On the renormalization group flow

[thephystudent]

I'm currently studying the Landau-Wilson model for critical phenomena (Statistical Mechanics, Kerson Huang) where the renormalization group is a central object. In the end, the calculations lead to a set of coupled differential equations that describe the (metaphorical) evolution of the parameters in the Landau energy functional as a function of the logarithm $\tau$ of the uv-cutoff. For a system with two such parameters the vector flow of the solutions can be plotted on a 2D-diagram. Now, I have two issues with this

1) The renormalization group is not a true group because there are no inverse operations. If you zoom out, you lose information and there is no unique way to zoom in again. But with these diferential equations, you can just take $\tau\rightarrow -\tau$ to invert the operations it seems to me? Is this apparent paradox a consequense of some approximations maybe?

2) It looks as if renormalization changes the temperature of the system (it corresponds to one of the eigenvalues of the linearized equations), is this correct? How is this consistent with temperature as an objective macroscopic quantity?

for the latter question, I asked a similar question a while ago (https://www.physicsforums.com/threads/is-rest-mass-a-subjective-quantity.866691/) in the context of quantum field theories, before I started studying the renormalization in more detail for critical phenomena. Is the answer similar: is there some limit that defines "the" temperature? Since most paths of the flow go to infinity, this would seem strange to me.

 

[radium]

The renormalization group is a group in the sense that any scale can be accessed from any other scale. However as you say, you are integrating out the high energy degrees of freedom (which correspond to short distance scales) so it is not really a group. You are doing this because these degrees of freedom are not really relevant to the low energy phenomenona and may contribute to UV divergences. For example, if you calculate the Casimir force with no cut off, it will diverge. But then you realize that you have included high frequency modes which would essentially just tunnel through the walls giving their spacing. You need to put in some regulator (not a hard cutoff) to get the correct answer at first order. Casimir showed that if you use a regulator of a certain form, you will always get the same finite answer.

The idea behind RG in phase transitions is the concept of universality. This means that the low energy behavior of such systems near criticality does not depend sensitively on the short distance physics (smaller than the lattice spacing etc)

 

[atyy]

https://en.wikipedia.org/wiki/C-theorem

https://arxiv.org/abs/1107.3987

https://arxiv.org/abs/1206.7004

 

[thephystudent]

https://en.wikipedia.org/wiki/C-theorem

https://arxiv.org/abs/1107.3987

https://arxiv.org/abs/1206.7004

So Information is lost indeed. But still I see a problem. If you make a state following the vector flow generated by the first order differential equations from the Landau-Wilson model, you get to a new, 'renormalized' state. And from this renormalized state, you can get back to the original unique state by reversing the arrows of the vector flow (that's just the way diff-eq's work). So from every renormalized state you can follow the 'inverse evolution'.
Does this just show the failure of the Landau-Wilson model?

 

[Demystifier]

I think the source of the confusion is the fact that two very different operations are both called "renormalization group". One is reversible (really a group) and defined by a differential equation. Another is irreversible (not really a group) and defined by ignoring details on small distances. Of course, the two operations are related, but they are not equivalent.

Suppose that some physical quantity depends on energies and momenta of various particles. How that quantity will change if all energies and momenta are changed by the same factor? This question can be answered by RG methods in the first sense.

How can a description of a system at large distances be simplified by ignoring fine details at small distances? This is what RG in the second sense is about.

This is very much like zooming a photograph by two different methods. One way is to first take a picture by a digital camera, and then put the picture on your computer and apply a zoom option of your viewer software. This kind of zooming is, of course, reversible. Another is to first make a zoom by lenses of the camera and then take the picture by the camera. This kind of zooming is not reversible. But both methods are called "zooming".

In the case of photograph, the reversible zooming is done by software, while irreversible zooming is done by hardware. By analogy, I would like to propose to refer to two kinds of RG as soft-renormalization group (SRG) and hard-renormalization group (HRG). Only soft-renormalization group is really a group.

 

[A. Neumaier]

So from every renormalized state you can follow the 'inverse evolution'.

The dynamics of partial differential equations cannot always be reversed. A familiar example is the dissipative heat equation, where only the forward dynamics makes mathematical sense.

