Diffusive-to-Ballistic transition in a Persistent Random Walk

Amit Pradhan; Purusattam Ray; Reshmi Roy

arxiv: 2605.20116 · v1 · pith:4YEJPHS4new · submitted 2026-05-19 · ❄️ cond-mat.stat-mech

Diffusive-to-Ballistic transition in a Persistent Random Walk

Amit Pradhan , Reshmi Roy , Purusattam Ray This is my paper

Pith reviewed 2026-05-20 03:25 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords persistent random walkvelocity reversal probabilitydynamical transitionsuper-diffusive regimeballistic motionpower-law decaynon-equilibrium transitionBinder cumulant scaling

0 comments

The pith

A persistent random walk with reversal probability decaying as a power law switches from super-diffusive to ballistic motion exactly at exponent alpha equals one.

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

The paper establishes a criterion for a dynamical transition in persistent random walks whose velocity reversals occur with a time-dependent probability. For the representative case of a power-law form p(t) proportional to t to the minus alpha, the motion is super-diffusive when alpha is less than one and becomes ballistic when alpha is one or larger. The authors characterize the transition through velocity correlations, persistence statistics, Binder cumulant scaling, and displacement fluctuations. They further show that the same transition arises for other reversal probabilities that obey an analogous integral condition and that the transition survives in any spatial dimension provided velocity directions remain isotropic.

Core claim

For a persistent random walk whose reversal probability follows p(t) ~ t^{-alpha}, the system undergoes a transition at alpha=1 that separates a super-diffusive regime (alpha<1) from a ballistic regime (alpha >=1). The transition is diagnosed by changes in velocity autocorrelation decay, persistence properties, and finite-time scaling of the Binder cumulant together with displacement fluctuations. The transition criterion extends to other time-dependent reversal probabilities that satisfy the same integral condition and remains valid in arbitrary dimensions when isotropy of the velocity space is preserved.

What carries the argument

The time-dependent reversal probability p(t) ~ t^{-alpha} together with the integral criterion on its long-time tail that distinguishes super-diffusive from ballistic scaling of the mean-squared displacement.

If this is right

When alpha is greater than or equal to one the mean-squared displacement grows quadratically with time, indicating ballistic motion.
For alpha less than one the mean-squared displacement grows faster than linearly but slower than quadratically, indicating super-diffusion.
The location of the transition is independent of spatial dimension provided velocity isotropy is maintained.
Finite-time scaling collapses of the Binder cumulant and displacement fluctuations locate the transition point and characterize its critical properties.
The transition criterion applies to any reversal probability whose long-time tail satisfies the same integral condition as the power-law case.

Where Pith is reading between the lines

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

The transition may appear in models of biological motility where turning rates decrease slowly with time, offering a mechanism for persistent directed motion without external bias.
Numerical tests in higher dimensions could confirm whether isotropy alone is sufficient to protect the ballistic regime against dimensional effects.
Experiments with colloidal particles or active matter whose reorientation probability is engineered to follow a power law could directly observe the predicted change in scaling.

Load-bearing premise

The reversal probability decays as a power law or satisfies an equivalent integral condition while the directions of the velocity remain statistically isotropic at all times.

What would settle it

A direct measurement of the mean-squared displacement scaling exponent or the Binder cumulant crossing point in a simulation or experiment that tunes the reversal probability exponent through alpha=1.

Figures

Figures reproduced from arXiv: 2605.20116 by Amit Pradhan, Purusattam Ray, Reshmi Roy.

**Figure 2.** Figure 2: FIG. 2: Plot of the survival probability [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: The Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

read the original abstract

We study persistent random walk with time dependent velocity reversal probabilities and identify a criterion for a non-equilibrium dynamical transition. As a representative example, we consider a power law reversal probability $p(t)\sim t^{-\alpha}$ and show that the system undergoes a transition at $\alpha=1$, separating a super-diffusive regime for $\alpha<1$ from ballistic regime for $\alpha \geq 1$. Using the results for velocity correlations and persistence statistics, together with finite time scaling of the Binder cumulant and displacement fluctuations, we characterize the transition and its properties in detail. We further argue that the transition is not limited to the power law form, but can also arise for several other time dependent reversal probabilities satisfying the same criterion. The transition persists in arbitrary spatial dimensions provided isotropy of the velocity space is preserved.

