Reddit reviews Thermodynamics and an Introduction to Thermostatistics

Grad student here (bachelors in physics with a minor in math, now doing computational chemistry).

TL;DR: Learn as much maths as possible.

There are physicists who hate when people say "physics is applied maths," but I'm not one of them.

It is.

9 times out of 10 when I tutor people, or would help my fellow classmates out with problems, the problem was a lapse in an understanding or correct application of the maths. The only thing left after you subtract maths from physics are experiments, intuition, and foundational concepts. The foundational concepts are not that hard to wrap your head around, if you don't fight them; intuition is almost always wrong at first, and just takes experience to build; experiments just need to be designed well.

The real machinery behind physics is maths, and it will always be better to know more maths than you need than to know less. If you are part of the lucky few who have a passion for pure maths, embrace it. You'll stand out. Many I graduated with resented having to learn more maths than they felt was "necessary".

I may be biased here, but I think the lack of insistence on robust and comprehensive maths education in physics curricula across the US is akin to intellectual malnourishment. The fact that some physics majors exit elite universities without having taken a class in, say, abstract algebra/group theory, real/complex analysis, or Fourier theory is mind-blowing. I don't know what physics they think they're preparing their students for, but it isn't 21st century physics.



Let me provide an example of the sort of thing I'm talking about: one way of constructing the foundation of Thermodynamics (the one laid out in Callen), is to put forward a few very simple and broadly applicable postulates:

(1) There exist particular states (called equilibrium states) of simple systems that, macroscopically, are characterized completely by the internal energy U, the volume V, and the mole numbers N_1, N_2, ..., N_r of the chemical components.

(2) There exists a function (called the entropy S) of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of constraint are those that maximize the entropy over the manifold of constrained equilibrium states.

(3) The entropy of a composite system is additive over the constituent subsystems. The entropy is continuous and differentiable and is a monotonically increasing function of the energy.

(4) The entropy of any system vanishes in the state for which (\partial U/ \partial S)_V,N = 0 (that is, at the zero of temperature).

With these four axioms, and a whole lot of maths cleverness, all of thermodynamics falls out. Everything. In fact, if you buy this book and study it (something I highly recommend), you'll be surprised to find that the exact formula for entropy that you are no doubt familiar with (S = k log(Omega)) is not even discussed until one of the very last chapters of the book, when Thermostatistics comes in to view. Macrostates and Microstates are not discussed. These postulates are viewed simply mathematically, and just happen to apply beautifully to an almost unbelievably wide array of systems, systems you might not even guess fall under the purview of thermodynamics. Studying the theory mathematically, from a simply axiomatic, rigorous point of view actually makes you a more** powerful physicist.

I really liked Callen.