Rigorous statistical thermodynamics?

[mordechai9]

I took classical (engineering) thermodynamics a few years ago, and this semester I am taking a statistical thermodynamics class from the physics department. We are using the book "Fundamentals of statistical and thermal physics" by Reif, which seems to be a pretty good book.

Unfortunately I get the feeling that I am playing around with a lot of poorly defined mathematics. The main problem is that differentials are constantly interchanged without any real discussion or justification of how you can break up the derivatives. Partial derivatives and full derivatives are often intermixed, and so on. It might just be my fault for not reading very carefully, but I feel like the author (and the subject) relies very heavily on "hand-wavy" type of differential manipulation.

I was wondering if people could recommend me some other books or resources that do a better treatment of making things more rigorous. Have other people noticed this kind of phenomenon with stat. mech.?

 

[ice109]

this is a fault of physics of general.

 

[Cincinnatus]

Well I've never studied physics, but I have looked at the relevant mathematics. You might want to look at books on stochastic analysis/ calculus / SPDEs etc and books on diffusion for mathematicians... There is a huge literature here but it's likely that most of it is neither accessible nor of interest to your usual undergraduate physics major...

How is your measure theory?

Zeev Schuss wrote a pretty short introduction to stochastic differential equations that you might be able to find a copy of somewhere. I think it is out of print though... He is also writing a longer book on diffusion, which I don't think has been published yet, though I've read (part of) a draft and it seems to be very good...

 

[mordechai9]

ice109- I don't really think so. There are certain books/subjects that deal with physical topics in a more rigorous way. For example, in classical mechanics, the math seems a lot more rigorous, because you are usually just taking derivatives or integrating other functions. Granted, there is still some manipulation of differentials when you change variables and stuff like that, but it's nowhere near as bad as it seems in my statistical mechanics book.


cincinnatus- I have studied some measure theory and I am fairly familiar with the mathematics that you mentioned. Stochastic DE's and PDE's, not quite as much. I am aware that most of the issue of interchanging differentials is related or based out of ODE and PDE theory. However, I'm afraid that if I open up a book on ODE's or PDE's it will be a very time consuming way of trying to understand Reif's manipulations. Instead, I'm more hoping to find a statistical mechanics book which takes on a more mathematical approach, with more mathematical discussion...

More suggestions? I'm sure there must be a book like this out there...

 

[Andy Resnick]

Feynman's "Statistical Mechanics" is fairly rigorous, but IIRC, what you describe is more a problem with thermodynamics than stat. mech. The lack of mathematical rigour in thermodynamics goes all the way back to Carnot, with sloppy use of differentials propagating and amplifying through the 19th and early 20th centuries. Reif, in particular, is guilty of this practice.

There has been some attempts to place thermodynamics on a rigorous mathematical formulation, the best I have seen is Truesdell's "Rational Thermodynamics" and related literature.

 

[mordechai9]

Interesting... as I suspected... I will have a look at that book, although it is checked out from library, they do seem to have something similar called "Lectures on Rational Mechanics" by Truedell.

 

[humanino]

Try "Statistical Mechanics: Rigorous Results" by David Ruelle (World Scientific, 1999)

 

[khemix]

I took classical (engineering) thermodynamics a few years ago, and this semester I am taking a statistical thermodynamics class from the physics department. We are using the book "Fundamentals of statistical and thermal physics" by Reif, which seems to be a pretty good book.

Unfortunately I get the feeling that I am playing around with a lot of poorly defined mathematics. The main problem is that differentials are constantly interchanged without any real discussion or justification of how you can break up the derivatives. Partial derivatives and full derivatives are often intermixed, and so on. It might just be my fault for not reading very carefully, but I feel like the author (and the subject) relies very heavily on "hand-wavy" type of differential manipulation.

I was wondering if people could recommend me some other books or resources that do a better treatment of making things more rigorous. Have other people noticed this kind of phenomenon with stat. mech.?

Having taken a course in theoretical thermodynamics, albeit from a different book, I don't remember any hand waiving differentials. What equations in particular are you referring to?

It may be that your math is not up to speed... remember, as an engineer you took an applied approach. Most of the differential equalities I seen could be derived from the chain rule. If you post some examples, I can show you how. Ofcourse, there were some that were just weird... particularily the variation ones with deltas. but as i recall, that was more in mechanics thant stat mech.

 

[Topher925]

I'm planning on auditing a statistical mechanics class later this year and professors have told me the same things when it comes to the mathematics of the subject. Kind of reminds of the rings and shells method of calc.