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

The paper claims a transition at α=1 separating super-diffusive from ballistic regimes in a time-dependent persistent random walk, but the regime assignment looks inconsistent with the velocity correlations implied by the survival probability.

read the letter

The main thing to know is that this work identifies a criterion for a dynamical transition in persistent random walks when the reversal probability varies with time. For the example p(t) ~ t^{-α} it places the change at α=1, with superdiffusion below and ballistic motion at and above that value, and it argues the same integral condition works for other time-dependent forms as well. The authors characterize the transition with velocity correlations, persistence statistics, Binder cumulant scaling, and displacement fluctuations, and they note it survives in any dimension under velocity isotropy. That package of tools is a reasonable way to study the problem and could be relevant for active-matter or biological-transport models where persistence changes over time. The generalization beyond power laws is the clearest new element here. The central soft spot is the mapping of regimes. When the reversal probability is p(t) ~ t^{-α}, the survival probability without reversal is a stretched exponential for α < 1. The velocity autocorrelation then decays fast enough that its time integral converges, which should produce normal diffusion rather than superdiffusion at long times. Ballistic behavior, with a nonzero long-time correlation, requires the integral of p(t) to converge and therefore α > 1. The paper's assignment therefore appears reversed or miscalculated, and this undercuts the main claim. The derivations would need to be checked directly to see where the mismatch arises. This paper is for readers working on non-equilibrium random walks and their applications in physics or biology. Someone already thinking about time-varying persistence might find the criterion useful if the transport exponents are corrected. I would send it for peer review so the derivations and regime identification can be examined in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript examines persistent random walks with time-dependent velocity reversal probabilities, using the power-law form p(t)∼t^{-α} as a representative example. It claims to identify a non-equilibrium dynamical transition at α=1 that separates a super-diffusive regime for α<1 from a ballistic regime for α≥1. The transition is characterized via velocity correlations, persistence statistics, finite-time scaling of the Binder cumulant, and displacement fluctuations. The authors further argue that the same integral criterion for the transition applies to other reversal probability forms and persists in arbitrary dimensions when velocity-space isotropy is preserved.

Significance. If the central claim were correct, the work would supply a general criterion for the onset of ballistic transport in persistent random walks with decaying reversal rates, with possible relevance to non-equilibrium transport problems. The combination of correlation functions with Binder-cumulant scaling provides a concrete route to locating the transition.

major comments (2)

[Abstract and §3 (velocity correlations)] Abstract and the section deriving the transport regimes: the assignment of super-diffusion to α<1 contradicts the standard evaluation of the long-time diffusion coefficient from the velocity autocorrelation. With p(t)∼t^{-α} and α<1 the survival probability decays as a stretched exponential, so that C(t) is integrable and the mean-square displacement is asymptotically linear in t (normal diffusion), not super-diffusive. The claimed transition at α=1 therefore rests on an incorrect identification of the resulting transport exponent.
[§5 (finite-time scaling)] The finite-time scaling analysis of the Binder cumulant and displacement fluctuations (presumably §5) presupposes the super-diffusive versus ballistic regimes derived earlier; once the transport exponents are corrected, the scaling forms and the location of the apparent transition must be re-derived.

minor comments (2)

[§2 (model definition)] Define the reversal probability p(t) and its relation to the persistence-time distribution explicitly at first use and maintain consistent notation thereafter.
[Figures 3–5] Label the axes and state the scaling collapse explicitly in all figures that display Binder cumulants or displacement fluctuations so that the claimed transition is immediately visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing detailed comments. We address each major comment point by point below. We agree that the transport regimes require correction and will revise the manuscript to reflect the proper asymptotic behaviors while preserving the core criterion for the transition.

read point-by-point responses

Referee: [Abstract and §3 (velocity correlations)] Abstract and the section deriving the transport regimes: the assignment of super-diffusion to α<1 contradicts the standard evaluation of the long-time diffusion coefficient from the velocity autocorrelation. With p(t)∼t^{-α} and α<1 the survival probability decays as a stretched exponential, so that C(t) is integrable and the mean-square displacement is asymptotically linear in t (normal diffusion), not super-diffusive. The claimed transition at α=1 therefore rests on an incorrect identification of the resulting transport exponent.

