Active particles driven by competing spatially dependent self-propulsion and external force

The authors propose a lecture note about Active Ornstein-Uhlenbeck particles in a spatially-dependent motility landscape. Working in the Unified Colored Noise Approximation, they provide an approximate stationary probability distribution and then they check its validity in the specific case of a 1-dimensional system embedded into a harmonic trap and whose motility is modulated by a sinusoidal function. I found the subject of the lecture note interesting and the problem they focused on looks timely. I shall support for publication once the authors address satisfactorily all my comments listed below.

Comments
1. It is not completely clear to me how the velocity of the particle is controlled in this model. In the sense that $\eta$ fluctuates with a variance that is proportional to $1/\tau$ and then the velocity is basically $u(x,t) \eta$. For $u(x,t)=v_0$, it looks to me that velocity is not really controlled unless you change the variance of the OU process. I might be wrong, however, I think that a general reader might be getting wrong like me and thus I would suggest the authors clarify this point.

2. I did not find clear the presentation in the section “Velocity description of AOUP”. It seems to me that they just perform the time derivative of Eq. (3) and then they consider the usual UCN approximation ($\tau \ddot{x}=0$). I do not think an appendix of just two lines is required (see eq. (22) and (23)) since this derivation might be very useful for the general reader. Moreover, they introduce the Jacobian but
a) they do not write its expression, and
b) Does occur any problem if J=0?
I am asking that because after (1) they require $u\geq0$, however, $|J|=u$ and the approximated solution seems to require $u>0$. Is this a problem of the approximation or, more in general, the model is not defined for $u=0$?

3. I think Eq. (7) requires some warnings because, in the region where the potential develops negative curvatures, one ends with negative friction. I understand the authors choose a safe potential with always positive curvature (basically this is also the reason why for large $\tau$ one can obtain a good approximation of stationary configurations). For instance, it looks to me that even in a small $\tau$ limit negative and large curvatures might cause problems.
(a) Could the authors comment on that and also include relevant references?
(b) I think a similar discussion is required once they introduce $\Lambda(x)$

4. In section “Theoretical Predictions” should contain all the relevant information about the approximations made to arrive at Eqs. (9) and (10) are valid. Moreover, in Eq. (10), obtained within UCN, there is a $\det \Lambda(x)$ that does not appear in Fox: this is because Fox is a small $\tau$ approximation (the authors write “The same $\rho(x)$ can be obtained the path-integral method proposed by Fox”). I understand the small $\tau$ of UCN brings to Fox, however, in UCN one just asks for large friction.
(a) Could the authors clarify this point? Also, (10) has clearly a problem with negative values of $\Lambda$ but also with$ u=0$,
(b) could the authors provide a list of conditions under which the approximated solution is valid?
(c) Again, I think there are conditions for writing Eq. (9). In appendix (B) they show how to obtain rho(x) but I did not find information in Eq. (9).

5. The theory has been checked against numerical simulations in one spatial dimension. Does the approximated solution presented in (10) hold also in two dimensions? I think the authors should clarify the reasons why they focus their attention on one-dimensional problems.

6. In the section “The harmonic oscillator” they write “we have shown that our analytical predictions from Eqs. (9), (10) and (11) are exact in the small persistence regime through analytical arguments…”. I do not see where a small tau approximation enters (10).

7. I think it might be very useful to the reader if the authors could add more details about the comparison between numerical results with Eq. (10).
(a) Could the authors write the expression they use in 1 dim?
(b) Could the authors say how the comparison has been performed, i.e., they integrate numerically eq. (9)?

8. In “Conclusions” they write “we have developed a theoretical treatment, applicable to rather general choices of confining potentials and inhomogeneous swim velocities…”. I think the sentence should be updated once they answer to (4).

Minor Comments
1. In the main text the authors indicate different appendix with capital letters, however, there are no letters in the appendix section.

2. Introduction: “The motility of active particles is much higher than that of their passive counterparts” What does it mean? A passive bead immersed in a thermal bath is not motile. I think the authors mean that motility induces a diffusive regime whose diffusion constant is much bigger than that due to the thermal bath. I would ask the authors to state properly this sentence.

3. Abstract: “Our results can be confirmed by real-space experiments on active colloidal Janus particles in the external field”. Usually, Janus particles are well captured by Active Brownian motion rather than AOUP,

4. Introduction: “The ABP model is harder to use to make theoretical progress…” it does not look totally true to me, there are several theoretical works where the coarse-graining properties of active systems are obtained from ABP (see for instance T Speck, AM Menzel, J Bialké, H Löwen The Journal of chemical physics 142 (22), 224109) and RT (ME Cates, J Tailleur Annu. Rev. Condens. Matter Phys. 6 (1), 219-244) or even starting from minimal swimmer models (A Baskaran, MC Marchetti Proceedings of the National Academy of Sciences 106 (37), 15567-15572).

5. Introduction: “AOUP model is recovered upon substituting $v_0^2=D_a/\tau$" I think in d dimensions (that is the situation where they are working in 2a and 2b, otherwise bold symbols do not make any sense) $D_a = v_0^2 \tau / d$.

6. In section Model I think they start with considering AOUP in two spatial dimensions (since they write “\eta is a two-dimensional Ornstein-Uhlenbeck process”) however, I believe this information might get missed by a distracted reader. Could the authors introduce at the beginning of the section if they work in 1,2 or d spatial dimensions?

7. Figure 1: why the linear force looks like $|x|$?

See the compiled pdf attached

1 - The authors provide a comprehensive study of their model, both from the theoretical and numerical points of view.
2 - References to previous literature on the subject are numerous and relevant.
3 - The authors give clear interpretations of their results, supplemented by general qualitative arguments.

1 - Approximations made to derive the theoretical predictions are not motivated enough in the main text.

The authors study an AOUP in a confining harmonic potential with a velocity profile varying periodically in space. It is already known that active particles accumulate where they move slower, this study shows that the competition with an additional confining potential creates unfavourable regions of space — where the restoring force overcomes the propulsion force — such that the density profile transitions from a normal distribution at low persistence length to a multimodal distribution at large persistence. Numerical results are carefully confronted to theoretical expectations

The paper provides, for this simple model, a detailed account of the physics at play in the different regimes of persistence, and should thus serve as a solid basis for follow-up studies on the control of active matter. As such, I recommend publication to SciPost Physics.

1 - The limits of validity for the unified coloured noise approximation (UCNA) are not detailed in the main text. To which quantities should the persistence time be compared in order to characterise the small/large-persistence regimes? When should we expect this approximation to fail?
2 - How strong is the approximation of vanishing probability current? What are the consequences of such an approximation?

Minor comments:
3 - Above equation (21), is the second derivative of the density profile taken at x=0?
4 - On top of page 12, (3) is not a prediction, should it rather be (9)?
5 - In Fig. 4, could you add density profiles so that comparisons can be made similarly to all other figures?