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Velocity Resetting of Inertial Run-and-Tumble Particles in Non-Newtonian Media: Velocity Distribution, Diffusion and First-Passage Time

T0 review · 2 major / 2 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read Velocity resetting at high rates produces a cusp singularity at zero velocity and makes long-time motion diffusive with diffusivity falling as r to the minus two, independent of the drag law, while the mean time to a target velocity has an

desk verdict The cusp and Deff ~ r^{-2} scaling look robust across drag laws from the trajectory-FP cross-checks, but the shear-thickening vs thinning distinction in optimal resetting rate rests on thinner numerical support for the MFPT curves. read the letter →

arxiv 2606.00560 v1 pith:24MEYNUX submitted 2026-05-30 cond-mat.soft

Subhanker Howlader , Sayantan Mondal , Prasenjit Das This is my paper · ORCID
classification cond-mat.soft
keywords run-and-tumbleparticlesvelocityresettingnon-Newtonianmediafirst-passagetimeeffectivediffusiondistributionshear-thickeningshear-thinning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies an inertial run-and-tumble particle whose velocity is reset to zero at constant rate r while it experiences nonlinear drag from a non-Newtonian medium. The steady velocity distribution is obtained from direct trajectories and from numerical solution of the Fokker-Planck equation; both methods agree. For large r the distribution develops a cusp at v equals zero and the particle diffuses at long times with an effective diffusivity that decays as r to the minus two, and these features hold for any drag function and any tumbling parameters. The mean first-passage time to a chosen target velocity, however, depends on the rheological character of the medium: an optimal resetting rate exists only when the medium is shear-thickening.

What carries the argument

Steady-state velocity distribution Ps(v) obtained from particle trajectories and numerical Fokker-Planck solution, together with mean first-passage time statistics under symmetric dichotomous noise and nonlinear drag g(v).

What would settle it

A simulation or experiment in which the effective diffusion coefficient fails to scale as r to the minus two for large r, or in which an optimal resetting rate appears in shear-thinning media, would falsify the reported results.

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Extended reading notes

Core claim

In the presence of velocity resetting at rate r the steady-state velocity distribution Ps(v) of the particle exhibits a cusp-like singularity at v=0 for large r, leading to diffusive behavior at long times with Deff decaying as r^{-2}, independent of the drag function g(v). The mean first-passage time to a target velocity vt depends on the medium type: an optimal r minimizes it in shear-thickening media but not in shear-thinning ones.

Load-bearing premise

The numerical solution of the Fokker-Planck equation and the trajectory sampling both faithfully represent the underlying stochastic process without extra approximations that would erase the difference in optimal resetting behavior between the two media types.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper examines an athermal inertial run-and-tumble particle in one dimension subject to velocity resetting at rate r while moving in a non-Newtonian medium with nonlinear drag g(v). The run-and-tumble motion is driven by symmetric dichotomous noise of strength Σ and flip rate λ. The authors solve the Fokker-Planck equation numerically and sample trajectories to obtain the steady-state velocity distribution Ps(v), which develops a cusp at v=0 for large r; they further report that the long-time motion is diffusive with effective diffusivity Deff scaling as r^{-2}, both features independent of the specific g(v), λ, and Σ. In contrast, the mean first-passage time to a target velocity vt depends qualitatively on the medium: an optimal resetting rate exists for shear-thickening g(v) but not for shear-thinning g(v).

Significance. If the reported distinction in mean first-passage time is numerically robust, the results demonstrate that the rheological character of the drag can qualitatively change the resetting-rate optimization for first-passage processes in active particles. The explicit cross-check between trajectory sampling and numerical Fokker-Planck integration for Ps(v) and the Deff scaling provides a concrete strength for those claims.

