Thermodynamics and an Introduction to Thermostatistics, by Herbert B. Callen

It is sort of a modern version of the Gibbs formulation. Entropy is postulated at the beginning, from which all Thermodynamics is formally derived. The scheme is somewhat abstract, but the book is easy to read. It points in the direction of statistical mechanics (specially near the end of the book).