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arxiv: 2606.22209 · v1 · pith:QU7D522Anew · submitted 2026-06-20 · ❄️ cond-mat.stat-mech · math-ph· math.MP

Giant Fluctuations in Self-Propelled Particles with Age-Dependent Switching

Shabnam Sohrabi , Farhad H. Jafarpour This is my paper

Pith reviewed 2026-06-26 10:53 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MP
keywords self-propelled particlesage-dependent switchinggiant fluctuationssemi-Markovian processcentral limit theorem breakdownballistic to diffusive transitionvariance scalingpersistent orientation
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The pith

At a critical switching parameter the displacement variance of these particles grows as T squared over log T.

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

The paper studies self-propelled particles that alternate between a downstream phase with fixed switching probability and an upstream phase whose reorientation hazard decays with the time already spent upstream. The time-averaged velocity acts as an order parameter that undergoes a continuous transition at a=1, separating a ballistic regime dominated by upstream persistence from a normal diffusive regime. Exactly at a=1 the position variance grows ballistically except for a logarithmic correction, Var(x_T) proportional to T squared over log T. This scaling arises because the slow decay of the reorientation probability produces long memory that marginally violates the central limit theorem. The result matters for any system in which persistence in one direction can produce unusually large fluctuations while still allowing net transport.

Core claim

The dynamics alternate between a Markovian downstream phase with constant switching probability r and a semi-Markovian upstream phase whose age-dependent hazard probability a/(b+c) decays with the internal clock c. The time-averaged velocity shows a continuous transition at a=1 that separates an upstream-dominated ballistic regime for a less than 1 from an ergodic diffusive regime for a greater than 1. At the critical point a=1 the variance of displacement scales as T squared over log T, demonstrating a marginal breakdown of the central limit theorem produced by the slowly decaying reorientation probabilities.

What carries the argument

The age-dependent hazard probability a/(b+c) in the semi-Markovian upstream phase, which controls the duration of persistent orientation and produces the critical scaling.

If this is right

  • The time-averaged velocity serves as the order parameter that jumps from ballistic to diffusive behavior across a=1.
  • Exact expressions for the propagator are obtained via generating-function methods and discrete-time moment recurrences.
  • For a less than 1 the mean displacement grows linearly with time while for a greater than 1 it grows diffusively.
  • The same mechanism produces unusually high-variance exploratory paths in any biased environment with slowly decaying reorientation.

Where Pith is reading between the lines

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

  • Similar giant-fluctuation scalings may appear in other persistent-motion models whenever the memory kernel decays slower than exponentially but still allows a critical point.
  • The marginal CLT breakdown suggests that weak power-law tails in waiting times can be tuned to produce intermediate regimes between normal diffusion and superdiffusion.
  • Generating-function techniques developed here can be reused for any semi-Markov process whose hazard rate depends on an internal age variable.

Load-bearing premise

The upstream phase is governed by a semi-Markovian process whose hazard probability takes the specific form a/(b+c) that decays with the internal clock c.

What would settle it

Measure the long-time variance of particle position in an experiment where the upstream reorientation probability is tuned to decay exactly as a/(b+c) and check whether it follows T squared over log T when a equals 1.

Figures

Figures reproduced from arXiv: 2606.22209 by Farhad H. Jafarpour, Shabnam Sohrabi.

Figure 1
Figure 1. Figure 1: Scaling of the second moment ⟨x 2 T ⟩ (solid symbols) and variance Var(xT ) (open symbols) for a broad range of the exponent a. In the sub￾critical (a = 0.100) and supercritical (a = 1.900) regimes, the variance follows the ordinary diffusive scaling ∼ T (dotted line). At the critical point a = 1.000, the variance exhibits a marked departure from normal diffusion, growing nearly ballistically. In the subcr… view at source ↗
Figure 2
Figure 2. Figure 2: Detailed view of the scaling behavior in the vicinity of the critical [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
read the original abstract

We investigate the transport and fluctuation properties of self-propelled particles whose motion is governed by an age-dependent phase-switching mechanism. The dynamics alternate between a Markovian downstream phase with a constant switching probability $r$ and a semi-Markovian upstream phase in which the age-dependent hazard probability $a/(b+c)$ decays with the internal clock $c$, representing persistent orientation. The time-averaged velocity, as an order parameter, shows a continuous transition at $a=1$ which separates an upstream-dominated ballistic regime ($a<1$) from an ergodic diffusive regime ($a>1$). Through generating-function methods and discrete-time moment recurrences, we derive exact expressions for the propagator and determine the long-time asymptotics of the mean displacement and variance. At the critical point $a=1$, the system exhibits giant fluctuations, with the variance scaling ballistically up to a logarithmic correction, $\mathrm{Var}(x_T) \propto T^2 / \log T$. These results demonstrate how slowly decaying reorientation probabilities lead to a marginal breakdown of the Central Limit Theorem, enabling unusually high-variance exploratory dynamics in biased environments.

