Boundary layers, transport and universal distribution in boundary driven active systems

1- The discussion is very thorough, no steps are skipped in the mathematical handling, and the results are very deeply analysed
2 - The dynamics as well as the steady-state profiles are studied
3- A novel boundary effect was identified when the average particle direction is kept fixed at the domain boundaries.
4 - A discussion of the impact of the paper's results to the broader field is had
5 - Closed analytical solutions are obtained
6- The authors have made an admirable effort to have the reader understand the material and their methods (e.g. Fig. 5).

1 - The paper is too detailed: the impact of the big results is lost because the reader must wade through many details of less importance
2 - No adimensionalization is performed and no units are mentioned.
3 - The paper makes a number of large-sounding claims that I believe when investigated closer, are not as insightful as they are made to be / are based on very loose analogies. An example is the supposed breaking down of Fourier's law, when in reality there is no temperature gradient or heat flows in the system. What the authors mean is that the flux of particles is not linearly proportional to the density gradient, which is well known in active system.
4 - I am unconvinced by a number of logical steps made during the discussion
5 - The notation is dense, with many quantities being defined, without a discussion of their physical significance

The paper has a clear and interesting research question and explores it thoroughly. The topic and results are of general interest to the active matter / statistical physics community. One notices that the authors have done their due diligence when investigating the consequences of their results, and I find the boundary layers that form due to the boundary conditions to be interesting results.

However, the paper is very dense, and often gets lost in details. I would suggest that a large chunk of the mathematical derivation be placed in the supplementary material, with the main results being mentioned in the main text.

I found some claims / results that strike me as weird: for example Eq. (67) describes the average orientation of the particles as a function of time. However (as noted by the authors), it does not depend on the initial average orientation. This is very strange, as I would expect this initial orientation to be the limit of Eq. (67) as time goes to zero.

As a minor point: a number of typos must be corrected (such as the lower case "j" and "c" in "janus particle" or "casimir effect", or the subscripted $(x,t)$ in the caption of Fig. 1 ).

While I believe expectations criteria #2 (Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work) is met, I can only recommend publication after heavy revision / clarification from the authors.

1- A discussion should be had on the real, physical realization of magnetization control. Do the authors know of a system where boundary magnetization can be controlled?

2 - In page 3, the authors mention "diffusive for large L" but have not yet introduced what L is.

3- the Milne length should be introduced: what is its physical meaning?

4- the authors should be careful when referring to particles travelling in opposite direction as "two species" as such a work invokes in the reader ideas of more qualitative differences (such as different values of $v$, for example).

5- Many variables are introduced in page 5, but very little discussion is had on why these quantities are important to define, and why they are defined that way. For example, what is the physical meaning of $\mu$ or $B_0$?

6- the "c" in "casimir effect", the "j" in "janus particle" should be capitalized. Furthermore, there is a typo in the caption of Fig. 1 where $(x,t)$ is in subscript.

7- statements such as "For $L$ large the current and the density gradient..." should be changed to indicate what scale is being used to compare $L$ with. Meaning, what is $L$ large in comparison to?

8- I think the analogy with the Fourier law and the Seeback effect is very stretched, and ends up overinflating the importance of the results. There is no temperature gradient along the system, and there is no heat flow. What the authors mean is that the flux of particles is not linearly proportional to the gradient of particle density. This is Fick's law, which is well-known to not apply to run-and-tumble particles.
A similar analogy states that there is a Seebeck effect, but there is no temperature drop and there is no voltage drop. What the authors means is that a state with zero current implies a balance between the diffusive flow and the active flow. Therefore, imposing magnetisation in zero net current must lead to a gradient in density. This is not surprising. The connection to the Seebeck effect involves a number of analogies to be made, and to be honest, I think hurts the paper more than it helps. It ends up leaving the reader feeling as if they were over-promised something.

9 - point II of page 6 is a bit jumbled. I understand what it is saying, but it's written in a very contrived way. What is a "fine-tuned kinetic boundary"? What does it mean to "restore the boundary condition"?

10 - What is $\overline{Q_0 + Q_1}$ in point V of page 6? I find the last sentence of this point to be also hard to understand

11 - Point VI of page 6 states that "the current governs the nature of the bulk profiles and not the boundary conditions". I do not understand this statement as boundary conditions should uniquely determine the solution.

12 - The discussion of page 6 would greatly benefit from references to Figs. 2 and 3. It is not clear what is meant with current reversal.

13 - A suitable adimensionalization must be made, or the authors must state their units. Statements such as "$v=D=1$" (caption of fig. 8) are very jarring.

14 - While the caption states that the results of Fig. 3 are amplified a factor of $10^4$, this must be made clear in the figure itself.

