Thread by @NigelGoldenfeld

Bravo to @stevenstrogatz for attempting the impossible task of presenting the renormalization group method for singular perturbation problems in one lecture!

Here's a short thread that he inspired me to write while taking a break from final exam grading & COVID-19 work for UIUC.

I hope it answers some of Steve's questions & perhaps those of the students in the class!

(1) Introductions. @stevenstrogatz - please meet Emeritus Professor @KunihiroTeiji who wrote the nice paper that was featured in your lecture! We've met in Urbana & Kyoto.

(2) Several times you said "renormalization" instead of "renormalization group". The two are actually different! And moreover, there is another asymptotic method sometimes called Pritulo's method of renormalization, discovered in 1962! (Discussed in e.g. Nayfeh's book.)

The difference is not just semantic. Renormalization is a process to remove infinities (originally in QFT), or secular terms (in differential equations). Renormalization group is a process to make this removal independent of the way the infinities are removed: the magical step!

(3) Renormalization starts with an equation that is singularly perturbed but the zeroth order is solvable. It has some constants of integration. The perturbation means that the constants of integration are no longer conserved. In my book, I show this for nonlinear diffusion.

In Barenblatt's equation, the perturbation destroys the conservation law that the integral of u(x,t) is a constant for all time. However, naive perturbation theory about the zeroth order in epsilon solution generates new solutions with the Green's function of the diffusion eqn.

But this Green's function has no way to know that the actual solution does not satisfy the conservation law. The secular terms in the perturbation theory arise because of this tension between the equation and the method of generating solutions perturbatively. Example below:

The renormalization process absorbs the divergence into the parameter m_0 in front of the solution. In the zeroth order equation, m_0 is a constant of integration. But it is not in the perturbed equation. So it should depend on time t or l, the extent of the initial condition.

I choose it to depend on l, keeping t fixed. Thus I write a renormalized version of the constant of integration as below. You can see that m and m_0 have the same units so the function Z is dimensionless. But it must depend on l. So I must introduce a new length scale, mu!

The idea is that as I take the limit l --> 0, the logarithmic singularity in the perturbation expansion will be magically cancelled out by the function Z. I can certainly find such a function to the order shown above, ie. first order. Here it is!

Let's check that it works. You can see that if I substitute into the perturbation theory solution, this form will cancel out the term in l at least to this order in epsilon.

Here is the algebra to show that this works.

This process is called renormalization. If I take the limit l --> 0, there is no divergence in my perturbation expansion. So, problem solved? No, not really. I was forced to introduce the length scale mu, by dimensional analysis essentially! How do I get rid of that?

(4) The next step is renormalization group. The length scale mu is basically a unit of length. The solution u(x,t) is independent of the units I use. So as I vary mu, the parameter m(mu) must somehow vary in such a way as to counteract the mu dependence in the logarithm.

This means that the function u(x,t), which appears to depend on mu in two places - once in m(mu) in front, and once in the logarithm that came from perturbation theory - actually does not depend on mu at all. What function m(mu) has this property? Differentiate u(x,t) w.r.t. mu

to get a differential equation for m(mu)! This equation is the envelope equation in @stevenstrogatz lecture. Now we know that we have a solution to the original equation, to order epsilon, which is finite and also independent of the units of length mu.

So what do we want to know? Well, we want to know how the solution decays for long times. The solution looks like m(mu) * Gaussian * {1 - stuff * log (stuff that depends on mu)}. We can choose mu any way we like because the solution won't depend on our choice by construction.

So let's choose mu to have the special value that makes the argument of the log be unity. Then because log (1) = 0, we just have the m(mu) * Gaussian. The stuff in the log depends on time t. So then m(mu) at that special value is actually a function of time t.

And it is easy to see that it goes like the formula below, taken from my PRL in 1990.

(5) In Steve's lecture, he mentioned or quoted from Kunihiro's paper that it is not obvious why a procedure that is known for scale invariance, has anything to do with solving differential equations. In the example I gave here, it actually does have self-similarity.

