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arxiv: 2606.24614 · v1 · pith:ZNEBYFSTnew · submitted 2026-06-23 · ❄️ cond-mat.soft · nlin.AO

Dynamics and stability of inertial flexible chains under follower activity

Sattwik Sadhu , Nitin Kriplani , Raghunath Chelakkot This is my paper

Pith reviewed 2026-06-25 21:53 UTC · model grok-4.3

classification ❄️ cond-mat.soft nlin.AO
keywords inertial active chainsfollower activityperiodic motionstability analysisflexible polymersnonequilibrium dynamicsbond length approximationchain conformation
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The pith

Inertial follower forces on flexible chains produce length-dependent periodic solutions whose stability is set by segment mass and activity level.

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

The authors investigate how significant inertia changes the motion of extensible chains of active particles that push or pull along their own contour. Simulations map the resulting patterns across chain length, particle mass, and activity strength. Separate analytic approximations are constructed for three-particle chains and for very long chains; both sets of formulas recover the observed bond lengths and angles to high accuracy. The short-chain theory also supplies explicit stability boundaries for the periodic orbit, while the long-chain theory locates the parameter region in which the circular solution loses structural stability.

Core claim

In the N=3 limit, closed-form expressions for bond lengths and angles are obtained and their linear stability is mapped as a function of mass and activity; in the N≫1 limit the same expressions remain accurate and the circular periodic solution is shown to become structurally unstable outside a bounded window of activity and mass.

What carries the argument

Follower activity, the force applied by each particle along the local chain tangent, coupled to inertial dynamics.

If this is right

  • Periodic motion in three-particle chains remains stable only inside a mass-activity region whose boundaries are given explicitly by the linear stability analysis.
  • For long chains the circular orbit loses structural stability once activity or mass crosses the identified threshold.
  • Bond lengths and angles along the chain can be predicted from the approximate formulas without integrating the full inertial equations in either limit.
  • The transition between ordered periodic states and other dynamical regimes is controlled by the same mass and activity parameters in both short and long chains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The analytic expressions could be tested by measuring average bond lengths in experiments on colloidal chains or robotic active filaments at controlled inertia.
  • Intermediate-length chains may exhibit a crossover between the two limiting behaviors that could be located by extending the same simulation protocol.
  • The structural instability identified for long chains suggests a route to chaotic or disordered motion that might be observable by increasing chain length while holding mass and activity fixed.

Load-bearing premise

The separate short-chain and long-chain approximations remain accurate across the full range of simulated masses and activities without requiring higher-order corrections.

What would settle it

Numerical integration at intermediate chain lengths or outside the stated activity-mass windows that shows bond lengths or angles deviating systematically from the derived expressions.

Figures

Figures reproduced from arXiv: 2606.24614 by Nitin Kriplani, Raghunath Chelakkot, Sattwik Sadhu.

Figure 1
Figure 1. Figure 1: FIG. 1: (a) Schematic showing the activity scheme used for our system under study (b) Schematic of the three bead system exhibiting circular [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Effect of inertia in the dynamics of a short chain with [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Stability Diagram showing the parameter region, where the 3-bead system is stable (amber) and unstable (sky blue). The x axis [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (a-b) Steady state chain configuration and trajectory for the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

The dynamics of flexible polymers and chains under follower activity is known to produce diverse nonequilibrium states. A prominent feature of such systems is the emergence of periodic motion arising from the coupling between internal activity and chain conformation. Recently, it has been shown that flexible and extensible chains of active particles exhibit rich dynamical patterns in the overdamped limit, where inertia is negligible. Here, we study the complex dynamics of a flexible and extensible chain of active particles under follower activity when inertia is significant. Using numerical simulations, we quantify the chain dynamics as a function of chain length ($N$), segment mass, and activity. To rationalize the numerical results, we develop theoretical descriptions in the limit of short chains ($N=3$) and long chains ($N \gg 1$). In both these limits, we derive approximate expressions for the bond lengths and bond angles along the contour, which show excellent agreement with the numerical results. In addition, for short chains, we derive the stability conditions for a periodic motion as a function of segment mass and activity. For long chains ($N\gg1$) we identify parameter regime in which the circular, periodic solution becomes structurally unstable. Our theoretical and numerical analysis provides insights into the emergence of ordered and periodic behaviour in active chains.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the inertial dynamics of flexible, extensible chains of active particles driven by follower forces. Numerical simulations map the dependence on chain length N, segment mass, and activity strength. Separate analytic approximations are constructed for the N=3 and N≫1 limits; these yield explicit expressions for bond lengths and angles that are reported to match the simulations, together with stability criteria for periodic orbits (N=3) and a regime of structural instability of the circular solution (N≫1).

