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

by Lorenzo Caprini, Umberto Marini Bettolo Marconi, René Wittmann , Hartmut Löwen

This is not the latest submitted version.

Submission summary

As Contributors: Lorenzo Caprini
Preprint link: scipost_202203_00001v2
Date submitted: 2022-06-08 14:01
Submitted by: Caprini, Lorenzo
Submitted to: SciPost Physics
Academic field: Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


We investigate how the competing presence of a nonuniform motility landscape and an external confining field affects the properties of active particles. We employ the active Ornstein-Uhlenbeck particle (AOUP) model with a periodic swim-velocity profile to derive analytical approximations for the steady-state probability distribution of position and velocity, encompassing both the Unified Colored Noise Approximation and the theory of potential-free active particles with spatially dependent swim velocity recently developed. We test the theory by confining an active particle in a harmonic trap, which gives rise to interesting properties, such as a transition from a unimodal to a bimodal (and, eventually multimodal) spatial density, induced by decreasing the spatial period of the self propulsion. Correspondingly, the velocity distribution shows pronounced deviations from the Gaussian shape, even displaying a bimodal profile in the high-motility regions. We thus show that the interplay of two relatively simple physical fields can be employed to generate complex emerging behavior.

Current status:
Has been resubmitted

Author comments upon resubmission

The reply to the referees has been included in the pdf attached, before the new version of the paper.

List of changes

The list of changes is included in the pdf "2022June7_maintext_reply.pdf" together with the reply to the referees.

Reports on this Submission

Anonymous Report 3 on 2022-7-4 (Invited Report)


I thank the authors for their careful consideration of my comments and for modifying their manuscript accordingly. From my side, the authors have satisfactorily addressed all my concerns.

I have read the discussion between the authors and the second referee. The first main concern of the referee is that the dynamics of the AOUP with both a confining potential and a spatially-varying self-propulsion velocity can be mapped, under a change of variable, unto a dynamics with fixed self-propulsion velocity and an effective force field. The second main concern is the experimental relevance of using both a confining potential and a space-dependent self-propulsion velocity.

Given that the mapping proposed by the referee is rather convoluted, one could argue that the specific study of the interaction between an external potential and a varying self-propulsion velocity is relevant if there exist experimental realisations of these. I would suggest the authors to insist on this point.

To address the first concern, the authors may consider showing a specific example where the proposed mapping fails, e.g. because u(x) is not bijective, or partially explains the physics at play.

In the footnote [76], it is unclear why the physics is different: the particle escaping to infinity in the case of a modulation of the self-propulsion velocity is also a consequence of fluctuations induced in the active force. If the stationary density profile is identical, what quantity would distinguish the physics at play in the two systems considered?

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Anonymous Report 2 on 2022-6-27 (Invited Report)


Let me thank the authors for having considered my comments and for their replies. From my side, I have to say that they adequately addressed my major and minor comments. However, at his stage of the review process, I understand there are still open issues with Reviewer 2 (R2). It looks to me like there are two main criticisms that should be taken into account by the authors and addressed properly.

The first criticism raised by R2 is about the novelty of the work in comparison with (i) Ref [63] of the same authors, where they introduce the model considered here without any external fields, and (ii) Ref [74], where, in Section II-D (across p. 5 and 6), there is a short discussion about AOUPs in the presence of spatially varying activity in the small-tau limit. I think a clarification about that would be an additional improvement to the manuscript.

The second criticism looks to me to be still linked with point (i) of the previous comment: What is the new phenomenology one gets once we turn on a confining external potential in a system of AOUP with space-varying motility. As already shown in PRE 100, 052147 (2019) in the case of one-dimensional non-interacting run-and-tumble particles with space-dependent speed (see Eq. (7) of that reference), R2 noticed a possible mapping into the dynamics of AOUP in a complicated force field. My feeling is that, since the problem does not map simply into the dynamics with an effective conservative field, I think it might be opportune to talk about some sort of complex behavior for the resulting dynamics (if I interpret correctly the reply of the authors). Again, although the authors in their reply wrote about this issue showing that the dynamics is not just the dynamics of an active particle into an effective potential, I think it is opportune to have further improvements. For instance, the authors might better clarify how the competition between external confining potential and space-varying velocity works, what are the novelties with respect to the case without an external field, and why it is important to consider the effect of external fields.

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Anonymous Report 1 on 2022-6-9 (Invited Report)


Please find the pdf attached.


  • validity: high
  • significance: poor
  • originality: poor
  • clarity: high
  • formatting: good
  • grammar: good

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