Authors: We thank the referee for this important observation. After verifying the velocity autocorrelation, we concur that for α < 1, the survival probability S(t) decays as a stretched exponential exp(−const × t^{1−α}), making C(t) integrable. Thus, the diffusion coefficient is finite, and the mean-square displacement grows linearly with time at long times, indicating normal diffusion. We will update the abstract, §3, and related sections to correctly classify the regime for α < 1 as normal diffusion. The transition at α = 1, where the survival probability transitions to a power-law decay leading to non-integrable correlations and ballistic behavior for α > 1, remains valid. This revision strengthens the manuscript by aligning with standard transport theory. revision: yes
Referee: [§5 (finite-time scaling)] The finite-time scaling analysis of the Binder cumulant and displacement fluctuations (presumably §5) presupposes the super-diffusive versus ballistic regimes derived earlier; once the transport exponents are corrected, the scaling forms and the location of the apparent transition must be re-derived.

Authors: We acknowledge that the scaling analysis in §5 was predicated on the original regime assignments. With the corrected regimes (normal diffusion for α < 1 and ballistic for α ≥ 1), we will re-derive the finite-time scaling forms for the Binder cumulant and displacement fluctuations. The location of the transition at α = 1 is determined by the integrability of the velocity autocorrelation, which is independent of the misidentified super-diffusive label. We will provide updated scaling functions and confirm that the Binder cumulant analysis still locates the transition correctly at α = 1. These changes will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from explicit reversal probability to velocity correlations without self-referential reduction

full rationale

The paper starts from an externally specified reversal probability p(t)∼t^{-α} (or analogous integral criterion) and computes velocity autocorrelation C(t) and persistence statistics directly from the survival probability and reversal process. The claimed transition location α=1 and the super-diffusive vs. ballistic regimes are obtained by examining the long-time behavior of the mean-squared displacement and the integral of C(t), using finite-time scaling of the Binder cumulant as an independent diagnostic. No parameter is fitted to the target transport exponent, no uniqueness theorem is imported from prior self-work, and the central mapping does not reduce to a renaming or tautological re-expression of the input p(t). The derivation remains self-contained against the stated assumptions of velocity-space isotropy.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available, so ledger is minimal and based on stated modeling assumptions.

axioms (1)

domain assumption Persistent random walk is defined by a time-dependent velocity reversal probability while preserving isotropy in velocity space.
This is the core modeling choice invoked to derive the transition.

pith-pipeline@v0.9.0 · 5670 in / 1089 out tokens · 38846 ms · 2026-05-20T03:25:38.931711+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 washburn_uniqueness_aczel unclear

?

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

when N(∞) remains finite ... ⟨x²(t)⟩∼t²; when N(t) grows ... ⟨x²(t)⟩∼∑1/p(u) (Eqs. 8,10)
IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction unclear

?

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

transition at α=1 separating super-diffusive (α<1) from ballistic (α≥1)

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

Works this paper leans on

32 extracted references · 32 canonical work pages

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ln t (Fig. 3), in excellent agreement with the analytical prediction. In addition, the displacement distribution at criticality becomes exceptionally broad (see Fig. S3 in [31]), directly reﬂecting the strong ﬂuctuations associated with the transition. The logarithmic scaling variable implies an exponen- tially large characteristic timescale t∗ (α ) ∼ e 1...

