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arxiv: 2606.00149 · v2 · pith:NELG7NCDnew · submitted 2026-05-29 · 🌊 nlin.AO · cond-mat.stat-mech· physics.data-an

Decomposition of Anomalous Diffusion in two-state random walks

Abhijit Bera , Kevin. E. Bassler This is my paper

Pith reviewed 2026-06-28 20:25 UTC · model grok-4.3

classification 🌊 nlin.AO cond-mat.stat-mechphysics.data-an
keywords anomalous diffusiontwo-state random walkLevy walkcontinuous-time random walkJoseph effectNoah effectMoses effectstochastic switching
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The pith

Two-state random walks switching between Lévy walks and rests produce Joseph, Noah, and Moses effects together through stochastic switching.

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

The paper studies two-state random walks that alternate between a continuous-time random walk rest phase and a Lévy walk motion phase, with power-law distributed times in each phase. It applies anomalous diffusion decomposition to demonstrate that these walks generically combine correlation effects, heavy-tailed increments, and aging effects. Classical Lévy walks alone produce only correlations, yet the Noah and Moses effects appear in the two-state case purely because of the random switching to the rest phase. This result indicates that state coupling can change which mechanisms generate anomalous diffusion in systems with intermittent dynamics.

Core claim

Two-state random walks that switch between a continuous-time random walk rest state and a Lévy walk motion state with power-law sojourn times exhibit a generic coexistence of Joseph (correlation), Noah (heavy-tailed increments), and Moses (aging) effects. Although classical Lévy walks possess only the Joseph effect, both Noah and Moses effects emerge in TSRWs solely due to stochastic switching with the CTRW phase. Coupling between dynamical states can therefore fundamentally reshape the mechanisms driving anomalous diffusion.

What carries the argument

Anomalous diffusion decomposition that isolates the separate contributions of Joseph, Noah, and Moses effects.

If this is right

  • TSRWs display all three effects at once for generic power-law exponents.
  • The Noah and Moses effects appear only when the CTRW rest phase is added via switching.
  • Transport in heterogeneous environments with intermittent switching follows this combined mechanism rather than pure Lévy walk scaling.
  • The minimal two-state setup captures how state coupling alters anomalous diffusion origins.

Where Pith is reading between the lines

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

  • The same decomposition could separate effects in multi-state systems with more than two phases.
  • Measurements of separate effect strengths in single-particle tracking data could identify the presence of hidden switching.
  • Varying the power-law exponents independently in each phase would test which exponent controls which effect.

Load-bearing premise

The decomposition cleanly isolates the three effects without extra coupling introduced by the switching rule between the two states.

What would settle it

A simulation or exact calculation of the decomposed moments for a TSRW that shows nonzero cross terms between the Joseph, Noah, and Moses contributions.

Figures

Figures reproduced from arXiv: 2606.00149 by Abhijit Bera, Kevin. E. Bassler.

Figure 1
Figure 1. Figure 1: a and fig. 1.b show the phase portrait of TSRW [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The phase portrait of (a) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Phase portraits of (a) the Noah exponent [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Overall phase diagram of the Two-State Random [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
read the original abstract

Two-state stochastic models, where motion alternates between distinct dynamical modes, are widely observed in complex systems. Here we study the Two-State Random Walk (TSRW), which switches between a continuous-time random walk (CTRW) rest state and a standard L'evy walk (LW) motion state, each with power-law distributed sojourn times. Using anomalous diffusion decomposition, we show that TSRWs exhibit a generic coexistence of Joseph (correlation), Noah (heavy-tailed increments), and Moses (aging) effects. Strikingly, although classical L'evy walks alone possess only the Joseph effect, both Noah and Moses effects emerge in TSRWs solely due to stochastic switching with the CTRW phase. Our results demonstrate that coupling between dynamical states can fundamentally reshape the mechanisms driving anomalous diffusion, offering a minimal yet powerful framework for transport in heterogeneous and intermittently switching 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

1 major / 0 minor

Summary. The manuscript examines two-state random walks (TSRWs) that alternate between a continuous-time random walk (CTRW) rest state and a Lévy walk (LW) motion state with power-law sojourn times in each. Using anomalous diffusion decomposition, the authors claim that TSRWs generically exhibit coexistence of Joseph (correlation), Noah (heavy-tailed increments), and Moses (aging) effects, with the Noah and Moses effects arising solely from the stochastic switching to the CTRW phase (in contrast to classical Lévy walks, which exhibit only the Joseph effect).

Significance. If the decomposition cleanly isolates the three effects, the result would illustrate how coupling between dynamical states can introduce new mechanisms for anomalous diffusion beyond those present in the individual components, offering a minimal model relevant to transport in heterogeneous or intermittently switching environments.

major comments (1)
  1. [Anomalous diffusion decomposition section] Anomalous diffusion decomposition section: the central claim that Noah and Moses effects emerge solely due to stochastic switching requires that the decomposition attributes these effects without residual coupling or misattribution arising from the alternating sojourn times themselves. The manuscript must include explicit validation (e.g., recovery of only the Joseph effect when the CTRW phase is removed) to confirm the decomposition separates the contributions as assumed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on the anomalous diffusion decomposition. We address the request for explicit validation below.

read point-by-point responses

  1. Referee: [Anomalous diffusion decomposition section] Anomalous diffusion decomposition section: the central claim that Noah and Moses effects emerge solely due to stochastic switching requires that the decomposition attributes these effects without residual coupling or misattribution arising from the alternating sojourn times themselves. The manuscript must include explicit validation (e.g., recovery of only the Joseph effect when the CTRW phase is removed) to confirm the decomposition separates the contributions as assumed.

    Authors: We agree that an explicit check of the decomposition in the pure Lévy-walk limit would strengthen the manuscript and directly confirm that the Noah and Moses effects are generated by the stochastic switching. Although the decomposition formulas are constructed to isolate the three contributions independently, we will add a new subsection (or supplementary figure) that analytically or numerically recovers only the Joseph effect when the CTRW rest state is removed (i.e., by taking the limit of vanishing CTRW sojourn probability or infinite mean rest time). This will demonstrate that the additional effects disappear without the alternating CTRW phase, thereby ruling out residual coupling from the sojourn-time alternation itself. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and provided text describe application of an anomalous diffusion decomposition to a two-state random walk model to attribute Joseph, Noah, and Moses effects to stochastic switching between CTRW and Levy walk phases. No equations, fitted parameters, self-citations, or uniqueness theorems are quoted that reduce the central claim to a definition or input by construction. The derivation appears self-contained against the model definition and external decomposition method, with no load-bearing self-referential steps evident.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted. The power-law sojourn times and the applicability of the anomalous diffusion decomposition are implicit but unexamined.

pith-pipeline@v0.9.1-grok · 5680 in / 1056 out tokens · 18430 ms · 2026-06-28T20:25:26.737964+00:00 · methodology

discussion (0)

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Reference graph

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