arxiv: 2603.13621 · v2 · submitted 2026-03-13 · ❄️ cond-mat.soft

Recognition: 1 theorem link

· Lean Theorem

Splitting probabilities of confined chiral active Brownian particles

Sarafa A. Iyaniwura , Zhiwei Peng

Authors on Pith no claims yet

Pith reviewed 2026-05-15 10:58 UTC · model grok-4.3

classification ❄️ cond-mat.soft

keywords chiral active Brownian particlessplitting probabilityFick-Jacobs approximationcorrugated channelconfined active matterbackward Fokker-Planckescape probabilityactive transport

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The pith

Channel geometry, activity, and chirality together set the probabilities that active particles escape through one boundary versus another.

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

Chiral active Brownian particles move forward while steadily rotating, so the chance they exit one end of a confined space rather than the other depends on how their persistent paths interact with the walls. The work solves the backward Fokker-Planck equation that directly yields these splitting probabilities. Exact formulas are obtained for a simple one-dimensional interval in several limiting regimes of activity and confinement. For two-dimensional channels whose width varies periodically, the Fick-Jacobs reduction produces an effective one-dimensional description when the channel is narrow; numerical solution of the full equation is used otherwise. The resulting probabilities change systematically with channel shape, propulsion speed, and the sign and strength of chirality.

Core claim

The splitting probability for chiral active Brownian particles obeys a boundary-value problem obtained from the backward Fokker-Planck operator of the underlying stochastic dynamics. Closed-form expressions exist for the one-dimensional interval in the limits of high activity, low activity, and strong confinement. In two-dimensional corrugated channels the Fick-Jacobs approximation integrates out the transverse coordinate to give an effective axial equation whose drift and diffusion coefficients incorporate both self-propulsion and chiral rotation; the equation is solved numerically when the aspect ratio is not small. The escape probabilities are thereby shown to depend explicitly on channel

What carries the argument

The backward Fokker-Planck equation for the splitting probability, reduced by the Fick-Jacobs projection onto the channel axis for narrow corrugated geometries.

Load-bearing premise

That transverse degrees of freedom can be averaged without large error once the channel aspect ratio becomes small, even when activity and chirality are present.

What would settle it

Full two-dimensional Brownian dynamics simulations of chiral active particles in a corrugated channel with aspect ratio 0.05, compared directly to the splitting probabilities predicted by the reduced one-dimensional Fick-Jacobs model.

Figures

Figures reproduced from arXiv: 2603.13621 by Sarafa A. Iyaniwura, Zhiwei Peng.

**Figure 2.** Figure 2: Plots of C0(x), defined in Eq. (19), as a function of the initial position of the particle x for different values of chirality (χ). In all cases, γ = 0.1. We note that, p1 does not contribute to hpRi due to symmetry. Using Eqs. (16) and (17), one can obtain the solution of p2. Writing p2 = C0(x) + A2(x) cos(2θ) + B2(x) sin(2θ), we obtain C0(x) = 1 4 Re " sech(√ γ + iχ) sinh(x √ γ + iχ) − x tanh(√ γ + iχ) (… view at source ↗

**Figure 3.** Figure 3: Comparison between the two-term asymptotic solution giv [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Plots of the splitting probability hpRi as a function of the particle’s initial position x. (a) hpRi for ABPs (χ = 0) at different values of P e. (b) hpRi for CABPs at different values of chirality χ (P e = 10). In all cases, γ = 0.1. conditions [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: (a) Comparison between the numerical (FEM) and leading- [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: Schematic illustration (not to scale) of an active particle in t [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

**Figure 7.** Figure 7: Splitting probabilities computed from the Fick–Jacobs theo [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗

**Figure 8.** Figure 8: (a) Comparison of the numerical solutions of the Fick–Jac [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