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E. Barkai and Y. Cheng, Aging continuous time random walks, J. Chem. Phys. 118, 6167 (2003)

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[32]

Diﬀusive-to-Ballistic transition in a Persistent Random Walk

See Supplemental Material for detailed additional der iva- tions along with other related discussions. 1 Supplemental Material for “Diﬀusive-to-Ballistic transition in a Persistent Random Walk” Amit Pradhan 1, Reshmi Roy 2 and Purusattam Ray 3 1Department of Physics, University of Calcutta, 92 Acharya P rafulla Chandra Road, Kolkata 700009 2Department of ...

work page

[1] [1]

3), in excellent agreement with the analytical prediction

ln t (Fig. 3), in excellent agreement with the analytical prediction. In addition, the displacement distribution at criticality becomes exceptionally broad (see Fig. S3 in [31]), directly reﬂecting the strong ﬂuctuations associated with the transition. The logarithmic scaling variable implies an exponen- tially large characteristic timescale t∗ (α ) ∼ e 1...

work page

[2] [2]

Renshaw and R

E. Renshaw and R. HENDER, The correlated random walk, J. Appl. Prob. 18, 403 (1981)

work page 1981

[3] [3]

J. W. Hanneken, D. R. Franceschetti, Exact distribution function for discrete time correlated random walks in one dimension, J. Chem. Phys. 109, 6533–6539 (1998)

work page 1998

[4] [4]

G. H. Weiss, Some applications of persistent random walks and the telegrapher’s equation , Physica A 311, 381 (2002)

work page 2002

[5] [5]

Kac, A Stochastic Model Related to the Telegrapher’s Equation, Rocky Mountain Journal of Mathematics 4, 497–509 (1974)

M. Kac, A Stochastic Model Related to the Telegrapher’s Equation, Rocky Mountain Journal of Mathematics 4, 497–509 (1974)

work page 1974

[6] [6]

Rossetto, The one-dimensional asymmetric persistent random walk , J

V. Rossetto, The one-dimensional asymmetric persistent random walk , J. Stat. Mech. 043204 (2018)

work page 2018

[7] [7]

Sadjadi, M

Z. Sadjadi, M. Miri, Persistent random walk on a one- dimensional lattice with random asymmetric transmit- tances, Phys. Rev. E 78, 061114 (2008)

work page 2008

[8] [8]

Svenkeson, B

A. Svenkeson, B. J. West, Persistent random motion with maximally correlated ﬂuctuations , Phys. Rev. E 100, 022119 (2019)

work page 2019

[9] [9]

Chupeau, O

M. Chupeau, O. B´ enichou, R. Voituriez,Mean cover time of one-dimensional persistent random walks , Phys. Rev. E 89, 062129 (2014)

work page 2014

[10] [10]

A. L. P. Livorati, T. Kroetz,C. P. Dettmann, I. L. Caldas, and E. D. Leonel, Transition from normal to ballistic dif- fusion in a one-dimensional impact system , Phys. Rev. E. 97, 032205 (2018)

work page 2018

[11] [11]

C´ enac, A

P. C´ enac, A. Le Ny, B. de Loynes and Y. Oﬀret,Persistent random walks. I. Recurrence versus transience , J. Theor. Probab. 31(1), 232 (2018). 6

work page 2018

[12] [12]

C´ enac, A

P. C´ enac, A. Le Ny, B. de L. and Y. Oﬀret, Persistent Random Walks. II. Functional Scaling Limits , J. Theor. Prob., 32(2), 633 (2019)

work page 2019

[13] [13]

Masoliver, J

J. Masoliver, J. M. Porra, G. H. Weiss, Solutions of the telegrapher’s equation in the presence of traps, Phys. Rev. A 45, 2222 (1992)

work page 1992

[14] [14]

Sadjadi, M

Z. Sadjadi, M. R. Shaebani and H. Rieger, and L. Santen, Persistent random walk approach to anomalous trans- port of self-propelled particles , Phys. Rev. E 91, 062715 (2015)

work page 2015

[15] [15]

H. C. Berg, Random Walks in Biology , Princeton Uni- versity Press, 1993

work page 1993

[16] [16]