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 self-propelled particles that alternate between a Markovian downstream phase (constant switching rate r) and a semi-Markovian upstream phase whose reorientation hazard is a/(b+c) with internal age c. Generating-function and discrete-time moment-recurrence methods are used to obtain exact expressions for the propagator. The time-averaged velocity undergoes a continuous transition at a=1 separating an upstream-dominated ballistic regime (a<1) from an ergodic diffusive regime (a>1). At criticality the variance is reported to scale as T²/log T, indicating giant fluctuations and a marginal violation of the central limit theorem.

Significance. If the derivations are correct, the work supplies an exactly solvable active-particle model in which a slowly decaying hazard produces a tunable transition to super-ballistic fluctuations. The explicit propagator and the T²/log T scaling constitute concrete, falsifiable predictions that could be tested in colloidal or bacterial systems with controlled persistence. The generating-function approach is a methodological strength.

major comments (2)
  1. [§3] §3 (or equivalent section deriving the moment recurrences): the claimed T²/log T scaling at a=1 is obtained from the solution of the recurrence for the second moment; the precise asymptotic analysis that isolates the logarithmic correction must be shown explicitly, because the hazard a/(b+c) at a=1 is marginally non-integrable and small changes in the cutoff can alter the log factor.
  2. [Model definition and §4] The definition of the order parameter (time-averaged velocity) and the location of the transition at a=1 rely on the specific functional form a/(b+c). It is not shown whether the same critical scaling persists for other slowly decaying hazards (e.g., 1/(b+c)^α with α≠1); this limits the generality of the “marginal breakdown of the CLT” claim.
minor comments (2)
  1. [Abstract and §2] The abstract states that the downstream phase is Markovian with constant probability r, yet the upstream phase is semi-Markovian; a brief remark on whether the overall process remains Markovian when both phases are considered together would clarify the scope of the generating-function method.
  2. [Throughout] Notation for the internal clock c versus global time t should be made uniform throughout; occasional use of c for both the age variable and a constant is confusing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation, and the constructive suggestions. We address the two major comments below.

read point-by-point responses

  1. Referee: §3 (or equivalent section deriving the moment recurrences): the claimed T²/log T scaling at a=1 is obtained from the solution of the recurrence for the second moment; the precise asymptotic analysis that isolates the logarithmic correction must be shown explicitly, because the hazard a/(b+c) at a=1 is marginally non-integrable and small changes in the cutoff can alter the log factor.

    Authors: We agree that an explicit asymptotic analysis is required. The recurrence for the second moment at a=1 yields a generating function whose singularity structure produces the T²/log T leading term; we will add a dedicated appendix (or subsection) that carries out the Tauberian analysis and the Euler-Maclaurin summation needed to isolate the logarithmic correction, confirming robustness against cutoff variations. revision: yes

  2. Referee: The definition of the order parameter (time-averaged velocity) and the location of the transition at a=1 rely on the specific functional form a/(b+c). It is not shown whether the same critical scaling persists for other slowly decaying hazards (e.g., 1/(b+c)^α with α≠1); this limits the generality of the “marginal breakdown of the CLT” claim.

    Authors: The hazard a/(b+c) is deliberately chosen as the exactly solvable marginal case (α=1) that permits closed-form generating functions. The transition at a=1 and the associated scaling are therefore specific to this functional form. We will revise the text to state explicitly that the marginal CLT breakdown is demonstrated for this critical hazard and that other exponents generally require different techniques; a short remark will be added noting the scope of the result. revision: partial

Circularity Check

0 steps flagged

Derivation self-contained from explicit hazard function; no circularity

full rationale

The model is defined by the explicit semi-Markovian hazard a/(b+c) in the upstream phase. The abstract and description state that generating-function methods and discrete-time moment recurrences are applied to this process to obtain the propagator, mean displacement, and variance asymptotics, including the T²/log T scaling at a=1. No equation reduces the claimed scaling to a fitted parameter, self-citation, or ansatz smuggled from prior work; the result is obtained by direct analysis of the stated dynamics. No load-bearing self-citations or uniqueness theorems are invoked in the provided text.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on the model definition of the two-phase dynamics and the mathematical solvability via generating functions; no new particles or forces are postulated.

free parameters (3)
  • a
    Controls the strength of age dependence in the upstream hazard; critical value a=1 separates regimes.
  • b
    Offset parameter in the hazard a/(b+c).
  • r
    Constant switching probability in the downstream Markovian phase.
axioms (2)
  • domain assumption The particle dynamics consist of alternating Markovian downstream and semi-Markovian upstream phases with the given hazard.
    Core modeling choice stated in the abstract.
  • standard math Generating functions and discrete-time moment recurrences suffice to obtain the exact propagator and long-time asymptotics.
    Standard technique invoked for the derivation.

pith-pipeline@v0.9.1-grok · 5738 in / 1493 out tokens · 40763 ms · 2026-06-26T10:53:03.992369+00:00 · methodology

discussion (0)

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Reference graph

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