Significance. If the limiting-case approximations remain accurate across the simulated parameter space, the work supplies concrete stability boundaries and identifies an inertial mechanism for loss of periodicity that is absent from the overdamped literature. The absence of free fitting parameters in the derived expressions and the direct comparison to full inertial simulations are strengths.

major comments (2)
  1. [Theoretical sections on N=3 and N≫1 limits] The stability conditions for N=3 and the structural-instability regime for N≫1 are obtained from truncated expressions for bond lengths and angles. The manuscript must demonstrate that the neglected higher-order inertial and follower-force couplings remain small throughout the explored ranges of segment mass and activity; otherwise the reported boundaries are not guaranteed to describe the full dynamics.
  2. [Comparison of theory and numerics] The claim of 'excellent agreement' between the approximate analytic expressions and the numerical trajectories is central to the paper. Quantitative error measures (e.g., L2 residuals on bond lengths/angles versus mass and activity) should be supplied for both limits so that the domain of validity of the truncations can be assessed.
minor comments (2)
  1. Figure captions should explicitly state the values of segment mass and activity used in each panel and indicate which curves correspond to the N=3 versus N≫1 analytic predictions.
  2. Notation for the follower force magnitude and the inertial parameter should be introduced once and used consistently; occasional redefinition of symbols slows reading.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments identify important points for strengthening the justification of our approximations. We will revise the manuscript to address both by adding the requested demonstrations and quantitative metrics.

read point-by-point responses

  1. Referee: [Theoretical sections on N=3 and N≫1 limits] The stability conditions for N=3 and the structural-instability regime for N≫1 are obtained from truncated expressions for bond lengths and angles. The manuscript must demonstrate that the neglected higher-order inertial and follower-force couplings remain small throughout the explored ranges of segment mass and activity; otherwise the reported boundaries are not guaranteed to describe the full dynamics.

    Authors: We agree that explicit verification of the truncation validity is required. In the revised manuscript we will add a dedicated subsection that estimates the magnitude of the neglected higher-order inertial and follower-force terms using the simulated trajectories. For each explored value of segment mass and activity we will compute the relative size of these terms with respect to the retained leading-order contributions, confirming that they remain small (below a stated threshold) inside the reported stability and instability regimes. revision: yes

  2. Referee: [Comparison of theory and numerics] The claim of 'excellent agreement' between the approximate analytic expressions and the numerical trajectories is central to the paper. Quantitative error measures (e.g., L2 residuals on bond lengths/angles versus mass and activity) should be supplied for both limits so that the domain of validity of the truncations can be assessed.

    Authors: We accept that quantitative error measures are needed to substantiate the agreement. The revised version will include new figures (or supplementary panels) that report L2 residuals and relative errors for bond lengths and angles, plotted versus segment mass and activity strength, separately for the N=3 and N≫1 analytic approximations. These metrics will be computed directly from the simulation data and will delineate the parameter domain where the truncations remain accurate. revision: yes

Circularity Check

0 steps flagged

No circularity: theoretical approximations derived independently and compared to simulations

full rationale

The provided abstract and context describe numerical simulations of inertial active chains followed by separate analytic approximations in the N=3 and N≫1 limits for bond lengths, angles, and stability boundaries. These approximations are stated to be derived from the governing equations in each limit and then shown to match the simulations. No equations, fitting procedures, or self-citations are quoted that would reduce the predictions to the input data by construction. The derivation chain therefore remains self-contained against external benchmarks (the simulations), with no load-bearing reduction to fitted parameters or prior self-work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the ledger is left empty pending access to the full derivations.

pith-pipeline@v0.9.1-grok · 5758 in / 1225 out tokens · 24110 ms · 2026-06-25T21:53:02.538263+00:00 · methodology

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