H. C. Berg, D. A. Brown, Chemotaxis in Escherichia coli analysed by Three-dimensional Tracking , Nature 239, 500–504 (1972)

work page 1972

[17] [17]

Tailleur, M

J. Tailleur, M. E. Cates, Statistical Mechanics of Inter- acting Run-and-Tumble Bacteria , Phys. Rev. Lett. 100, 218103 (2008)

work page 2008

[18] [18]

B´ enichou, C

O. B´ enichou, C. Loverdo, M. Moreau, and R. Voituriez, Intermittent search strategies , Rev. Mod. Phys. 83, 81 (2011)

work page 2011

[19] [19]

B´ enichou and R

O. B´ enichou and R. Voituriez, From ﬁrst-passage times of random walks in conﬁnement to geometry-controlled kinetics, Phys. Rep. 539, 225 (2014)

work page 2014

[20] [20]

M. C. Marchetti et al., Hydrodynamics of soft active mat- ter, Rev. Mod. Phys. 85, 1143 (2013)

work page 2013

[21] [21]

Bechinger et al., Active particles in complex and crowded environments , Rev

C. Bechinger et al., Active particles in complex and crowded environments , Rev. Mod. Phys. 88, 045006 (2016)

work page 2016

[22] [22]

Elgeti, R

J. Elgeti, R. Winkler, and G. Gompper, Physics of mi- croswimmers—single particle motion and collective be- havior: a review , Rep. Prog. Phys. 78, 056601 (2015)

work page 2015

[23] [23]

A. P. Solon et al., Pressure and Phase Equilibria in In- teracting Active Brownian Spheres, Phys. Rev. Lett. 114, 198301 (2015)

work page 2015

[24] [24]

Zaburdaev, S

V. Zaburdaev, S. Denisov, and J. Klafter, L´ evy walks, Rev. Mod. Phys. 87, 483 (2015)

work page 2015

[25] [25]

Metzler and J

R. Metzler and J. Klafter, The random walk’s guide to anomalous diﬀusion: a fractional dynamics approach , Phys. Rep. 339, 1 (2000)

work page 2000

[26] [26]

Metzler et al., Anomalous diﬀusion models and their properties: non-stationarity, non-ergodicity, and agein g at the centenary of single particle tracking , Phys

R. Metzler et al., Anomalous diﬀusion models and their properties: non-stationarity, non-ergodicity, and agein g at the centenary of single particle tracking , Phys. Chem. Chem. Phys. 16, 24128 (2014)

work page 2014

[27] [27]

Barkai, Y

E. Barkai, Y. Garini, and R. Metzler, Strange kinetics of single molecules in living cells , Phys. Today 65, 29 (2012)

work page 2012

[28] [28]

Fedotov, Subdiﬀusion, chemotaxis, and anomalous ag- gregation , Phys

S. Fedotov, Subdiﬀusion, chemotaxis, and anomalous ag- gregation , Phys. Rev. E 83, 021110 (2011)

work page 2011

[29] [29]

Barkai and Y

E. Barkai and Y. Cheng, Aging continuous time random walks, J. Chem. Phys. 118, 6167 (2003)

work page 2003

[30] [30]

J. H. Jeon and R. Metzler, Fractional Brownian motion and motion governed by the fractional Langevin equation in conﬁned geometries , Phys. Rev. E 81, 021103 (2010)

work page 2010

[31] [31]

J. H. P. Schulz, E. Barkai, R. Metzler, Aging Eﬀects and Population Splitting in Single-Particle Trajectory Aver- ages, Phys. Rev. Lett. 110, 020602 (2013)

work page 2013

[32] [32]

Diﬀusive-to-Ballistic transition in a Persistent Random Walk

See Supplemental Material for detailed additional der iva- tions along with other related discussions. 1 Supplemental Material for “Diﬀusive-to-Ballistic transition in a Persistent Random Walk” Amit Pradhan 1, Reshmi Roy 2 and Purusattam Ray 3 1Department of Physics, University of Calcutta, 92 Acharya P rafulla Chandra Road, Kolkata 700009 2Department of ...

work page