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Nonexistence of motility induced phase separation transition in one dimension
by Indranil Mukherjee, Adarsh Raghu, P. K. Mohanty
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Submission summary
| Authors (as registered SciPost users): | Pradeep Kumar Mohanty |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202209_00024v2 (pdf) |
| Date submitted: | 2022-11-24 15:44 |
| Submitted by: | Mohanty, Pradeep Kumar |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
We introduce and study a model of hardcore particles obeying run-and-tumble dynamics on a one-dimensional lattice, where particles run in either +ve or -ve x-direction with an effective speed v and tumble (change their direction of motion) with a constant rate ω. We show that the coarse-grained dynamics of the system can be mapped to a beads-in-urn model called misanthrope process where particles are identified as urns and vacancies as beads that hop to a neighbouring urn situated in the direction opposite to the current. The hop rate is same as the magnitude of the particle current; we calculate it analytically for a two-particle system and show that it does not satisfy the criteria required for a phase separation transition. Nonexistence of phase separation in this model, where tumbling dynamics is rather restricted, necessarily imply that motility induced phase separation transition can not occur in other models in one dimension with unconditional tumbling.
Author comments upon resubmission
Thank you for providing us an opportunity to resubmit the article to SciPost. We also thank all the referees for their valuable comments and suggestions. In the revised version we have incorporated all their suggestions. The article has gone through a major revision, with restructuring of the texts and clarification of the issues raised by the referees.
Separately, we have prepared replies to all the referees, addressing their queries point by point. Replies to all three referees are uploaded to SciPost (as pdf files) just below the respective referee comments. We have also added the summary of changes there.
We sincerely hope that you will find the revised article suitable for publication in SciPost.
Thanking you again for your kind consideration.
With regards,
Indranil Mukherjee, Adarsh Raghu, and P. K. Mohanty
List of changes
Summary of changes
===============
The article has gone through a "major revision" following the valuable comments of the referees. It is difficult to
provide details as in the revised manuscript, the whole structure and almost all the paragraphs are modified.
Only some important changes are listed below.
1. Restricted tumbling dynamics is now given separately as Eq. (2) followed by a longer discussion.
2. Fig. 1 is modified - exact and coarse-grained urn models are now described more clearly in Fig1(b).
3. Discussions on Matrix Product Ansatz (MPA) is described in the APPENDIX. We hope it helps the readers to arrive at the results and conclusions of the article without bothering much about the detailed mathematical steps of MPA.
4. New references [15], [16] and [23] are added.
Current status:
Reports on this Submission
Anonymous Report 3 on 2023-1-27 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202209_00024v2, delivered 2023-01-27, doi: 10.21468/SciPost.Report.6619
Report
Dear Editor,
The authors have improved their manuscript to address many of the
comments made by the referees. That said, the reading remains
difficult, with many statements that can easily be
confusing. Furthermore, I think the results do not match Scipost
editorial policy which aims at publishing articles which "provide
details on groundbreaking results obtained in any (sub)specialization
of the field".
Overall, this is an interesting technical contribution that maps a
slightly peculiar model of run-and-tumble particles onto an urn model
and then employs an approximation to rule out phase separation in
1d. I list below several parts which I still find confusing and should
probably be improved prior to publication. Once this is done, I would
support publishing this article in a specialized journal. Whether
Scipost Physics wants to do so is an editorial choice---I have no
strong opinion on that matter.
-/ Restricted tumbling. Particles can tumble only when the site on
their right is occupied. This should be stated explicitly in the
abstract. The current presentation is confusing.
-/ "In this article, we argue and show explicitly using 1D lattice
models of RTP that indeed MIPS can not occur in 1D". I think this
statement, which is repeated several times, is too strong. The
authors show this for a particular model. I agree this has
consequences beyond the sole model studied here but the authors
themselves discuss a counter example when the rates scale with the
system size. I do not think such generic and vague statements are
particularly useful. (The same goes for "a proof of nonexistence of
MIPS in our model necessarily guarantees its nonexistence in any
other model that has more liberal tumbling dynamics.", etc.)
-/ The authors speak about a "left <-> right" symmetry on page 2. I am
not sure to what they refer but their restricted tumbling rule breaks
the symmetry i -> L-i. Their model is thus not endowed with what I
would naively call a "Left-right symmetry".
-/ Please define $m_k$ in the caption of Figure 1. It is not possible
to understand it the first time it is cited without this definition.
-/ The authors mention several times that their coarse-graining
amounts to "averaging" the current over "internal degrees of
freedom". It's not clear what measure they use to do so
(flat?). Please write a clear mathematical definition of the
coarse-graining procedure. What is done is really not clear at the
moment (the second column of page 2 should be much clearer given the
importance of this coarse-graining/approximation).
-/ Equation (3) and above: is there a comma missing between \sigma_k
and m_k? This notation is not clear to me.
-/ Equation (6). Can the normalization Q be computed?
-/ Right after Equation (9). I suggest using a more careful
phrasing. All the computations so far are for N=2 particles. It has
been said several time in the literature that two-body interactions
are not sufficient to account for MIPS (see, e.g.,
[Phys. Rev. Lett. 126, 038002 (2021)]). Have the authors ruled out
more than that at this stage?
-/ Fig 2a: in the text, $w$ is said to vary from 0.03 to 10. In the
caption, it is said to vary from 0.05 to 10. Please clarify. Also,
the authors should use a color code allowing the reader to associate
the curves to the corresponding values of $w$.
Anonymous Report 2 on 2023-1-26 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202209_00024v2, delivered 2023-01-26, doi: 10.21468/SciPost.Report.6609
Report
I thank the authors for their effort in re-writing the manuscript and also for detailed responses to my comments.