major comments (2)
  1. [Section on mean first-passage time (likely §4)] The headline claim that an optimal resetting rate exists only in shear-thickening media rests on the numerical evaluation of MFPT(r). No details are supplied on the discretization scheme, grid resolution, absorbing-boundary implementation, number of trajectories, or convergence tests used for the MFPT computation (in contrast to the explicit trajectory-vs-FP comparison stated for Ps(v)). This omission directly affects the reliability of the media-type distinction.
  2. [Abstract and MFPT discussion] The statement that Ps(v) and Deff ~ r^{-2} hold 'irrespective of the specific form of g(v)' is supported by the reported numerical agreement, but the MFPT results are presented only for two representative g(v) forms without a systematic scan over additional functional forms or parameter values to confirm the qualitative contrast persists.
minor comments (2)
  1. [Abstract] The abstract states that results hold 'irrespective of ... the values of λ and Σ' yet the figures appear to use fixed λ and Σ; a brief statement clarifying the range explored would improve clarity.
  2. [Numerical methods and results sections] Error bars or convergence diagnostics are not mentioned for the trajectory-sampled Ps(v) or Deff even though the abstract highlights agreement with the numerical FP solution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive feedback on our manuscript. The comments highlight important aspects of the numerical methodology and the scope of the MFPT analysis that require clarification. We address each major comment point by point below and will revise the manuscript accordingly to improve transparency and robustness.

read point-by-point responses

  1. Referee: [Section on mean first-passage time (likely §4)] The headline claim that an optimal resetting rate exists only in shear-thickening media rests on the numerical evaluation of MFPT(r). No details are supplied on the discretization scheme, grid resolution, absorbing-boundary implementation, number of trajectories, or convergence tests used for the MFPT computation (in contrast to the explicit trajectory-vs-FP comparison stated for Ps(v)). This omission directly affects the reliability of the media-type distinction.

    Authors: We agree that the manuscript lacks sufficient detail on the MFPT numerics, which is a valid concern for assessing the reliability of the shear-thickening versus shear-thinning distinction. In the revised version, we will add a dedicated subsection describing the numerical procedure: the finite-difference discretization of the time-dependent Fokker-Planck equation, the spatial grid resolution and domain size, the implementation of absorbing boundary conditions at the target velocity vt (including how probability flux is removed), the number of independent trajectory realizations used for cross-validation, and the convergence tests with respect to grid size and time step. These additions will directly address the reliability issue without altering the reported qualitative results. revision: yes

  2. Referee: [Abstract and MFPT discussion] The statement that Ps(v) and Deff ~ r^{-2} hold 'irrespective of the specific form of g(v)' is supported by the reported numerical agreement, but the MFPT results are presented only for two representative g(v) forms without a systematic scan over additional functional forms or parameter values to confirm the qualitative contrast persists.

    Authors: The two representative forms were chosen because they exemplify the distinct rheological classes (shear-thickening with g(v) increasing faster than linear, shear-thinning with g(v) increasing slower than linear), and the MFPT behavior traces to the sign of the second derivative of g(v) near the origin. While this provides a clear illustration, we acknowledge that a broader exploration would strengthen the claim. In the revision we will therefore include MFPT(r) curves for two additional g(v) forms (a different power-law exponent in each class and a saturating nonlinear drag) over a range of Σ and λ values, confirming that the existence of an optimal resetting rate remains confined to the shear-thickening class. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: results obtained from independent numerical trajectory integration and FP solution with explicit cross-checks

full rationale

The paper states that the steady-state velocity distribution Ps(v) is computed directly from particle trajectories and compared to numerical solution of the Fokker-Planck equation, with reported agreement. Deff scaling as r^{-2} follows from the long-time diffusive behavior observed in the same simulations. MFPT(r) is evaluated from the underlying stochastic process for the two forms of g(v). None of these steps reduce by definition or fitting to the target quantities; the media-type distinction in optimal resetting rate is an output of the numerics rather than an input. No self-citations or ansatzes are invoked as load-bearing premises. This is the standard case of a self-contained numerical study.

