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On perturbation around closed exclusion processes
by Masataka Watanabe
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Submission summary
| Authors (as registered SciPost users): | Masataka Watanabe |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202406_00015v1 (pdf) |
| Date submitted: | 2024-06-07 12:42 |
| Submitted by: | Watanabe, Masataka |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We derive the formula for the stationary states of particle-number conserving exclusion processes infinitesimally perturbed by inhomogeneous adsorption and desorption. The formula not only proves but also generalises the conjecture proposed in [Phys. Rev. E 97, 032135] to account for inhomogeneous adsorption and desorption. As an application of the formula, we draw part of the phase diagrams of the open asymmetric simple exclusion process with and without Langmuir kinetics, correctly reproducing known results.
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
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024-8-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202406_00015v1, delivered 2024-08-02, doi: 10.21468/SciPost.Report.9519
Strengths
A mathematically rigorous formulation is obtained for the weak perturbation limit under the presence of adsorption and desorption.
Weaknesses
The short path to general readers who are not interested in mathematical derivation but interested in physical phenomena is missing.
Report
The degenerated perturbation is considered among the stationary states of exclusion process, each of which conserves the number of particles, by the perturbative effect of adsorption and desorption. As a result, a rigorous mathematical expression of the stationary state is obtained. For the case of ASEP, the author shows that the domain wall phase appears between the high and low density phases. The article is well structured, and self contained. I recommend the publication of this article. The authors can optionally consider the following minor points.
(a) In several equations, such as (3.4), period is missing at the end of the equation.
(b) The scaling form of the domain wall structure to the limit a -> 0 and b -> 0 can be considered. In other word, how is the straight slope of the density in the coexisting phase shown in Fig. 1 recovered from the curve shown in Fig. 3 in that limit?
(c) Possibilities of numerical crosscheck either by Monte Carlo or tensor network simulations can be discussed.
Requested changes
The submitted article can be published even as it is. Additionally, I recommend to check grammatical structure of each sentences.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Anonymous (Referee 1) on 2024-7-4 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202406_00015v1, delivered 2024-07-04, doi: 10.21468/SciPost.Report.9340
Strengths
1. It presents an original approach to open ASEP by using perturbation theory.
2. It provides a proof of an earlier conjecture on the steady state of ASEP with Langmuir kinetics.
Weaknesses
1. The perturbative approach does not cover the entire parameter space of the model but valid only for infinitesimally small input/output rates.
Report
The subject of the manuscript is the one-dimensional asymmetric simple exclusion process (ASEP), which is a paradigmatic model of non-equilibrium driven systems and, in particular, of boundary induced phase transitions. The Author considers the case, where the particle non-conserving terms are small perturbations to the closed (particle number conserving) ASEP, the stationary state of which is exactly known. As the perturbation mixes the different degenerate steady states of the closed system, even the zeroth order of the resulting steady state is non-trivial. The Author applies degenerate perturbation theory and derives the steady state in the zeroth order of perturbation. Thereby an earlier conjecture of Ref. 11 on the steady state of the ASEP with Langmuir kinetics is proved in a somewhat more general form. The general result is then applied to construct the phase diagram of the ASEP with boundary input and output terms, as well as for the ASEP with Langmuir kinetics (bulk non-conserving terms), which is a much studied problem.
The method of applying perturbation theory to closed ASEP is original and may potentially be applied to other variants of ASEP. The results are presented clearly. I recommend the publication of the manuscript in SciPost Physics after the Author has considered the following points.
Requested changes
1. The calculations are presented in general clearly and in a self-contained way. Except of one point in the proof, which is, however, hard to follow and reference 14 is not much helpful here. This is the statement in Eq. (3.6). This part of the proof could be better explained.
2. It seems to me that there is a typo in Eq. 4.7 and in Eq. 4.8.
Eqs. 4.9 contain the ratio of binomial factors on the right-hand side (the first factor on the right-hand side). Using the definitions of q_L in Eq. 4.4 and the definition of A_N in Eq. 4.6, it seems to me that the same ratio of binomial factors should also appear in Eq. 4.7 and 4.8, but it is missing.
3. In the Introduction and Discussion, the papers in Ref. 10 and 12 are cited and discussed in which a continuum (or hydrodynamic) description is applied to the ASEP with Langmuir kinetics. Although the starting point of deriving the hydrodynamic limit is indeed a mean-field approximation, there is a later publication, Popkov et al., Phys. Rev. E 67, 066117 (2003), in which the authors argue for the validity of the hydrodynamic limit as the infinite ASEP has a product measure steady state. I bring this paper to the Author's attention.
Recommendation
Ask for minor revision
Author: Masataka Watanabe on 2024-07-07 [id 4605]
(in reply to Report 1 on 2024-07-04)Thank you very much for the careful reading of the manuscript and the valuable feedbacks!
Here are the replies to the referee's comments.
(1) Thank you for encoiuraging me to do it. Rereading my manuscript, indeed this part was not very clearly written. I changed the draft so that it is self-contained. I added a paragraph on page 5, saying
"Let us also present the strategy of the proof.
We will be finding an eigenvector of a non-normal matrix in perturbation theory, starting from degenerate vacua.
As the eigenvector we are looking for is the stationary state of the perturbed Markox matrix, we have
\begin{align}
(M_0+\epsilon H)\lvert{\tilde{S}}\rangle=0, \quad \lvert{\tilde{S}}\rangle\equiv \lvert{S}\rangle+\epsilon \lvert{v}\rangle+O(\epsilon^2),
\end{align}
where $\lvert{S}\rangle\in K_0$ and $\lvert{v}\rangle\in K_1$.
At order $O(\epsilon)$, the equation reduces to
\begin{align}
H\lvert{S}\rangle=M_0\lvert{v}\rangle\in K_1.
\end{align}"
(2) This was a bad explanation of mine. We are meant to sum over (3.8) on the LHS of (3.6), so we use the fact that $\sum_{(n)_N}q_{L_0}[(n)_{N_0}]=\binom{L_0}{N_0}_q$. Then we will get the q-binomials in the final expressions. I changed the draft accordingly.
(3) This is very important. Thank you! I changed some parts of the draft where I over-emphasized the mean-field approximations. I also cited related papers I found, as well as Phys. Rev. E 67, 066117 (2003).
I added the following on page 3,
"One can also derive the hydrodynamic equation by separating slow diffusion modes from fast transport modes as in \cite{cond-mat/0302208}.
Such analysis indeed gives the correct phase diagram at large volume, even though it does not account for fluctuations and so it does not constitute algorithmic computations of physical quantities. (However see \cite{1803.06829} for the application of fluctuating hydrodynamics to an exclusion process.)"
and on page 14,
"It would also be important to justify the hydrodynamic description theoretically.
This could be either justifying the mean-field approximation or continuing the idea developed in \cite{cond-mat/0302208}.
In terms of the former, one could for example compute the two-point functions perturbatively in $\epsilon$;
If they factorise in the thermodynamic limit, the mean-field approximation is justified at least perturbatively.
In this context, it might be worthwhile to rewrite the open ASEP-LK in the language of one-dimensional (non-Hermitian) spin chains.
The mean-field approximation can then be justified when the model flows to the free fixed-point in the infrared.
In terms of the latter, it would be interesting to come up with a model which is strongly-coupled in the infrared, where the mean-field approximation cannot be justified but the hydrodynamic description is available \cite{1801.08952,2104.14650,2208.02124,2405.19984}."