However, we did not know this at the beginning and it was not an assumption we made. The logic of the method works just as well when the equation has some other Lie group symmetry, not scale invariance. That was the point of our work.

(6) RG is famous for explaining universality, following Kadanoff and Wilson. What does this have to do with singular perturbation theory? When we look at asymptotics, we seek the parts of the solution which remain at long times, independent of the initial conditions.

In other words, asymptotics is (are?) the universal part of the solutions of the differential equation at long times! In statistical mechanics, we use RG to find the universal parts of solutions independent of small scales. It's the same concept!

(7) If you want to understand this properly, Chapter 10 of my book is as pedagogical as I know how to explain it, in the simplest case of similarity solutions which have anomalous scaling. As Steve mentioned, the big deal is that the method requires no insight at the outset .

And it generates solutions that work well even when the small parameter epsilon ~ unity. Here's an example of an awful switchback problem that we solved in our 1994 PRL (but the graph is from our long 1996 paper).

I hope that this adds some background and rationale for what undoubtedly seems like a very strange way to solve differential equations when you first encounter it. I think I explained all the conceptual points that @stevenstrogatz mentioned seemed confusing.

Some of the applications of this that I have worked on are to low Reynolds number fluid mechanics as @stevenstrogatz mentioned. I found an old video of me at a summer school, which might be helpful to anyone wanting to know more.

Nigel out.

Let me add some more information. The reason RG works is that because naive perturbation theory is forced to use the unperturbed operator and expands about the wrong zeroth order solution, it can only represent the true symmetries by divergent terms.

If summed, these terms would be finite and satisfy the symmetries of the problem. But we only know the terms to some finite order in perturbation theory. RG uses the information in the divergent terms to figure out what starting point would be finite.

This is accomplished in my example by the steps that balance the m(mu) with the logarithmic divergence from the perturbation theory.

You can see that by taking the original perturbation theory & guessing that it is actually an exponential. RG shows that this is true (for this example), no guess. The point is that expansion of the exact formula about the wrong (Gaussian) starting point has divergent terms:

So to summarise: secular divergences are perturbation theory's way of telling you that something is wrong with the zeroth order solution. Remember: it is individual terms in the perturbation series that diverge, not the whole series (if fully summed).

The case of asymptotics beyond all orders is more complex and I do not fully understand it. In our 1996 paper, we showed examples of that, in the guise of WKB. There is more that can be done there.

These RG methods are not just a curiosity but practically useful. Here are two non-traditional (for applied mathematicians) areas where there is work to which I would like to draw your attention.

Inspiralling binary compact objects in gravitation, with possible applications to gravitational wave detection by LIGO of lower mass objects than the original observations. Here's a remarkable calculation, using RG to calculate the post-Newtonian dynamics

journals.aps.org/prd/abstract/1…

An old friend from when I was a postdoc, Daniel Boyanovsky, pioneered applications of RG in time-dependent QFT at finite temperature: e.g. cosmology and astrophysics.

See here for a nice review, including tutorial in appendix. He calls it Dynamic RG.

doi.org/10.1016/S0003-…

@stevenstrogatz used the envelope formulation by @KunihiroTeiji in his class. What is the precise connection to RG as I have described it? Above I let t be fixed and varied mu. If I had done it the other way, I would have got this formula

This describes the result of using the renormalized (i.e. finite) perturbation theory around the time t*. You can see that this perturbation expansion is of limited validity, restricted to a neighbourhood where t is near to t*, so that the logarithmic correction is small.

The expansion defines tangent manifolds to the true solution shown in green. Both the red and the blue tangent curves would be good approximations for t in the vicinity of t* and t** respectively. But since t* is arbitrary, we can construct the envelope of these tangent curves.

And that is the green curve, i.e. the true solution to this order in epsilon.

The interpretation of RG in this way connects it to another construction known as "equation-free" methods, where local approximations are patched together numerically.

en.wikipedia.org/wiki/Equation-…