Assumptions & free parameters 2 free parameters · 2 assumptions · 0 invented entities

Based on abstract only: the model assumes a one-dimensional athermal inertial particle whose drag is captured by an arbitrary nonlinear g(v), whose run-and-tumble motion is represented by symmetric dichotomous noise of strength Σ and flip rate λ, and whose velocity is reset to zero at constant rate r; the Fokker-Planck description is taken as the correct continuum limit of the underlying stochastic process.

free parameters (2)
  • resetting rate r
    Varied as control parameter; central claims concern its large-r asymptotics.
  • noise strength Σ and flip rate λ
    Model parameters whose specific values are stated not to affect the reported Deff scaling.
assumptions (2)
  • domain assumption The Fokker-Planck equation derived from the stochastic process is the appropriate continuum description whose numerical solution can be compared directly to trajectory statistics.
    Invoked when the authors state they solve the FP equation numerically and compare to trajectories.
  • domain assumption The non-Newtonian drag can be represented by a general nonlinear function g(v) without specifying its exact form for the diffusion scaling claims.
    Stated explicitly when claiming results hold irrespective of the specific form of g(v).

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Cite this review

Pith. "Pith review of Velocity Resetting of Inertial Run-and-Tumble Particles in Non-Newtonian Media: Velocity Distribution, Diffusion and First-Passage Time." pith.science (2026). https://pith.science/paper/24MEYNUX

@misc{pith2026260600560,
  author       = {Pith},
  title        = {Pith review of: Velocity Resetting of Inertial Run-and-Tumble Particles in Non-Newtonian Media: Velocity Distribution, Diffusion and First-Passage Time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24MEYNUX}},
  note         = {Machine review of arXiv:2606.00560}
}
abstract

We study the dynamics of an athermal inertial run-and-tumble particle moving through a non-Newtonian medium in $d=1$, where the particle's velocity $v$ is reset to zero at a constant rate $r$. The drag force from the non-Newtonian medium is represented by a nonlinear velocity-dependent function $g(v)$. The run-and-tumble dynamics is modeled by a symmetric dichotomous noise with strength $\Sigma$ and flipping rate $\lambda$. We begin with the Fokker-Planck (FP) equation for the velocity distribution $P(v,t)$ of the particle. In the presence of resetting, however, the FP equation does not yield a closed-form solution even in the steady state. We therefore compute the steady-state velocity distribution $P_s(v)$ directly from particle trajectories and compare it with the numerical solution of the FP equation, finding good agreement between the two approaches. For sufficiently large $r$, $P_s(v)$ shows a cusp-like singularity at $v=0$ and the particles display diffusive motion at long times. The effective diffusion coefficient $D_{\mathrm{eff}}$ decays as $r^{-2}$ in the large-$r$ regime. These results hold irrespective of the specific form of $g(v)$ and the values of $\lambda$ and $\Sigma$. However, the mean first-passage time exhibits a strong dependence on the nature of the medium as the resetting rate $r$ is varied. In shear-thickening media, there exists an optimal resetting rate that minimizes the time required to reach the target velocity $v_t$. In contrast, no such optimal resetting rate is observed in shear-thinning media.

Figures

Figures reproduced from arXiv: 2606.00560 by the authors.

Figure 1
Figure 1. Plot of mean-squared velocity ⟨v 2 ⟩r(t), as a function of time t for λ = 0.5 at different resetting rates, as indicated. The data points represent numerical results obtained by solving Eq. (5), while the solid lines denote the steady-state mean-squared velocity, ⟨v 2 ⟩ ss r , computed from Eq. (33). Data points and solid lines of the same color correspond to the same parameter values, with panel (a) showing the she… view at source ↗
Figure 2
Figure 2. Plot of steady state velocity distribution functions [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure analogous to Fig. 2 for the shear-thinning medium with Σ = 0 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Plot of mean-squared displacement ⟨x 2 ⟩(t), as a function of time t for λ = 0.5 at different resetting rates, as indicated. Panel (a) corresponds to the shear-thickening case at Σ = 3, while panel (b) shows the shear-thinning case at Σ = 0.6. The dotted lines labeled …
Figure 5
Figure 5. Figure 5: Log-log plot of the effective diffusivity [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Plot of mean first-passage times, τr(v0), for non-Newtonian media. (a) Plot of τr(v0) versus vt − v0 for r = 2 and (b) Log-log plot of τr(v0) versus r for vt = 0.5 for the shear￾thickening medium with Σ = 3 and different values of λ, as mentioned. Similarly, (c) Plot o…

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