The current manuscript is a much improved version of its predecessor. The presentation, as it stands now, is much easier to follow. Transporting the technical discussion to the Appendix has also helped the manuscript. (There are still various typos in the text, e.g., sate (state) or loose (lose), etc.)
The discussion in the current version makes clear several important things, such as why in the coarse-grained description one can neglect internal degree of freedom of the urns; or how the simulations were performed to infer about u(m).
Largely, I am in agreement with the subject matter of the text but I also feel that an important point was raised by one of the Referees regarding symmetry-breaking in the model.
Let's consider the model where a tumbling can occur only if the particle has neighbors on both sides (instead of one side). It seems that for such a situation there can still be a mapping to a beads-in-urn system : (a) the number of beads in each urn is now equal to the total number of vacancies on both sides of the particle, (b) leftwards hopping of the particle (corresponding to urn_k) to a vacant site is equivalent to a bead hopping from urn_(k-1) to urn_(k+1), (c) total number of beads is conserved and equal to half the sum of beads in all urns. Indeed, in the long-time limit any cluster can break as none of the four rates p+, p-, q+, q- are zero but whether or not the system becomes homogeneous for any density $0 <\rho < 1$ might be an open question.
Nevertheless, as I said previously, the manuscript has strong potential to lead to further work on both steady-state and transient problems in one dimension. Analysis of hardcore active systems at a microscopic level is not an easy task. I find the article to be a welcome contribution in this regard. Hence, I recommend publication.
Author: Pradeep Kumar Mohanty on 2023-02-02 [id 3301]
(in reply to Report 2 on 2023-01-26)
We thank Referee 2 for recommending publication of this article in SciPost. In the revised version we have addressed all your comments.
Symmetry: An alternative model, with same run dynamics and tumbling occurring only when particles are assisted from left, leads to the same steady state as our model where tumbling occurs when particles are assisted from right (this can be seen easily by mapping both models to corresponding Urn models). Thus, although dynamics of our model violates CP symmetry (run-dynamics obeys it whereas tumbling does not), the steady state remains invariant under the CP transformation.
Symmetric tumbling: A model with same run dynamics and tumbling occurring only when particles are assisted from both left and right is being studied. We find exact steady states for four RTPs on a lattice. One can explicitly show that MIPS transition is absent there (work is in progress).
Anonymous Report 1 on 2022-12-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202209_00024v2, delivered 2022-12-19, doi: 10.21468/SciPost.Report.6341
Report
The authors have made several revision to the manuscript in response to the reviewer comments (which were largely consistent between reviewers). Overall these have had the effect of clarifying the manuscript somewhat, and although the reader is still required to invest some effort in unpacking the mappings between the different models investigated, all the information one needs is there.
One of my questions in the first report however remains unresolved. This pertains to the presence, or otherwise, of CP symmetry in the dynamics. I agree with the authors that CP means: flip the spins, and reverse their order on the lattice. Under this transformation, configurations like $++$ turn into $--$ whilst $+-$ and $-+$ do not change (since spin flip is the same as order reversal here).
I agree that when $p_{\pm}=q_{-+}$ the particle hop dynamics exhibit CP symmetry. But, unless I am grossly misreading either the model dynamics or the notion of CP symmetry, the tumble dynamics do not. Consider for example the move $++ \to -+$ that takes place at rate $\omega$. Under CP transformation, this becomes $-- \to -+$$ which is not included in Eq (2) in the text. Therefore I do not believe that the tumble dynamics satisfy this symmetry. Actually, one can probably understand this already from the statement that tumbling is assisted from the right: under the P transformation right turns into left.
So I guess there are three questions.
(1) Does this model break CP symmetry or not?
(2) If so, does this matter? (Previously I suggested not, because it's the restriction that is important to the condensation argument, not symmetry; but nevertheless I feel that symmetry breaking is not something one does lightly)
(3) How does flipping being assisted from the right facilitate the mappings described in the main text? Does the mapping to the urn model depend on it?
Requested changes
Please address the point re CP symmetry set out above.
Author: Pradeep Kumar Mohanty on 2023-02-02 [id 3300]
(in reply to Report 1 on 2022-12-19)We thank Referee 1 for recommending publication of this article in SciPost. The question regarding symmetry is clarified in the article. An alternative model, with same run dynamics and tumbling occurring only when particles are assisted from left, leads to the same steady state as our model where tumbling occurs when particles are assisted from right (this can be seen easily by mapping both models to corresponding Urn models). Thus, although dynamics of our model violates CP symmetry (run-dynamics obeys it whereas tumbling does not), the steady state remains invariant under the CP transformation.
Author: Pradeep Kumar Mohanty on 2023-02-02 [id 3302]
(in reply to Report 3 on 2023-01-27)We thank Referee 3 for his/her useful comments and suggestions. We have incorporated all the suggestions in the revised manuscript.Those requires additional explanations/reply are mentioned below.
Counter example (MIPS in 1D ): In the counter examples, the tumbling rates are downplayed by a factor
inversely proportional to the system size, which vanishes in the thermodynamics limit. Exsitence of MIPS is
not surprising here as in absence tumbling, the system trivially goes to a jammed/absorbing-state.
Symmetry: An alternative model, with same run dynamics and tumbling occurring only when particles are assited from left, leads to same steady state as our model where tumbling occurs when particles are assited from right (this can be seen easily by mapping both models to corresponding Urn models). Thus, although dynamics of our model violates CP symmetry (run-dynamics obeys it whereas tumbling does not), the steady state remains invariant under the CP transformation.
Coarse-graining: The word coarse-graining is used here to denote 'an approximate macroscopic description'; it is not a strict mathematical procedure, one like, coarse-graining of lattice dynamics to obtain a effective continuum description.
Normalization Q: Explicit form of Q is mentioned just above Eq. (5).