15 - I am unclear on exactly what simulations were performed in section 3.1. Appendix A seems to be entirely analytical, no?

16 - the large $L$ limit of section 3.1 should be done more rigorously, as it neglects some terms of order $e^{-\mu L}$, but not others. Why are only some terms OK to ignore?

17 - It should be clarified that $A_n^+$ and $A_n^-$ are new quantities that have not been previously defined

18 -Why are the eigenvalues expected to be close to zero when $L$ is large ( section 4.1.1)?

19 - Why is it OK to neglect $e^{-k_a L}$ when $L$ is large, but not $e^{-k_b L}$ (page 13)?

20 - Is $| P_{tr}>$ in Eq. 62 a complex number? According to its definition, it should stay real.

21 - I don't understand why the magnetization of Eq. (71) does not tend to $m_0$ in the limit of $t \rightarrow 0$.

22 - The discontinuity of Fig. 9 is a bit disturbing to me. How compatible is it with the boundary conditions? What steps have been made to ensure there is no error (numerical or otherwise)? If you take the limit of $D \rightarrow 0$, do you see a continuous sharpening of the peak or do you always have a jump?

23 - What does "boundary layer" mean in Eq. 72?

24 - What are the definitions of $P_{BM}$ and $P_{BL}$ in page 20?

Ask for minor revision

1 - Introducing the concept of active motion in channels connected to active particle reservoir as a playground to explore novel features of nonequilbrium transport
2 - A number of novel analytical results, both for steady state statistics and relaxational dynamics
3 - Interesting physical insight and identification of mechanisms/features that may be feasibly generalised to a broader class of problems

1 - Notation could be more transparent (see requested changes)
2 - Algebra-heavy sections in main text can be hard to navigate on a first read
3 - A microscopic definition of boundary interactions is not provided, preventing numerical verification of some of the results and potentially obscuring the applicability of the derived results

This work presents a systematic and thorough exploration of (non-interacting) 1D run-and-tumble motion in the presence of boundary driving by “active” reservoirs, which are mathematically defined to impose fixed density and polarity boundary conditions. A number of novel results are derived, including (1) the steady state distribution; (2) the eigenspectrum and time-dependent distribution, with explicit results mostly being obtained in the large system size and long relaxation time limit; (3) a proposed universal form of the large time distribution on the semi-infinite line (with an absorbing boundary) for “macroscopically-diffusive” processes with an exponentially correlated noise, which is shown to hold both for equilibrium (inertial) motion and non-equilibrium self-propelled motion. Interesting parallels are drawn with traditional transport theory, including the definition of an effective Milne length, the violation of Fick’s law in the presence of non-zero boundary polarity and a Seebeck-like effect. The impact of bare diffusion on the establishment of boundary layers in both the steady state and transient distributions is also thoroughly analysed. Indeed, the authors put an admirable amount of effort into extracting as much of the relevant physics from their analytical results as possible (with the downside of the text being at times quite verbose). I believe that this work meets the criteria for publication in SciPost Physics, insofar as it will certainly stimulate further research into the non-trivial features of boundary-driven transport in active matter systems. I thus recommend publication after minor adjustments, as detailed below.

1 - Define meaning of a Milne length in physical terms in the context of this work when first introduced
2 - A microscopic mechanism to enforce the boundary conditions discussed in this work would be a valuable addition and allow for future numerical validation/exploration. I wonder if this could be done in a similar vein to [Roberts & Pruessner, PRR 4(3), 033234 (2022)].
3 - Below Eq.(9), could the authors say a few words about the physical interpretation of the of the various effective parameters?
4 - In point V (page 6), what is the meaning of $\overline{Q_1 + Q_1}$?
5 - In Sec.4, I find the distinction between $a_n$ and $\alpha_n$ visually confusing, perhaps a different symbol for either of the two would help? Also in Eq.(20), it should be clarified why $\alpha_n$ is not absorbed into the definition of $| A_n \rangle$.
6 - Below Eq.(34), the two eigenvalues $\lambda=0$ and $\lambda=-2\omega$ are identified. The authors write that the second "corresponds to the tumble dynamics", by which I guess they mean that it is associated with the relaxation of spatially homogeneous polarity fluctuations (phrasing could be clarified). Could the authors comment more explicitly about why only the bands of eigenvalues around these two are deemed relevant for further analysis?
7 - Below Eq.(67): "the last but one term falls...". Is this a typo?
8 - Sec.5 should perhaps more accurately be titled "A proposed universality..."
9 - Above Eq.(74): "to write the distribution at large times..." should include a reference to the equation being referenced, I assume (72)

Ask for minor revision