Number fluctuations distinguish different self-propelling dynamics

Carine Douarche; Sophie Marbach; Tristan Cerdin

arxiv: 2604.02872 · v1 · submitted 2026-04-03 · ❄️ cond-mat.soft · cond-mat.stat-mech

Number fluctuations distinguish different self-propelling dynamics

Tristan Cerdin , Sophie Marbach , Carine Douarche This is my paper

Pith reviewed 2026-05-13 18:55 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.stat-mech

keywords number fluctuationsself-propelled particlesreorientation dynamicsactive matternonequilibrium suspensionsfluctuation statisticsdynamical parameters

0 comments

The pith

Number fluctuations over time distinguish different self-propelled particle models through reorientation dynamics.

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

The paper develops a theory to learn the dynamical parameters of self-propelled particle models from the statistics of number fluctuations N(t) in virtual observation boxes. Unlike traditional trajectory analysis, N(t) statistics distinguish between models by sensing subtle differences in reorientation dynamics that govern re-entrance events in boxes. A sympathetic reader would care because the approach works from collective signals rather than single-particle tracks and therefore applies to dense nonequilibrium suspensions where individual tracking is difficult. The method turns static fluctuation measurements into a dynamic probe of reorientation rules.

Core claim

In nonequilibrium suspensions, the time-dependent number fluctuations N(t) inside virtual observation boxes encode enough information to distinguish and parameterize different self-propelled particle models. The distinction arises because reorientation dynamics control the frequency and statistics of particle re-entries into the boxes, producing model-specific signatures in the fluctuation signal.

What carries the argument

The number fluctuation signal N(t) measured inside fixed virtual boxes, which registers re-entrance events controlled by particle reorientation rules.

If this is right

Models with distinct reorientation dynamics generate distinguishable N(t) statistics.
Dynamical parameters can be extracted directly from fluctuation measurements without full trajectory reconstruction.
The approach remains usable in dense suspensions where single-particle tracking fails.
Dynamic N(t) signals add time-resolved information beyond what static number fluctuations provide.

Where Pith is reading between the lines

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

The same fluctuation data could be used to test whether real experimental systems follow one reorientation rule over another.
Extensions might examine how interaction strength alters the inversion accuracy when density is no longer fixed.
The method could be combined with existing structure-factor measurements to separate structural from dynamical contributions in active matter.

Load-bearing premise

Differences in reorientation dynamics dominate the observed N(t) statistics and can be inverted to recover model parameters without being masked by density or interaction effects.

What would settle it

Two models with different reorientation rules but otherwise identical parameters would produce identical N(t) statistics.

Figures

Figures reproduced from arXiv: 2604.02872 by Carine Douarche, Sophie Marbach, Tristan Cerdin.

**Figure 1.** Figure 1: Probing self-propelled particle models. (a) Simulated trajectories of 3 models – Active Brownian (ABP), Run and tumble (RTP) or Active Ornstein-Uhlenbeck Particles (AOUP) – overlayed with virtual observation boxes of size L = 3 µm. Here v = 3 µm s−1 , Dr = 1 s−1 , and for illustrative purposes we turn off translational diffusion. (b) The mean-squared displacements for many particles are indistinguishable… view at source ↗

**Figure 3.** Figure 3: Limit regimes of advection or diffusion captured by scaling laws. Number fluctuations for ABPs, same data as in Fig. 2a, with (a) advective rescaling in time and (b) diffusive rescaling. (c-d) Schematics illustrating the probability distribution clouds to find a particle in a box after some time t (yellow) given it started in the box initially (blue) in a purely (c) advective or (d) diffusive case, with D… view at source ↗

**Figure 2.** Figure 2: Number fluctuations ⟨∆N 2 (t)⟩ exhibit 3 distinct regimes in time, for (a) Active Brownian particles, (b) Run and Tumble particles, and (c) Active OrnsteinUhlenbeck particles. Box sizes are the same for all 3 plots, and go from small (in light color) to large boxes (in dark color). Parameters are the same as in Fig. (1)b. Symbols: simulation; lines: theory – see text. Gaussian, as is the case for AOUPs… view at source ↗

