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arxiv: 2510.27536 · v4 · submitted 2025-10-31 · ❄️ cond-mat.stat-mech

Velocity modulus diffusion of self-propelled spherical and circular particles: A generalized Langevin approach

Pedro J. Colmenares This is my paper

Pith reviewed 2026-05-18 03:11 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords self-propelled particlesgeneralized Langevin equationOrnstein-Uhlenbeck processvelocity modulus diffusionharmonic confinementBrownian motionactive matterstochastic modeling
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The pith

Internal self-propulsion causes temporary fluctuations in the velocity magnitude of confined Brownian particles that decay over time.

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

This paper establishes a framework for calculating the average velocity magnitude of an accelerated self-propelled Brownian particle diffusing in a thermal fluid while confined by a harmonic potential. The dynamics are split into an internal self-velocity generated by independent Ornstein-Uhlenbeck processes and the particle motion described by a modified generalized Langevin equation. A reader would care because this separation clarifies how directed propulsion interacts with random thermal forces in bounded settings such as microfluidic traps. The model reveals that the internal mechanism produces spontaneous fluctuations in the diffusive velocity magnitude. These fluctuations decay at long times as expected for diffusive systems, and the approach is illustrated with derivations and simulations for both spheres and disks.

Core claim

The velocity modulus of self-propelled spherical and circular particles is analyzed by partitioning the dynamics into two stochastic processes. The internal self-velocity is generated by a set of independent Ornstein-Uhlenbeck processes that remain independent of the external harmonic field. This internal velocity initializes the diffusion process modeled by a modified generalized Langevin equation in the thermal bath. The system exhibits spontaneous fluctuations in the diffusive velocity magnitude due to the internal mechanism, but these fluctuations decay at long times. The internal propulsion velocity magnitude is derived in spherical coordinates, with accompanying simulations for the 3D.

What carries the argument

The two-process model consisting of internal self-velocity from independent Ornstein-Uhlenbeck processes feeding into a modified generalized Langevin equation for the particle's diffusion under harmonic confinement.

Load-bearing premise

The internal self-velocity is generated by a set of independent Ornstein-Uhlenbeck processes that remain completely independent of the external harmonic field and the thermal bath interactions.

What would settle it

An experiment measuring the velocity magnitude of a self-propelled particle in a harmonic trap over long times and finding persistent fluctuations that do not decay would falsify the model's prediction.

Figures

Figures reproduced from arXiv: 2510.27536 by Pedro J. Colmenares.

Figure 1
Figure 1. Figure 1: FIG. 1. The susceptibility [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Diffusion VM for the sphere. The curves are identi [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Diffusion VM for the disk. The curves are identified [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Simulated averaged propelled VM of the sphere for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Simulated averaged propelled VM of the disk for var [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

This research presents a framework for describing the average velocity magnitude of an accelerated, self-propelled Brownian particle diffusing in a thermal fluid and confined by a harmonic external potential. The system is immersed in a thermal bath of harmonic oscillators at a constant temperature, where the bath constituents also interact with the external field. The dynamics are investigated for both a sphere and a disk, partitioned into two distinct stochastic processes. The first process describes the coarse-grained, time-dependent internal self-velocity generated by a set of independent Ornstein-Uhlenbeck processes, independent of the external field. This internal mechanism provides the initial velocity for the particle to diffuse within the fluid, which is modeled via a modified generalized Langevin equation as the second process. We find that the system exhibits spontaneous fluctuations in the diffusive velocity magnitude due to the internal mechanism; however, as expected, these momentary fluctuations decay at long times. Finally, the internal propulsion velocity magnitude in spherical coordinates is derived, accompanied by simulations of the different magnitudes for both the sphere and the disk, the latter following established equations in polar coordinates.

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 develops a generalized Langevin framework for self-propelled Brownian particles (sphere and disk) confined by a harmonic potential in a thermal bath of oscillators. Dynamics are partitioned into an internal self-velocity generated by independent Ornstein-Uhlenbeck processes (decoupled from the external field) that supplies the initial velocity, and a separate modified GLE describing diffusion in the fluid. The central claims are that spontaneous fluctuations in the diffusive velocity modulus arise from the internal mechanism but decay at long times, together with an explicit derivation of the internal propulsion velocity magnitude in spherical coordinates and supporting simulations for both geometries.

