# Statistical mechanics of coupled supercooled liquids in finite dimensions

### Submission summary

 As Contributors: Ludovic Berthier · Benjamin Guiselin Preprint link: scipost_202111_00060v1 Date accepted: 2022-01-26 Date submitted: 2021-11-29 09:07 Submitted by: Guiselin, Benjamin Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational Statistical and Soft Matter Physics Approaches: Theoretical, Computational

### Abstract

We study the statistical mechanics of supercooled liquids when the system evolves at a temperature $T$ with a field $\epsilon$ linearly coupled to its overlap with a reference configuration of the same liquid sampled at a temperature $T_0$. We use mean-field theory to fully characterize the influence of the reference temperature $T_0$, and we mainly study the case of a fixed, low-$T_0$ value in computer simulations. We numerically investigate the extended phase diagram in the $(\epsilon,T)$ plane of model glass-forming liquids in spatial dimensions $d=2$ and $d=3$, relying on umbrella sampling and reweighting techniques. For both $2d$ and $3d$ cases, a similar phenomenology with nontrivial thermodynamic fluctuations of the overlap is observed at low temperatures, but a detailed finite-size analysis reveals qualitatively distinct behaviors. We establish the existence of a first-order transition line for nonzero $\epsilon$ ending in a critical point in the universality class of the random-field Ising model (RFIM) in $d=3$. In $d=2$ instead, no phase transition is found in large enough systems at least down to temperatures below the extrapolated calorimetric glass transition temperature $T_g$. Our results confirm that glass-forming liquid samples of limited size display the thermodynamic fluctuations expected for finite systems undergoing a random first-order transition. They also support the relevance of the physics of the RFIM for supercooled liquids, which may then explain the qualitative difference between $2d$ and $3d$ glass-formers.

Published as SciPost Phys. 12, 091 (2022)

We first would like to thank the two Referees for their careful reading of the manuscript. We have revised our manuscript according to the comments made by the two Referees. A detailed response with a description of the changes is given below the Referees' reports. We have rewritten or clarified aspects of the manuscript pointed out by the Referees and added some supplemental information and results.

On the other hand, considering the huge simulation effort combined with the development of an efficient numerical strategy which has allowed us to simulate as large as possible, as many as possible, as low-temperature as possible systems with present-day computer resources, going much beyond what had been achieved before, we do not think that it is reasonable let alone feasible to ask for still more data.

We hope that our manuscript will now be suitable for publication in SciPost.

### List of changes

- partial rephrasing of the Abstract, the Introduction and the Conclusion along with new references,
- definition of the p-spin model in the main text and modification of the caption of Fig. 2,
- clarification of the justification of the shape of isotherms in the canonical and Gaussian ensembles (Fig. 3 vs Fig. 16)
- rephrasing of the presentation of the results for the two-dimensional system,
- new data for the two-dimensional system representing the size evolution of the disconnected susceptibility [Fig. 8(b)],
- new subsection IV-D with associated new results regarding the size evolution of the probability distribution of the overlap at maximum overlap variance when T=T_0 for both two- and three-dimensional systems,
- clarification of the meaning of the curves in Fig. 12 in Appendix A,
- withdrawal of Fig. 13(d) and the associated discussion of the fate of the phase diagram for T_0>T_cvx,
- rephrasing of subsection A-4,
- clarification of subsection A-5,
- correction of Eq. (A33),
- discussion of the tethered Monte Carlo method in Appendix B in connection to the method used in the present study.

### Submission & Refereeing History

Resubmission scipost_202111_00060v1 on 29 November 2021
Submission 2105.08946v1 on 3 June 2021

## Reports on this Submission

### Strengths

The analysis is solid and of high quality.

### Weaknesses

The results presented are in the most part an extension of previous works. There are not many new ideas here.

### Report

I am satisfied with the answers provided by the authors and the changes introduced in the text.

### Requested changes

No further change is requested.

• validity: top
• significance: good
• originality: ok
• clarity: high
• formatting: good
• grammar: excellent