**Figure 4.** Figure 4: Number correlation functions CN (t) distinguish different models. (a) Number correlations from simulations (symbols) and theory (lines) for different box sizes; (b) Pstay(t) (open) and Preturn(t) (full symbols) from simulations, taking L = 5 µm. (inset) Lin-log scale of the same plot keeping just ABP data. Color codes for particle models are the same as in (a) and the gray color represents a passive case w… view at source ↗

read the original abstract

In nonequilibrium suspensions, static number fluctuations $N$ in virtual observation boxes reveal remarkable structural properties, but the dynamic potential of $N(t)$ signals remains unexplored. Here, we develop a theory to learn the dynamical parameters of self-propelled particle models from $N(t)$ statistics. Unlike traditional trajectory analysis, $N(t)$ statistics distinguish between models, by sensing subtle differences in reorientation dynamics that govern re-entrance events in boxes. This paves the way for quantifying advanced dynamic features in dense nonequilibrium suspensions.

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

N(t) statistics give a workable way to tell reorientation rules apart in active particles, but the mapping looks fragile once interactions enter.

read the letter

The paper's real move is to treat the time series of particle counts inside a virtual box as a signal that carries information about reorientation dynamics rather than just static structure. That is new. Static number fluctuations have been used before, but turning the fluctuations into a tool that separates models by how particles turn around is a fresh angle, especially for systems where full trajectories are hard to get. The abstract frames the claim cleanly: differences in re-entry statistics driven by reorientation rules show up in the moments of N(t). If the derivations hold, this could be a practical observable for dense active matter where tracking is messy. Credit for keeping the focus on an experimentally accessible quantity instead of another simulation-heavy comparison. The soft spot is exactly the one the stress-test flags. The theory appears to start from non-interacting Langevin or telegrapher equations, yet the target is dense suspensions. Once excluded volume or alignment interactions change effective persistence and crossing rates, the link from bare reorientation parameters to N(t) moments is no longer one-to-one. The manuscript does not appear to supply a decoupling argument or a scaling check that would show the distinction survives at finite packing fraction. That gap makes the central claim harder to trust without further work. The paper is for people working on active colloids or bacterial suspensions who already measure local densities and want an extra handle on dynamics without single-particle tracking. It is worth sending to referees because the idea is concrete and the potential payoff is clear, even if the interaction robustness needs to be tightened. A serious review would likely ask for explicit tests with and without interactions plus error bars on the extracted parameters.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a theory for extracting dynamical parameters of self-propelled particle models from the statistics of time-dependent number fluctuations N(t) inside virtual observation boxes. It argues that N(t) distinguishes models by sensing differences in reorientation dynamics that control re-entrance events, providing an alternative to trajectory-based analysis for dense nonequilibrium suspensions.

Significance. If the central mapping holds, the work offers a practical route to infer reorientation parameters from fluctuation data in dense active systems where full tracking is impractical. The focus on re-entrance statistics as a probe of intrinsic dynamics is a potentially useful addition to the active-matter toolkit, provided the separation from interaction effects can be established.

major comments (2)

[Theory section (derivation of N(t) from reorientation rules)] The abstract states that the theory targets dense nonequilibrium suspensions, yet the derivation begins from non-interacting Langevin or telegrapher equations for trajectories inside the virtual box. No decoupling argument or scaling analysis is supplied to show that the mapping from bare reorientation parameters to N(t) moments remains injective once excluded-volume or alignment interactions are restored at finite packing fraction.
[Results on model distinction] The claim that N(t) statistics uniquely sense reorientation dynamics (rather than effective persistence modified by interactions) is load-bearing for the distinction between models. The manuscript provides no numerical test or analytic bound demonstrating that interaction-induced changes in crossing probabilities do not erase or mimic the signatures of different bare reorientation rules.

minor comments (2)

[Introduction] Notation for the virtual box size and the precise definition of re-entrance events should be introduced earlier and used consistently.
[Figures] Figure captions should explicitly state the packing fractions used in the simulations so readers can assess the dense-regime regime.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised concern the extension of the non-interacting derivation to dense interacting systems, which we address by clarifying the assumptions and committing to additional analysis and tests in the revision.

read point-by-point responses

Referee: [Theory section (derivation of N(t) from reorientation rules)] The abstract states that the theory targets dense nonequilibrium suspensions, yet the derivation begins from non-interacting Langevin or telegrapher equations for trajectories inside the virtual box. No decoupling argument or scaling analysis is supplied to show that the mapping from bare reorientation parameters to N(t) moments remains injective once excluded-volume or alignment interactions are restored at finite packing fraction.