Significance. If the decoupling assumption is valid, the separation into independent stochastic processes offers a tractable route to analytic expressions for velocity-modulus statistics in confined active particles and reproduces the expected long-time decay of fluctuations. The work supplies concrete derivations and numerical illustrations that could serve as a baseline for more coupled active-matter models.

major comments (2)
  1. [Abstract and modeling description] Abstract and partitioning of dynamics: the statement that internal self-velocity is generated by 'independent Ornstein-Uhlenbeck processes, independent of the external field' makes the decay of |v| fluctuations true by construction inside the decoupled model. Because the bath oscillators themselves couple to the same external harmonic potential, a consistent fluctuation-dissipation relation for the velocity modulus would generally require cross terms or back-action; the manuscript does not derive or test the consequences of restoring even weak coupling.
  2. [Derivation of internal propulsion velocity magnitude] Derivation of internal propulsion velocity magnitude: the Ornstein-Uhlenbeck correlation time and amplitude appear as free parameters whose values are not shown to be fixed by independent external data. If these parameters are subsequently adjusted to match the same velocity statistics the model is intended to predict, the central expressions reduce to a tautology rather than a genuine derivation from the stated equations.
minor comments (2)
  1. Notation for spherical versus polar coordinates is introduced without an explicit comparison table or side-by-side equations, making it difficult to verify consistency between the sphere and disk cases.
  2. Simulation details (integration scheme, time-step, ensemble size, and exact parameter values used for the OU processes) are mentioned but not supplied in sufficient detail for independent reproduction.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, indicating planned revisions where appropriate to improve clarity while maintaining the focus of the work.

read point-by-point responses

  1. Referee: Abstract and partitioning of dynamics: the statement that internal self-velocity is generated by 'independent Ornstein-Uhlenbeck processes, independent of the external field' makes the decay of |v| fluctuations true by construction inside the decoupled model. Because the bath oscillators themselves couple to the same external harmonic potential, a consistent fluctuation-dissipation relation for the velocity modulus would generally require cross terms or back-action; the manuscript does not derive or test the consequences of restoring even weak coupling.

    Authors: We agree that the observed decay of |v| fluctuations is a direct consequence of the decoupling assumption in our framework. This separation is introduced deliberately to isolate the internal self-propulsion (modeled via independent OU processes) from the external confinement and bath dynamics, enabling closed-form analytic expressions for the velocity-modulus statistics. While the bath oscillators do interact with the harmonic potential, our model treats the internal velocity as an additive driving term without back-action on the bath. We acknowledge that restoring even weak coupling would generally introduce cross terms and modify the fluctuation-dissipation relations. In the revised manuscript we will add an explicit discussion of this modeling assumption, its range of validity (e.g., when self-propulsion arises from internal forces largely decoupled from the bath response), and the expected qualitative effects of weak coupling, without attempting a full re-derivation of the coupled case. revision: partial

  2. Referee: Derivation of internal propulsion velocity magnitude: the Ornstein-Uhlenbeck correlation time and amplitude appear as free parameters whose values are not shown to be fixed by independent external data. If these parameters are subsequently adjusted to match the same velocity statistics the model is intended to predict, the central expressions reduce to a tautology rather than a genuine derivation from the stated equations.

    Authors: The correlation time and amplitude of the OU processes are indeed phenomenological parameters that encode the persistence and strength of the internal self-propulsion mechanism. They are not derived from the confined dynamics within this work; rather, they are inputs that can be fixed independently, for instance by fitting the velocity autocorrelation function measured in unconfined active particles or by reference to microscopic models of the propulsion. The subsequent derivations of the time-dependent velocity modulus then follow rigorously from the stated stochastic differential equations once these parameters are specified. We will revise the manuscript to state this distinction more clearly, emphasizing that the parameters are set by the self-propulsion properties alone and that the analytic expressions constitute genuine predictions for the confined statistics rather than a post-hoc fit. revision: yes

standing simulated objections not resolved
  • Deriving or numerically testing the consequences of restoring coupling (including cross terms) between the internal self-velocity and the bath oscillators that also feel the external harmonic potential.

Circularity Check

0 steps flagged

Derivation is self-contained; no reduction to inputs by construction

full rationale

The paper explicitly introduces the internal self-velocity as generated by a set of independent Ornstein-Uhlenbeck processes that are decoupled from the external harmonic field and thermal bath. The statement that fluctuations in the diffusive velocity magnitude decay at long times is presented as expected from this internal mechanism. This follows directly from the standard decay properties of OU processes and does not require fitting parameters to the target statistics or invoking self-citations. The derivation of the internal propulsion velocity magnitude in spherical coordinates is a direct calculation from the stated model equations rather than a tautological renaming or self-referential fit. No load-bearing step reduces the central claims to the modeling assumptions by construction; the assumptions remain independent and falsifiable outside the fitted values.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that internal propulsion can be modeled as a set of independent Ornstein-Uhlenbeck processes decoupled from the external potential and bath; no free parameters are explicitly named in the abstract, but the model necessarily introduces at least the correlation times and amplitudes of those processes.

free parameters (1)
  • Ornstein-Uhlenbeck correlation time and amplitude
    These set the statistics of the internal self-velocity and must be chosen or fitted to produce the reported fluctuations.
axioms (1)
  • domain assumption The thermal bath consists of harmonic oscillators that interact with the external harmonic field.
    Invoked to justify the modified generalized Langevin equation for the fluid interaction.

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Reference graph

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