 

[thephystudent]

I think I found the solution: the (ordinary) differential equations only describe evolution of the coupling constants r and u in the Landau free energy, there is no ambiguity in defining them for a system with higher resolution. They do not correspond to a single particular microscopic realisation of the system.

 

[thephystudent]

Regarding my second question, a bit of further research shows that a system slightly above the critical temperature will have an increasing temperature when renormalizing. While I find this a reasonable thing on intuitive grounds, I'm still wondering how this can coincide with the temperature a standard thermometer will measure when measuring the system.

 

[atyy]

So Information is lost indeed. But still I see a problem. If you make a state following the vector flow generated by the first order differential equations from the Landau-Wilson model, you get to a new, 'renormalized' state. And from this renormalized state, you can get back to the original unique state by reversing the arrows of the vector flow (that's just the way diff-eq's work). So from every renormalized state you can follow the 'inverse evolution'.
Does this just show the failure of the Landau-Wilson model?

I agree with your answer in post #7. Kardar makes the same point in his notes http://ocw.mit.edu/courses/physics/...pring-2014/lecture-notes/MIT8_334S14_Lec7.pdf (around the bottom of p42).

 

[Demystifier]

The dynamics of partial differential equations cannot always be reversed. A familiar example is the dissipative heat equation, where only the forward dynamics makes mathematical sense.

Can you recommend a reference where dissipative differential equations are explained in a simple (not rigorous) way?

 

[A. Neumaier]

Can you recommend a reference where dissipative differential equations are explained in a simple (not rigorous) way?

It is easy to see that the backward heat equation is ill-posed by looking at the Fourier transform of a solution. The high frequency Fourier modes explode violently, the more strongly the higher the frequency. Thus arbitrarily tiny contributions of sufficiently high frequency contributions will produce in arbitraily short time arbitrarily large changes in the solution. This makes the system intuitively intractable since one can never determine experimentally the presence or absence of these fluctuations. However, it needs some rigor to argue why this implies that the IVP is not solvable in a meaningful way - since one cannot find a Banach space where the IVP would have a unquely solvable IVP.

The intuitive part of the above argument should be explained in many places since it is a basic property of heat equations.

 

[Demystifier]

Thus arbitrarily tiny contributions of sufficiently high frequency contributions will produce in arbitraily short time arbitrarily large changes in the solution.

Is this a kind of chaotic behavior? If not, what's the difference between this and chaotic behavior?

 

[A. Neumaier]

Is this a kind of chaotic behavior? If not, what's the difference between this and chaotic behavior?

It is much worse than chaotic. Chaotic behavior is still well-determined at short times. arbitrary perturbations of size $epsilon$ will grow in time $t$ to size at most $e^{\lambda t}\epsilon$, where $\lambda>0$ is the Lyapunov exponent. Therefore given a data accuracy $\epsilon$ one can still get reasonable predictions for time $O(|\log \epsilon|/\lambda)$. (For example weather is chaotic, and the limit prediction time with initial data and Earth topography known everywhere to a few decimal digits is about 14 days. The dynamics of the well-known Lorenz attractor is derived from an extremely simplified weather model, and one can see its short term deterministic behavior.) But for the backward heat equation, there is no such uniform bound since $\lambda$ is proportional to the frequency and there are always frequencies that make it blow up arbitrarily badly in arbitrarily short time. This makes it unphysical.

 

[atyy]

https://people.maths.ox.ac.uk/trefethen/pdectb/backward2.pdf

https://mat.iitm.ac.in/home/mtnair/public_html/back-heat.pdf

 

[A. Neumaier]

https://people.maths.ox.ac.uk/trefethen/pdectb/backward2.pdf

https://mat.iitm.ac.in/home/mtnair/public_html/back-heat.pdf

good articles.

Note th the backward heat equation is used in application not as a process happening in time but as a process for reconstructing (after appropriate regularization) information lost in a coarsening or dissipative process. This can be done only to a limited extent, only approximately, and only if there is known qualitative smoothness information about the target to be reconstructed. For reconstruction and regularization in general, see my survey article here.

 

[Demystifier]

https://people.maths.ox.ac.uk/trefethen/pdectb/backward2.pdf

This is exactly what I meant when I asked for a simple non-rigorous reference. Thanks!