**Figure 9.** Figure 9: Plots of the splitting probability hpRi as a function of the normalized y coordinate y/(εh) at different values of the axial coordinate: (a) x = 0, (b) x = 0.5, and (c) x = 0.9, computed numerically from the full PDE model [Eq. (27)]. The physical parameters are given by P e = 1, γ = 0.1, χ = 0, and ε = 1. In (a), all curves collapse onto a single line; the legend for the different lines is shown in (b). v… view at source ↗

read the original abstract

Active particles exhibit self-propulsion, leading to transport behavior that differs fundamentally from passive Brownian motion. In confined or structured domains, activity strongly influence escape probabilities and first-passage behavior. Understanding these effects is essential for describing transport in biological microenvironments, microfluidic devices, and heterogeneous media. In this work, leveraging the backward Fokker--Planck equation, we investigate the splitting probability of chiral active Brownian particles in confined domains, focusing on both a one-dimensional interval and a two-dimensional corrugated channel. Analytical solutions are derived for the one-dimensional case in various asymptotic regimes. In corrugated channels with small aspect ratios, we develop a Fick--Jacobs reduction that yields effective transport equations along the axial direction, whereas for finite aspect ratios, the splitting dynamics are characterized numerically. We demonstrate how channel geometry, particle activity, and chirality modulate the likelihood of escape through different boundaries. Our results provide quantitative predictions for the transport of active matter in complex environments and highlight the interplay between confinement and activity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Analytical 1D splitting probabilities for chiral active particles look solid, but the Fick-Jacobs channel reduction needs more error control.

read the letter

The main takeaway here is that the authors derive analytical expressions for the splitting probabilities of chiral active Brownian particles in a one-dimensional interval across different asymptotic regimes, and then apply a Fick-Jacobs reduction to handle narrow two-dimensional corrugated channels. They do a clean job on the 1D case. Starting from the backward Fokker-Planck equation, they solve for the probability of exiting left or right under varying activity, chirality, and confinement strengths. Those closed forms let you see directly how the rotational diffusion and self-propulsion bias the escape routes. The numerical work for finite-aspect-ratio channels also gives a practical check on when the approximations hold. The weaker part is the Fick-Jacobs reduction for the channels. Standard passive derivations assume fast transverse equilibration, but here the particles have persistent directed motion and ongoing rotation. That persistence can keep transverse velocities correlated longer than the channel width would suggest, especially at moderate activity levels. Without explicit bounds on the error or comparisons to full 2D simulations across a range of parameters, it's hard to know how accurate the effective 1D predictions are for real geometries. The abstract claims the reduction works for small aspect ratios, but the stress from activity might require more justification. This paper is aimed at researchers working on active matter in confined spaces, like those studying bacterial navigation or particle sorting in microfluidics. It offers quantitative tools that could be plugged into larger models without too much computation. I would send it out for peer review. The 1D analytics are a useful addition, and the reduction is a reasonable first step even if it needs tightening. A referee could push on the validity range of the approximation and suggest validation tests.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates splitting probabilities of chiral active Brownian particles confined in one-dimensional intervals and two-dimensional corrugated channels. Leveraging the backward Fokker-Planck equation, it derives analytical solutions for the 1D case in various asymptotic regimes. For corrugated channels at small aspect ratios, a Fick-Jacobs reduction yields effective axial transport equations, while finite aspect ratios are treated numerically. The work shows how geometry, activity, and chirality modulate escape probabilities through different boundaries.

Significance. If the central results hold, the paper supplies quantitative predictions for non-equilibrium escape and transport of active particles in structured confinements, with relevance to biological microenvironments and microfluidic devices. The combination of exact 1D asymptotics and a controlled reduction for channels is a methodological strength that could enable further analytic progress in active-matter first-passage problems.

major comments (2)

[Fick-Jacobs reduction for corrugated channels] Fick-Jacobs reduction section: the reduction to effective 1D axial equations assumes rapid transverse relaxation to a local equilibrium. For chiral ABPs the combination of persistent self-propulsion and rotational dynamics can sustain non-equilibrium transverse currents even at small aspect ratios; without explicit error bounds, higher-order corrections, or direct comparison against full 2D numerics across the activity and chirality parameter range, the quantitative accuracy of the predicted splitting probabilities remains uncertain.
[One-dimensional interval] One-dimensional analytical solutions: the abstract states that solutions are obtained in various asymptotic regimes, yet no explicit expressions, validity ranges, or limiting cases (e.g., vanishing chirality or activity) are provided. This omission makes it impossible to verify whether the reported modulation by chirality is recovered consistently in the appropriate limits.

