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Unveiling the anatomy of mode-coupling theory
by I. Pihlajamaa, V. E. Debets, C. C. L. Laudicina, L. M. C. Janssen
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Submission summary
| Authors (as registered SciPost users): | Ilian Pihlajamaa |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2307.03443v2 (pdf) |
| Date submitted: | 2023-10-12 11:51 |
| Submitted by: | Pihlajamaa, Ilian |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
The mode-coupling theory of the glass transition (MCT) has been at the forefront of fundamental glass research for decades, yet the theory's underlying approximations remain obscure. Here we quantify and critically assess the effect of each MCT approximation separately. Using Brownian dynamics simulations, we compute the memory kernel predicted by MCT after each approximation in its derivation, and compare it with the exact one. We find that some often-criticized approximations are in fact very accurate, while the opposite is true for others, providing new guiding cues for further theory development.
Author comments upon resubmission
We have carefully considered the comments of the referees and adapted the manuscript accordingly. Hereby, we would like to resubmit this manuscript for publication in SciPost physics.
Sincerely,
on behalf of all authors,
Ilian Pihlajamaa
List of changes
- we have rewritten the section on the projection on doublets as referee 2 suggested, and added a corresponding section in the appendix
- we have revised the referencing in the introduction
- we have added references for the works of Masters & Madden, Charbonneau et al., and Schofield & Oppenheim in relation to offdiagonal theories
- we have made minor changes to fix typographical and grammatical errors as pointed out by the referees and that we found ourselves.
Please consult the pdf attached to the replies to the referees for the marked changes in the document.
Current status:
Reports on this Submission
Anonymous Report 2 on 2023-10-26 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2307.03443v2, delivered 2023-10-26, doi: 10.21468/SciPost.Report.7997
Report
I would like to thank the authors for many changes that they made in the paper. These changes properly addressed the comments made by the referees.
Before the paper is accepted I would like the authors to consider the following additional remark. I am not trying to force them to make changes. If the authors indeed do not make any changes, I hope that this remark may still be useful either for them of for the broader community.
I hope the authors noted structural similarity of Eq. (29) and the last, un-numbered equation in Appendix A.
More importantly, both equations hint that there might be a fundamental problem with separating the standard mode-coupling approximation, which is usually presented as factorization and replacement of the projected dynamics by the standard dynamics, done at the same time, into separate approximations.
First, let me recall that it is commonly believed that for Brownian systems without hydrodynamic interactions the time integral of the irreducible memory function vanishes as k^2 at small wavevectors. This result is preserved by the standard mode-coupling approximation (on the
fluid side of the glass phase diagram). Both Eq (29) and the last equation of Appendix A exhibit contributions to the irreducible memory functions that are proportional to the intermediate scattering function. It seems to me that at small k, the time integrals of these contributions will scale as k^0 (note that the intermediate scattering functions are multiplied by terms \propto k^2), violating the standard conventional wisdom mentioned above. It is possible that these contributions will be canceled by the contributions from other terms.
To conclude, I suggest that while the present paper focuses at the behavior of the memory and intermediate scattering functions for wavevectors near the structure factor peak, it would be interesting if future work checked also behavior of these functions at small wavevectors.
Report 1 by Saroj Nandi on 2023-10-17 (Invited Report)
Report
The authors have adequately addressed most of my comments, except the issue at the lower temperature results (point 6 in the last report). I agree with the authors that this is hard and the analysis even at higher temperature is interesting.
I do not have any further comment and recommend publication of the manuscript.
Author: Ilian Pihlajamaa on 2023-10-26 [id 4068]
(in reply to Report 2 on 2023-10-26)We appreciate the constructive feedback provided by the referee. However, we respectfully hold a different perspective regarding the fundamental issue raised. The referee highlights the scaling of the time integral of the memory kernel for small k, emphasizing a scaling of k^2.
In our understanding, this scaling pertains to the memory kernel itself, rather than its time integral, as supported by Equation 4.12c in Götze's "Complex Dynamics of Glass-Forming Liquids" (2009). This interpretation can also be derived directly from the standard mode-coupling kernel (as presented in our Equation 19), noting that the vertices each exhibit linear scaling in k for small k. As Götze points out, this originates from the linear relationship in k for the fluctuating force, required for mass conservation.
In light of these considerations, we maintain that each term in our Equation (29) adheres to the correct scaling behavior. Specifically, given that <h h(t)> ~ d^2F(k, t)/dt^2 ~ k^4 and <\rho h(t)> ~ dF(k, t)/dt ~ k^2, each term aligns with the expected scaling.
Nevertheless, we acknowledge the referee's point regarding the small k limit, which presents a very interesting avenue for future exploration. For example, the breakdown of the Stokes-Einstein relation in this regime indeed holds significant promise for further investigation, and we appreciate the valuable suggestion in this regard.
We thank the referee for their thoughtful evaluation and look forward to addressing any additional concerns or questions they may have.
Author: Ilian Pihlajamaa on 2023-10-27 [id 4072]
(in reply to Ilian Pihlajamaa on 2023-10-26 [id 4068])After more consideration we realize that referee 1 is correct in their claim that the integral of the memory kernel should scale as k^2. It can be seen from the same equation in Götze that we mentioned in our previous comment. We apologize for the confusion.
Together, equations 26, 29, and 30 must yield the correct scaling of the time integral--they are exact after all. Perhaps as the referee suggests the spurious O(k^0) is exactly cancelled, although this is not trivial to see from the respective equations.
We shall add a short discussion on this topic to the appendix of the manuscript, and thank the referee for their excellent critical and constructive comments.