## Tuesday, April 30, 2013

### Students should learn the method of steepest descent

Today I gave a lecture to a Solid State Physics class on "Magnetic quantum oscillations and mapping out the Fermi surface." I basically follow chapter 14 of Ashcroft and Mermin.

The central result is Onsager's 1952 equation that the period of the magnetic oscillations is related to extremal areas of the Fermi surface perpendicular to the magnetic field.

[Aside: This is an amazing result because it only involves fundamental constants and so the interpretation of the experiments is not "theory laden", a rare thing in condensed matter].

There is one point I struggle to explain: why extremal areas?
Ashcroft and Mermin have a figure to justify this. It is lost on me, no matter how many times I read it and stare at the pictures.

Does anyone know a clear and convincing way to demonstrate this?

The only way I know how to get this result of extremal areas is to do a very fancy calculation (Lifshitz-Kosevich) which evaluates the magnetisation (or thermodynamic potential or partition function), summing over all the Landau levels and integrating over the momentum direction parallel to field. One then evaluates the last integral using the method of steepest descent (saddle point approximation), which then picks out the extremal areas of the Fermi surface.
However, this is way beyond what one should be doing at this level (final year undergraduate).
Furthermore, the students said they have never encountered the method of steepest descents before.
That is reasonable since neither did I as an undergraduate in Australia.
I only learnt it in graduate school after learning how to evaluate Feynman path integrals by this method. Only later did I learn it also applied simple one-dimensional integrals!

Should undergraduate students learn the method of steepest descent?
In mathematics and/or physics?

1. In my opinion - yes, and I was taught it at the end of my undergraduate at the university. I find two reasons for that. First, it provides a simple and intuitive way to estimate many integrals. Second - it gives the students that feeling of solving physical problems by simple approximations - somewhat very useful in their subsequent academic life.

2. start with integrals of exp(-f(x)) and undergrads can 'get' the simple 1D version... so since it's an integral and a taylor series...yes

3. For a really great text, I agree entirely about Fig. 14.5, if that's the one you mean. I remember staring at that for far too long last year when I wanted to gain some intuition for this. It seems to be an unusual way of demonstrating a divergent density of k-states, if that's even what it is trying to get at.

Much easier to work in 2D.

I learnt the method of steepest descent when I taught 3rd year statistical mechanics in Germany. Certainly in Australia I'd never even heard of it. I agree with Andrey, it is intuitive and, so long as you don't try to follow the rigour of the wikipedia link, quite simple. I don't think I explained it well though, because the students found it very difficult and didn't quite appreciate the beauty of the approximations.

Perhaps the significance and value of simplifying approximations doesn't sink in until you've tried to solve a bunch of problems, struggled, become discouraged, and then found that perfect approximation that makes the whole thing tractable. This tends to happen more at PhD level I suppose, because an unsolvable problem in undergrad will always be clarified during the next class. Even for a PhD, perhaps, the problems are cherry picked, somewhat, and this still doesn't happen a lot of the time. I don't know.

4. It is quite an amusing fact that Mark Kac, a renowned mathematician (the Ferymann-Kac path integral formula bears his name), had never heard of the method of steepest descent until his physicist colleague, Theodore Berlin, explained it to him. (See http://dx.doi.org/10.1063/1.3051173 for Kac's reminiscence.)

Back then they were collaborating on a project about phase transition, the outcome of which would be later known as the spherical model. The spherical model is now recognized as a prototype of large N theories, and the method of steepest descent becomes exact in the limit of large N.

5. Also, I would like to point out that the method of steepest descent is not equivalent to the saddle-point or the stationary-phase approximation. It is relatively easy to understand the saddle-point approximation for undergraduates with working knowledge in calculus. However, it takes a solid background in complex variables to understand the method of steepest descent.

I did my undergraduate study in China. I remember that we learned the method of steepest descent in the undergraduate "methods of mathematical physics" course. But I have to say the undergraduate mathematical physics course in Chinese universities (at least those reputable ones) are roughly equivalent to the first year graduate mathematical physics course in the US.

6. It's taught in MATH4107. Should probably be third year.