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Resolving phase transitions with Discontinuous Galerkin methods
by Eduardo Grossi, Nicolas Wink
This is not the current version.
|As Contributors:||Eduardo Grossi · Nicolas Wink|
|Arxiv Link:||https://arxiv.org/abs/1903.09503v2 (pdf)|
|Date submitted:||2019-04-24 02:00|
|Submitted by:||Wink, Nicolas|
|Submitted to:||SciPost Physics|
We demonstrate the applicability and advantages of Discontinuous Galerkin (DG) schemes in the context of the Functional Renormalization Group (FRG). We investigate the $O(N)$-model in the large $N$ limit. It is shown that the flow equation for the effective potential can be cast into a conservative form. We discuss results for the Riemann problem, as well as initial conditions leading to a first and second order phase transition. In particular, we unravel the mechanism underlying first order phase transitions, based on the formation of a shock in the derivative of the effective potential.
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Anonymous Report 1 on 2019-6-17 (Invited Report)
1. The paper seeks to introduce and discuss the applicability of Discontinuous Galerkin
methods in solving functional renormalization group flow equations. This could in principle be a useful goal.
1. Although the basic goal of introducing Discontinuous Galerkin methods to a new audience
could be useful if achieved, it is not clear that the results would be significant enough for
2. Unfortunately the introduction to Discontinuous Galerkin methods is too brief to be
understood without already having knowledge of these methods, although it is possible to
fish out some introductory material buried much later in the paper.
2. The paper is applied only to the large-N O(N) models, apart from some brief speculations
at the end of sec. III. Such models are very special - in fact exactly soluble in this context also as the authors themselves review in appendix B. One is left with the impression that these
methods are only applicable to such first order non-linear equations, as appears also to be
true of methods based on a weak solution on which these methods appear to be closely
based. In this case, one is already better off with using the method of appendix B coupled with numerical inversion.
3. The number of typos, mis-spellings, grammatical inaccuracies and other evidences of lack
of care, is so great that it would be impractical to list more than a sample.
4. There are a number of places where the mathematical development and/or physical
reasoning seems also to be suspect.
The authors' stated intention is "to close the cap [sic: should be "gap"] between the FRG and DG community". As someone from the FRG community, I found sec. IB far too brief to achieve this ambition. Even before this introduction starts there is reference to "conservation
equations" which one gradually realises means something quite specific here (normal use of
the word "conservation" would imply something -like a charge- is held constant) and to a
"weak formulation". These terms are never explained (to be joined by many others e.g. "CFL" conditions and "BDF" methods on p5, NB next sentence: "persevering"->preserving). On the other hand above eqn (23) you actually write "the first derivative of the potential is a
conserved quantity". At face value you seem to be telling me that the object in eqn (23) is a
constant with respect to t? Reference is made to intuition about relative rates of diffusion
versus convection, but the reader is never inducted into this way of thinking i.e. what precisely
this means for the FRG equation and how it relates to physics the reader from the FRG
community may have seen before.
The derivation of eqn (9) is given as integrating by parts eqn (7) to get (8) and then
integrating by parts back again. Since (9) is different from (7) there is clearly more to this, but readers are left to try to figure out why for themselves, or go learn about these methods from
somewhere where it is better explained. Eqn (10) is pulled out of a hat (NB "state the we work" -> state that we work), together with (11) (now in general dimension and not one dimension as earlier? why? - given that the rest of the paper is in one dimension).
Another example: eqn (27) on p5 is not explained, just quoted. Then it turns out it is derived
at the end of appendix C1! This can easily be missed unless one picks up on the one reference
on another matter at the end of sec. IIC on p11. Equation (27) is integrated in eqn (28). It
would appear that the authors assume that $u_L$ and $u_R$ are held constant, although the authors are silent on this point. When is such an assumption justified? (Incidentally in the next
paragraph I am not sure what word you actually meant by "prepositions").
Some more examples: eqn (35) is claimed to be generic for any coupling. Maybe so in large N but not generally. (Next section "It is instructing" -> It is instructive.) Why are Legendre
polynomials used? (App A. NB "Strum-Liouville"-> Sturm-Liouville, here & throughout.)
In many places two different forms of a sentence appear together, such as "are given can
easily be obtained", "convenient [with] respect compared to others" in App A. Many verbs
incorrectly conjugated e.g. "chose"-> choose, "condense" -> condensed, "outline" ->
outlined etc. Wrong words e.g. "underling" -> underlying, "limes" -> limits etc.
The authors need to go through the paper line by line e.g. using an editor that will prompt for all the grammatical inaccuracies. With some substantial improvements, not only to this but
the whole logic and presentation, this paper might find a useful home in a more pedagogically
minded journal. Sifting through this paper in its present form, I saw no evidence to make me
think that the work is suitable for this journal.