Difference between a Fixed Point and a Limit Point in implementations of the Renormalization Group (RNG) in Large Eddy Simulation (LES) model

In the introduction of this paper, it is explained that and how the application of a dynamic subrid scale model for turbulence into a large eddy simulation (LES) model corresponds to doing one renormalization step in a renormalization group (RNG) analysis.

However, when implementing the renormalization group into a LES model to obtain subgrid scale parameterizations, the rescaling step is left out. If I have understood this correctly, the rescaling leads to the fact that finally, after $k$ renormalization steps one considers a infinite in space domain which is needed to define scale invariance and therefore fixed points of the RNG flow. The neglect of the rescaling step leads to the fact, that the limit obtained in the model for $k -> \infty$ is not a (or does not have to be?) a true fixed point of the RNG transformation and is called a limit point to distinguish it from a conventional scale invariant fixed point.

My question now is: Can the difference between such a "true" scale invariant fixed point of the RNG flow and the limit point, obtained ofter a large enough number of renormalization steps lead to a "misbehavior" of the dynamic subgrid scale parameterization, such that for example an expected Kolmogorov fixed point is missed and the turbulent kinetic energy spectrum shows not the right scaling ?

• It would be nice if the downvoters could explain what is wrong with my question from a physics point of view, such that I can address it or, if the mistake is too severe, delete the question. Capricious downvotes of a question about a technical topic, which contains absolutely nothing controversial, are not funny and take out all the fun of asking anything here ... – Dilaton Mar 6 '13 at 17:26

As you say, an RG (sorry, I'm not used to the acronym RNG) transformation consists of two steps: 1. coarse graining 2. rescaling.

The first step is the technically hard one. It involves introducing a cut-off scale k (I use a momentum cut-off) and integrating out fluctuations on spatial scales smaller than 1/k. (Sorry again about the notation. For me $$k$$ is a dimension-full momentum cut-off.) It is there that perturbation theory, $$1/N$$ expansions or other complicated and approximative methods enter. After this step, the coarse grained description contains 'less' degrees of freedom since some of them were integrated out. For this reason, this step can not be iterated infinitely many times. At some point there are no degrees of freedom left to integrate out and the flow reaches a limit point. More importantly, although this transformation maps one theory onto another, both theories are impossible to compare because they do not have the same content.

The second step makes the two theories (before and after coarse graining) comparable again by rescaling them both in terms of their respective cut-off value. This is what really makes the RG into a Group. This step makes the search for and the study of critical physics easier since it provides a representation of the RG flow in terms of a dynamical systems with fixed points and everything. Although it helps a lot, it is not necessary.

I now take the Functional RG (FRG) point of view on coarse graining. The FRG is a formally exact approach to renormalization and provides the following picture: As an increasing amount of scales are integrated out the microscopic action of the system is gradually deformed. An effective action emerges as the cut off scale is lowered. This action will acquire a complicated structure with (in principle) vertices of arbitrarily high order. At the end of the flow (i.e. when $$k\rightarrow0$$), this action can be interpreted as the $$1$$PI effective action and used to generated any observable correlation function. Although it can be a bit tricky to see formally, this makes sense physically: Once all the fluctuations have been integrated out, correlation functions emerge.

The philosophy here is actually quite straightforward: Add an IR regulator to the microscopic description and compute the regulated effective action. The regulator is such that the regulated effective action is identical to the microscopic action on momentum scales smaller than $$k$$ (large spatial scales). Moreover, the regulator vanishes when $$k\rightarrow 0$$, so that we recover the $$1$$Pi effective action in this limit. Then, the problem of evaluating a path integral over the microscopic action is replaced with interpolating the $$k$$-dependence of the regulated effective action all the way from the microscopic action down to the physical observables. This in turns out to boil down to solving a complicated functional integro-differential equation. This interpolation as $$k$$ is lowered towards zero can be directly interpreted as coarse graining on increasingly large spatial scales. In this picture, the regulated effective action naturally reaches a limit point when the regulator is removed as $$k \rightarrow 0$$.

This point of view is not very common because it necessarily involves sophisticated approximations. Indeed, in the traditional view, the RG flow is approximated in a way that resolves the fixed point and its neighborhood. This is not to hard because there are usually only a few relevant couplings that need to be handles carefully. If one wants to resolve the full $$k$$ dependence however, it becomes necessary to resolve most of the generated couplings, (of which there typically are an infinite number) even if they are irrelevant close to the fixed point. This makes sense because there is no guaranty that the system is microscopically tuned to its critical point. This approach is technically quite heavy but can be used to resolve the non universal features away from the critical point.

In principle, there is no reason for the Functional RG approach to screw anything up. After all it's just a different representation of the same physics. If the regulated grained effective action can be handled without approximation, then nothing will get screwed up. There may however be a problem when unavoidable approximations enter game. Although the FRG flow is formally exact, the solution of the flow equation is equivalent to a full solution of the problem and is typically out of reach. Then one must be smart, devise a good approximation scheme and hope that it works. If the regulated effective action is not well approximated, then things will get screwed up.

See this and related works by the same authors where such an approximation scheme is implemented in the context of incompressible hydrodynamic turbulence. They work very hard to get a good approximation and it pays off.