Authors: We agree that the derivation is performed in the non-interacting limit to isolate the effect of reorientation rules on re-entrance statistics. The abstract's reference to dense suspensions reflects our expectation that the signatures remain observable when interactions are moderate. In the revision we will add a scaling analysis and perturbative argument showing that the mapping from bare parameters to N(t) moments stays approximately injective for packing fractions below a crossover value where single-particle reorientation still dominates crossing events. We will also cite effective-medium approaches from the active-matter literature to support the range of validity. revision: partial
Referee: [Results on model distinction] The claim that N(t) statistics uniquely sense reorientation dynamics (rather than effective persistence modified by interactions) is load-bearing for the distinction between models. The manuscript provides no numerical test or analytic bound demonstrating that interaction-induced changes in crossing probabilities do not erase or mimic the signatures of different bare reorientation rules.

Authors: This is a fair criticism; the current manuscript establishes the baseline distinction for non-interacting particles. To substantiate the claim for interacting cases we will include, in the revised manuscript, Brownian-dynamics simulations of the models at moderate packing fractions together with an analytic bound based on short-time independence of crossing events. These additions will demonstrate that the ordering of N(t) moments between models is preserved under weak to moderate interactions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; theory derives N(t) from reorientation dynamics independently

full rationale

The paper develops a theory to extract dynamical parameters from N(t) statistics by modeling re-entrance events governed by reorientation rules in virtual boxes. The abstract and description present this as a forward derivation from standard Langevin/telegrapher-type equations for particle trajectories, without any quoted steps that fit parameters to a data subset and then rename the output as a prediction, or that reduce the central mapping to a self-citation or self-definition. The distinction between models is claimed to arise from differences in intrinsic reorientation that affect crossing probabilities, and no load-bearing equation is shown to be equivalent to its inputs by construction. This is the expected non-finding for a paper whose core claim remains an independent theoretical mapping.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5377 in / 869 out tokens · 42883 ms · 2026-05-13T18:55:36.581871+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 washburn_uniqueness_aczel unclear

?

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

We derive analytical theory... ⟨ΔN²(t)⟩=2⟨N⟩P_out(t) ... P_out(t)≃4vt/πL ... for ABP and RTPs ... 4√π⟨N⟩vt/L for AOUPs
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

MSD of all 3 models is captured by a single equation ... ⟨Δr²(t)⟩=4(Dt + v²/2Dr)t + ...

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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[1] [1]

M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrody- namics of soft active matter, Reviews of modern physics 85, 1143 (2013)

work page 2013

[2] [2]

Ramaswamy, The mechanics and statistics of active matter, Annual Review of Condensed Matter Physics1, 323 (2010)

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work page 2010

[3] [3]

Bechinger, R

C. Bechinger, R. Di Leonardo, H. L¨ owen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Rev. Mod. Phys.88, 045006 (2016)

work page 2016

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work page 1983

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F. W. Dahlquist, P. Lovely, and D. E. Koshland, Quanti- tative analysis of bacterial migration in chemotaxis, Na- ture New Biology236, 120 (1972)

work page 1972

[6] [6]

M. J. Schnitzer, Theory of continuum random walks and application to chemotaxis, Phys. Rev. E48, 2553 (1993)

work page 1993

[7] [7]

Saragosti, V

J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin, and P. Silberzan, Directional per- sistence of chemotactic bacteria in a travel- ing concentration wave, Proceedings of the Na- tional Academy of Sciences108, 16235 (2011), https://www.pnas.org/doi/pdf/10.1073/pnas.1101996108

work page doi:10.1073/pnas.1101996108 2011

[8] [8]

Vicsek, A

T. Vicsek, A. Czir´ ok, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett.75, 1226 (1995)

work page 1995

[9] [9]

Tailleur and M

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work page 2008

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J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian, Self-motile colloidal particles: from directed propulsion to random walk, 6 Physical Review Letters99(2007)

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