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Resolving phase transitions with Discontinuous Galerkin methods
by Eduardo Grossi, Nicolas Wink
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Submission summary
| Authors (as registered SciPost users): | Eduardo Grossi · Nicolas Wink |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/1903.09503v4 (pdf) |
| Date accepted: | 2023-08-28 |
| Date submitted: | 2022-11-17 15:21 |
| Submitted by: | Wink, Nicolas |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
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.
List of changes
Includes changes suggested by the referee.
- Most notably, updated the convergence figure from some intermediate RG-time to the final RG-time considered throughout the work
- Other changes include typo corrections and extensions to some explanations (see the answer to the referee report for a detailed list)
Published as SciPost Phys. Core 6, 071 (2023)
Reports on this Submission
Anonymous Report 1 on 2023-7-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1903.09503v4, delivered 2023-07-02, doi: 10.21468/SciPost.Report.7431
Strengths
As outlined in previous reports, the strengths are that the paper
1. Introduces discontinuous Galerkin methods into the context of the FRG.
2. Results on the O(N) model in the large N limit can be benchmarked against the analytic solution, which illustrates the stability of the numerical methods.
3. Many interesting insights resulting from numerical and analytical investigations are formulated and will be of interest to a large community of researchers.
Weaknesses
1. The main weakness is for me the restriction to the large-N limit which is clearly an idealization for any real physics system. But given that the paper contains anyway enough new material this is acceptable.
Report
The paper has gone through several stages of refereeing and was significantly improved during the process. It is overall very interesting and a substantial step forward for the aim to find reliable numerical methods to solve functional renormalization group equations. In fact, it has already triggered a set of new developments in the field, despite the fact that it is until now only available as a preprint.
I would therefore like to recommend publication of the manuscript in SciPost Physics.
Requested changes
I have just one suggestions that the authors may take into account: The formulation of the flow equation for the effective potential in the form of a local conservation law plays a significant role for the whole paper. However, the physical significance of this conservation law is not discussed very much. I would find it useful to have also an integrated form (with respect to the field space variable rho) stated and briefly discussed.
Here are some typos that came to my attention:
page 3: the numerical fluxes are define ... -> ...are defined...
page 4: an immense variety of physical system at different energy scale... -> ... physical systems at different energy scales...
page 5: stability preserving scheme Runge-Kutta scheme -> stability preserving Runge-Kutta scheme
Author: Nicolas Wink on 2023-07-10 [id 3792]
(in reply to Report 1 on 2023-07-02)Hi,
we thank the referee for the report.
While we do agree that some comments regarding the interpretation of conservation may have been appropriate in the original manuscript, we do not think that it is timely anymore.
The question has been addressed to a various degree in quite some follow-up publications:
Grossi, Ihssen, Pawlowski, Wink - Phys.Rev.D 104 (2021)
Koenigstein et al - Phys.Rev.D 106 (2022) 6, 065012
Koenigstein et al - Phys.Rev.D 106 (2022) 6, 065013
Steil, Koenigstein - Phys.Rev.D 106 (2022) 6, 065014
Ihssen, Pawlowski, Sattler, Wink - https://arxiv.org/abs/2207.12266
Ihssen, Sattler, Wink - Phys.Rev.D 107 (2023) 11, 114009
There are more, but these should be the most relevant ones for this question.
The integrated form itself is not very illuminating, since the equation is derived by taking a derivative and the integral simply yields the original flow equation [e.g. (22)], evaluated at the boundaries of the chosen integration domain.
Anonymous on 2023-07-12 [id 3804]
(in reply to Nicolas Wink on 2023-07-10 [id 3792])The answer of the authors is reasonable and I would suggest to keep this as it is now.