minor comments (2)

[Abstract] Abstract: the summary of results would be strengthened by inclusion of at least one key reduced equation or the leading-order dependence of the splitting probability on the chirality parameter.
[Fick-Jacobs reduction] Notation: the definition of the effective diffusion coefficient and drift term after the Fick-Jacobs reduction should be stated explicitly, including any dependence on the local channel width and the rotational diffusion constant.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major point below and describe the revisions we will implement.

read point-by-point responses

Referee: [Fick-Jacobs reduction for corrugated channels] Fick-Jacobs reduction section: the reduction to effective 1D axial equations assumes rapid transverse relaxation to a local equilibrium. For chiral ABPs the combination of persistent self-propulsion and rotational dynamics can sustain non-equilibrium transverse currents even at small aspect ratios; without explicit error bounds, higher-order corrections, or direct comparison against full 2D numerics across the activity and chirality parameter range, the quantitative accuracy of the predicted splitting probabilities remains uncertain.

Authors: We agree that the Fick-Jacobs approximation for chiral active Brownian particles merits additional scrutiny, since persistent self-propulsion can in principle maintain transverse fluxes. In the revised manuscript we will add a systematic comparison of the reduced one-dimensional predictions against full two-dimensional Brownian dynamics simulations performed over a representative range of activity and chirality values. We will also include a discussion of the approximation's validity domain together with quantitative error estimates derived from these comparisons. revision: yes
Referee: [One-dimensional interval] One-dimensional analytical solutions: the abstract states that solutions are obtained in various asymptotic regimes, yet no explicit expressions, validity ranges, or limiting cases (e.g., vanishing chirality or activity) are provided. This omission makes it impossible to verify whether the reported modulation by chirality is recovered consistently in the appropriate limits.

Authors: The explicit closed-form expressions for the splitting probabilities in the various asymptotic regimes, together with their validity ranges, are derived in Section II of the manuscript. In the limit of vanishing chirality the expressions reduce to the known results for achiral active particles, while the zero-activity limit recovers the equilibrium Brownian splitting probability. We will revise the abstract to mention these expressions explicitly and will add a concise summary table of the limiting cases in the main text to facilitate direct verification. revision: yes

Circularity Check

0 steps flagged

Derivation from standard backward Fokker-Planck with no self-referential reductions or fitted predictions

full rationale

The paper's central derivations begin from the standard backward Fokker-Planck equation applied to the chiral active Brownian particle model, yielding analytical solutions in 1D asymptotic regimes and a Fick-Jacobs reduction to effective 1D axial equations for small aspect ratios in corrugated channels. These steps use established mathematical reductions without fitting parameters to subsets of data, without renaming known results as new predictions, and without load-bearing self-citations that reduce the claims to prior author work by definition. The modulation of splitting probabilities by geometry, activity, and chirality follows directly from solving the resulting transport equations, keeping the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the backward Fokker-Planck equation applied to chiral active Brownian motion and the validity of the Fick-Jacobs approximation for narrow channels; these are standard techniques in the field with no new entities introduced.

axioms (1)

domain assumption Particle dynamics follow the overdamped Langevin equations for chiral active Brownian motion with constant speed and rotational diffusion
Standard model assumed for active particles; invoked implicitly to set up the backward Fokker-Planck equation.

pith-pipeline@v0.9.0 · 5465 in / 1407 out tokens · 62008 ms · 2026-05-15T10:58:14.330089+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean; IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction; washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Usq · ∇ pR + Dx∇²pR + Ω ∂pR/∂θ + Dθ ∂²pR/∂θ² = 0 (Eq. 2); Fick-Jacobs reduction yielding effective 1D axial equations for small aspect ratios (Eq. 31)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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