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Large Deviations in the Symmetric Simple Exclusion Process with Slow Boundaries: A Hydrodynamic Perspective
by Soumyabrata Saha, Tridib Sadhu
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Tridib Sadhu · Soumyabrata Saha |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2310.11350v2 (pdf) |
| Date accepted: | 2024-07-23 |
| Date submitted: | 2024-05-01 08:14 |
| Submitted by: | Saha, Soumyabrata |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We revisit the one-dimensional model of the symmetric simple exclusion process slowly coupled with two unequal reservoirs at the boundaries. In its non-equilibrium stationary state, the large deviations functions of density and current have been recently derived using exact microscopic analysis by Derrida, Hirschberg and Sadhu in J. Stat. Phys. 182, 15 (2021). We present an independent derivation using the hydrodynamic approach of the macroscopic fluctuation theory (MFT). The slow coupling introduces additional boundary terms in the MFT-action, which modifies the spatial boundary conditions for the associated variational problem. For the density large deviations, we explicitly solve the corresponding Euler-Lagrange equations using a simple local transformation of the optimal fields. For the current large deviations, our solution is obtained using the additivity principle. In addition to recovering the expression of the large deviations functions, our solution describes the most probable path for these rare fluctuations.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Published as SciPost Phys. 17, 033 (2024)
Reports on this Submission
Strengths
This article presents in a coherent and clean way the derivation of large deviations from the macroscopic fluctuation functional
Weaknesses
The weakness is that some of the points have been treated in previous literature, for which references are carefully given
Report
The criteria are easily met
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #2 by Anonymous (Referee 2) on 2024-7-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2310.11350v2, delivered 2024-07-10, doi: 10.21468/SciPost.Report.9369
Report
Outside equilibrium, the steady state of a physical system and its fluctuations are sensitive to the boundary conditions. A natural question arises: how sensitive or robust are the fluctuations, particularly the associated large deviation functions, to the details of the boundary?
In the manuscript currently under review, the authors address this problem for the Symmetric Exclusion Process (SEP). Specifically, they apply Macroscopic Fluctuation Theory (MFT) to compute the large deviation functions for density and current in the SEP with slow boundaries.
While these quantities had been previously computed using the exact solution of the microscopic dynamics, it is quite important to be able to recover these results using an approach like MFT.
This approach has the potential to be broadly applicable, particularly to models that are not exactly solvable at the microscopic level. Therefore, it is relevant to understand, as done in the paper, how specific boundary conditions at the microscopic level are reflected in the MFT
formalism.
The paper is carefully written, and has been a pleasure to read. It contains original results that deserve publication.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Soumyabrata Saha on 2024-07-16 [id 4626]
(in reply to Report 2 on 2024-07-10)We sincerely thank the referee for their valuable comments regarding our Submission.
Report #1 by Anonymous (Referee 3) on 2024-7-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2310.11350v2, delivered 2024-07-05, doi: 10.21468/SciPost.Report.9345
Strengths
In the context of the symmetric simple exclusion process with open boundaries, the hydrodynamic approach of macroscopic fluctuation is employed systematically to rederive fluctuation results obtained by earlier more
model specific methods. The detailed computations are explained in the appendices, which thereby serve as a blueprint for other models.
Weaknesses
One might argue that there are too little novel results. But in my judgement this is balanced by demonstrating a method which has the potential of being applied
to other models, for which it is unlikely to have such exact identities as for the SSEP.
Report
The article is very well and carefully written. It is accessible to a larger readership. At first sight the long list of appendices is somewhat unusual. But at second reading, such separation has the advantage that the necessary computations are being spelled out in detail.
The article amply meets the level of SciPost and I recommend publication in its present form.
Requested changes
The notion "marginal coupling" is not optimal. There is a risk of confusion.
The term "marginal boundary coupling" would be already much better.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Soumyabrata Saha on 2024-07-16 [id 4624]
(in reply to Report 1 on 2024-07-05)
We sincerely thank the referee for their valuable comments. We would like to address the concerns raised by the referee regarding a perceived minor weakness in our work. Some of the results derived in our current submission have been previously treated in the existing literature using different model-specific microscopic methods, and there is concern about "too few novel results."
It is indeed true that the final expressions for the large deviation functions (LDF) of density and current were derived using microscopic techniques by Derrida, Hirschberg, and Sadhu in 2019 [48]. However, what was not previously understood is how the rare fluctuations are generated, specifically the optimal paths of evolution that give rise to these observed fluctuations for density and for current.
Earlier works by Bertini et al. [39] using the Macroscopic Fluctuation Theory (MFT) verified the LDF of density in the limiting case of fast boundary coupling by addressing the Hamilton-Jacobi equations, which again do not provide information about the optimal paths of evolution.
Our work achieves these by an exact solution of the Euler-Lagrange equations using a novel local transformation. This contrasts with a two-step non-local transformation used in [40,41] for a similar solution, but only in the fast coupling limit. Furthermore, our solution method is generalisable to a broader class of model systems beyond the Symmetric Simple Exclusion Process (SSEP). This, in our understanding, is an important advancement considering recent attention to the integrability [18] of the Euler-Lagrange equations for SSEP, where an explicit solution at arbitrary times remains challenging.
From a technical point of view, the solution for the marginal (and slow) boundary coupling boundary condition is non-trivial. It also addresses how the boundary conditions in the fast coupling limit naturally emerge, which were previously argued heuristically. These novelties have already been emphasised in the third paragraph on page three of our Submission.
Lastly, our derivation of the MFT action is independent and does not assume any a priori conditions imposed on the system. This provides a more robust technique that can be applied to a variety of models, extending beyond the SSEP.
Author: Soumyabrata Saha on 2024-07-16 [id 4625]
(in reply to Report 3 on 2024-07-15)We sincerely thank the referee for their valuable comments. We would like to address the concerns raised by the referee regarding a perceived weakness in our work. Some of the results derived in our current submission have been previously treated in the existing literature using different model-specific microscopic methods.
It is indeed true that the final expressions for the large deviation functions (LDF) of density and current were derived using microscopic techniques by Derrida, Hirschberg, and Sadhu in 2019 [48]. However, what was not previously understood is how the rare fluctuations are generated, specifically the optimal paths of evolution that give rise to these observed fluctuations for density.
Earlier works by Bertini et al. [39] using the Macroscopic Fluctuation Theory (MFT) verified the LDF of density in the limiting case of fast boundary coupling by addressing the Hamilton-Jacobi equations, which do not provide information about the optimal paths of evolution.
Our work achieves this by an exact solution of the Euler-Lagrange equations using a novel local transformation. This contrasts with a two-step non-local transformation used in [40,41] for a similar solution, but only in the fast coupling limit. Furthermore, our solution method is generalisable to a broader class of model systems beyond the Symmetric Simple Exclusion Process (SSEP). This, in our understanding, is an important advancement considering recent attention to the integrability [18] of the Euler-Lagrange equations for SSEP, where an explicit solution at arbitrary times remains challenging.
From a technical point of view, the solution for the weak coupling boundary condition is non-trivial. It also addresses how the boundary conditions in the fast coupling limit naturally emerge, which were previously argued heuristically. These novelties have already been emphasised in the third paragraph on page three of our submission.
Lastly, our derivation of the MFT action is independent and does not assume any a priori conditions imposed on the system. This provides a more robust technique that can be applied to a variety of models, extending beyond the SSEP.