# Is our professor expecting too much?

1. Apr 18, 2012

### Lavabug

I'm a 3rd year physics undergrad, currently taking a first compulsory course in statistical mechanics. The course follows R. Kubo's text of the same title along with additional lecture notes given in class(theory on ensembles and such).

The highest level mathematics any of my classmates have taken is PDE's/integral transforms & ODEs/complex analysis. The prior courses in linear algebra and calculus were a lot more rigorous and proof-based (Spivak/Apostol for example). We have dealt with/proven special function relations like the Gamma, Beta and erf functions in our last calculus courses and have had exposure to Legendre, Bessel, Hermite, ... etc. polynomials as a basis for DE solutions in both physics and math courses.

However now we are being faced with problems that require math tools that are unknown to anyone in the class. Among them are several problems that involve determining the thermodynamic weight (in other words: state density) for a system using combinatorics (never studied it before, at best a few of us know that you can order 3 books in 3 spaces in 3! ways. No idea how to generalize that to N books with "up or down" degree of freedom into M spaces), proving Stirling's formula for logs of factorials, proving Euler-McLaurin's formula, among a few other things.

I've been asking around and none of my classmates know how to do these things... but we're still plowing through the assigned problems (mostly from Kubo's book) without stopping other than to discuss the physical significance of a solution (which is great, but no physical reasoning is going to get me the correct factorial expression for the state density).

It is worth mentioning that the undergrad program just got restructured and one compulsory "math methods for physicists" (group theory, integral equations, combinatorics & probability) course got dropped from the curriculum, however my professor seems to be up to date on our curriculum so I don't think this is unintentional. Is this excessive?

What should I do to get up to speed in these areas, combinatorics in particular (as it has shown up the most in problems so far)? Should I be expressing these concerns to my professor about a month away from the final exam? (we started doing problems quite late into the course, the first half was all theory).

Last edited: Apr 18, 2012
2. Apr 18, 2012

### AlephZero

If you are a third year undergrad, you should be long past needing to be spoon fed every little bit of information you need.

You can probably learn the basics of "permutatiosns and combinations" in about a day. When I was at school, we covered those topics at about age 14.

It took me about 15 seconds to find a one-page proof of the Stirling formula on the web.

IMO, stop whining and try doing some work instead.

3. Apr 18, 2012

### opaka

Try asking your professor if there is any supplementary texts that he could recommend you read to get a handle on the necessary mathematics.

4. Apr 18, 2012

### Lavabug

You could do without the snide remarks and insinuating that I'm lazy and actually point me to something helpful on the subject instead. I've already found a proof for Stirling's formula but that is besides the point (as aside from initially proving it as an exercise, we just apply it everywhere else). I find myself lacking to derive the proof on my own.

My typical online resources for "quick and easy math refreshers" (khan academy, paul's online math notes) have proven to be way too shallow for the level of combinatorics used in the problems, I could use a pointer to something that will allow me to pick it up. Any textbook in particular?

Last edited: Apr 18, 2012
5. Apr 18, 2012

### bromden

For combinatorics, you should be able to find a good treatment in any probability theory text. I used "A First Course in Probability" by Ross. Though it's not the greatest book, it does the job. Honestly, just go to the library and thumb through a few books with similar call numbers as this book and see which one suits you the best.

I'm surprised that your statistical mechanics text doesn't have any discussion of the derivation of the Stirling approximation, at least in an appendix. I think my old statistical mechanics book, by Ashley Carter, did, but I'm not sure. Again, your best bet is to go to the library and thumb through texts with similar call numbers as the one you're using for your course.

I've never used either of the references you talk about, but I think it's safe to say you're now at the level where there isn't a single website with a good, unified treatment of the topics you're studying. Getting used to supplementing your required text with multiple additional sources (other books, lecture notes, websites, etc.) is going to be something you have to get used to, especially if you go to graduate school, so you might as well get started on it now.

Good luck!

6. Apr 18, 2012

### Steely Dan

I went through a similar process in my undergraduate statistical mechanics course. Many physics students don't have a formal introduction to probability theory or combinatorics (I certainly didn't), so their first introduction to statistical mechanics can be difficult. And the course is never supposed to be easy, and it's always a little unpleasant until you get used to it. That being said, statistical mechanics is a framework steeped in the mathematical language of both combinatorics and distribution functions, so it's definitely something to become more comfortable with before you move on to the graduate version of the course (if you're aiming that way). I don't think it's unreasonable for your professor to do this: part of the whole purpose of undergrad stat mech is to get you to learn the math.