Nonexistence of motility induced phase separation transition in one dimension

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 $w$.

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.

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?

Please address the point re CP symmetry set out above.