# Somewhat Rigorous thermodynamics?

1. Aug 18, 2014

### whyevengothere

Any thermodynamics book that's somewhat mathematical,but is still for beginners?

2. Aug 18, 2014

### ShayanJ

I guess you feel things are not firm enough in thermodynamics, specially the second law.Right?

3. Aug 18, 2014

### whyevengothere

Yes ,any good book?

4. Aug 18, 2014

### ShayanJ

Its not a matter of book. That's how thermodynamics is.
I think your problem is only about planck's statement of the 2nd law of thermodynamics i.e. :
I had the same problem. I always asked myself what is entropy? why should it always increase? why this statement seems that much right that they accepted it as an axiom?
I then found Caratheodory's principle:
Which provides a geometrical formulation of Thermodynamics using special kinds of manifolds and so makes thermodynamics more mathematical. But at the end, Caratheodory's principle is just an axiom accepted based on observations (in fact based on its equivalence to other statements of the 2nd law) and so is as infirm as Planck's statement.
But then I found the book "Entropy Demystified: The Second Law Reduced to Plain Common Sense" by Arieh Ben-Naim which makes second law natural to intuition.
The whole point is that, in a complex enough system, the underlying simple laws, when applied in a tremendous number of times because of the huge number of particles, give rise to so much a chaotic evolution that the system seems to evolve randomly and this randomness causes something to increase in the evolution and reach a maximum at the end and we call that thing, entropy. To see how this happens, read the book I mentioned.

Last edited: Aug 18, 2014
5. Aug 18, 2014

### whyevengothere

I'm asking for a thermodynamics book , with quality on par with Kleppner or Purcell, if there is any.

6. Aug 18, 2014

### megatyler30

Although it might not be accesible, you can try Callen's Thermodynamics (from physics perspective, not necessarily "rigorous" but I've heard good things). I have tried to look for a rigorous thermo book too. Another one, from the math perspective is A First Course in the Mathematical Foundations of Thermodynamics by Owen. Also, I have not read these books, only looked through then. Currently, Im taking statistical mechanics, which is much better, for example why must net Entropy increase? Well Entropy doesn't have to increase, it's only very very very unlikely for it to decrease, because an increase in entropy corresponds to a big increase in available quantum states and it's unlikely for whatever you are talking about to take up exactly only the starting quantum states. Feel free to ask if you have any questions. Or if you need suggestions for statistical mechanics.

Also, on the easier side but most likely will be even less rigorous, Modern Thermodynamics with Statistical Mechanics by Henle looks better than most of the easier books.

7. Aug 18, 2014

### jkl71

Last edited by a moderator: May 6, 2017
8. Aug 19, 2014

### ShayanJ

That's not a book on thermodynamics. Its only on entropy and 2nd law. It tries to make 2nd law intuitive through a series of dice games.

9. Aug 24, 2014

### dextercioby

H. Callen's text is the standard textbook on Thermodynamics. I cannot imagine a better text.

10. Sep 11, 2014

### DrDu

What I don't like in Callen is the deductive approach, which is probably its major strenght at the same time. He postulates entropy and shows that it is consistent will all observations. The advantage is that you get a clear picture of the theory. The disadvantage is that you don't get a feeling which observations made the introduction of entropy necessary and what alternatives where discarded on what grounds.

What is very readable, though not a book, is the article by Lieb and Yngvason:
http://arxiv.org/pdf/cond-mat/9708200
Maybe the mathematically most concise formulation of thermodynamics. Lieb is quite famous as he has solved many open problems in statistical and quantum mechanics, among others.

11. Sep 11, 2014

### vanhees71

Well, entropy is a somewhat difficult subject. For me the best approach is information theory a la Shannon and Jaynes. For sure, entropy is more general than to be an equilibrium quantity and it's more than just being a mathematical construct with temperature as the integrating factor in the formula
$\mathrm{d}U=T \mathrm{d} S-p \mathrm{d} V.$
I guess, however, this is the wrong forum to discuss about the foundations of thermodynamics, which in my opinion should be taught as statistical physics from the very beginning.

12. Sep 11, 2014

### whyevengothere

I think what am asking is if wether or not there is a thermodynamics book for the mathematician.

13. Sep 11, 2014

### megatyler30

I know that but it seems like no one else echo posted does (besides you obviously). I posted lecture notes intended for mathematicians and two of the better non-mathematician books with respect to rigor.

14. Sep 11, 2014

### ShayanJ

So, Thermodynamics for mathematicians!
I don't think the mainstream formulation can be stated that rigorously. So I only can suggest you any book on Carathéodory's formulation. His formulation uses manifold geometry and so is harder to learn, but it seems to be more rigorous.
I don't know of a thermodynamics book that focuses on it.

15. Sep 11, 2014

### jdlawlis

I have found Gaskell's "Introduction to Metallurgical Thermodynamics" to be excellent, with clear sections on statistical thermodynamics and excellent sections on entropy changes due to phase mixing, and a mathematical treatment of everything from integrating the specific heat with respect to temperature to find enthalpy, Carnot cycles, fugacities and activities, to reaction equilibria in multi-component systems. Unfortunately, I think it is out of print, but it appears that used versions are still available.

16. Sep 11, 2014

### mattt

Agree. Callen's is a wonderful book, I strongly recommend this book. If the OP wants an even more mathematical approach, maybe he would like this paper: http://arxiv.org/pdf/0705.3790v1.pdf that follows Callen's lines but it all presented in a purely mathematical way.

17. Sep 11, 2014

### atyy

Things are firm in thermodynamics. The second law is given plain English statements like you cannot transfer heat from a cold to a hot body, or you cannot convert heat solely into work. The brilliant work of Clausius turned these statements into a quantitative second law, much in the same way that Einstein began from the constancy of the speed of light in all inertial frames and ended up with Lorentz transformations.

A free source that gives the derivation is Kardar's notes http://ocw.mit.edu/courses/physics/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2007/lecture-notes/ [Broken] (lectures 1-4).

Last edited by a moderator: May 6, 2017
18. Sep 11, 2014

### atyy

This is a matter of taste. I am not a professional physicist, so this is an amateur's viewpoint. I would take classical equilibrium thermodynamics as one of the great intellectual structures. To learn it from the point of view of statistical mechanics, instead of a complete subject in itself, is to miss the amazing derivation of Planck - it is amazing because all the classical thermodynamic laws are correct - the Stefan-Boltzmann law and Wien's law all come from classical thermodynamics. There was nothing wrong with it. What was wrong was classical statistical mechanics. Planck's derivation is amazing because it showed that it was statistical mechanics that needed to become quantum, while there was nothing that needed to be corrected about thermodynamics.

Also, in more recent times there is the analogy that Jacobson drew between the Einstein field equations and classical thermodynamics. I think the idea of "entanglement thermodynamics" is promising, but we shall have to wait and see whether that really works out. But here again, it is classical thermodynamics as something that can stand on its own without statistical mechanics, but is complementary to it, that one needs to appreciate the ideas.

This is heretical, but I would even venture that kinetic theory, with Boltzmann's flawed derivation of irreversibility, is more important than statistical mechanics:) Of course, that's just what someone who likes Bohmian mechanics would say:P

Last edited: Sep 11, 2014
19. Sep 11, 2014

### whyevengothere

I found this: Thermodynamics:
A Dynamical Systems Approach
Wassim M. Haddad, Vijay Sekhar Chellaboina, & Sergey G. Nersesov and didn't like it
is there something similar but better written ?

20. Sep 11, 2014

### HallsofIvy

Staff Emeritus
"Somewhat Rigorous"? Is that anything like "somewhat virgin"?