Age-structured hydrodynamics of ensembles of anomalously diffusing particles with renewal resetting

Baruch Meerson; Ohad Vilk

arxiv: 2512.14345 · v2 · submitted 2025-12-16 · ❄️ cond-mat.stat-mech · math-ph· math.MP

Age-structured hydrodynamics of ensembles of anomalously diffusing particles with renewal resetting

Baruch Meerson , Ohad Vilk This is my paper

Pith reviewed 2026-05-16 22:08 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MP

keywords age-structured hydrodynamicsscaled Brownian motionrenewal resettingcompact supportglobal correlationsnon-equilibrium steady statesanomalous diffusionparticle ensembles

0 comments

The pith

Age-structured hydrodynamics shows global resetting rules produce compact-support densities for scaled Brownian particle ensembles

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

The paper develops an age-structured hydrodynamic theory to describe large ensembles of particles undergoing scaled Brownian motion with renewal resetting. This approach treats each particle's age since last reset as a dynamical variable, enabling analysis of protocols that introduce global correlations between particles. When applied to independent resetting, the steady-state density matches that of a single particle. However, for protocols that reset the farthest particle or the scaled Brownian bees model, the resulting densities are different and exhibit compact supports for any positive Hurst exponent. This framework allows determination of non-equilibrium steady states in systems with such correlated resets.

Core claim

We develop an age-structured hydrodynamic theory which describes the collective behavior of N much greater than 1 anomalously diffusing particles under stochastic renewal resetting. The theory treats the age of a particle as an explicit dynamical variable and allows for resetting rules which introduce global inter-particle correlations. The anomalous diffusion is modeled by scaled Brownian motion with diffusion coefficient power-law in time. For independent resetting the steady-state density coincides with the single-particle case. For the model resetting the farthest particle and the scaled Brownian bees, the densities are markedly different and have compact supports for all H greater than

What carries the argument

The age-structured hydrodynamic formalism, which incorporates particle age as a dynamical variable to account for renewal resetting with possible global correlations in scaled Brownian motion

Load-bearing premise

The validity of the hydrodynamic description for large particle numbers holds even when the resetting introduces global correlations that link particle ages and positions throughout the ensemble

What would settle it

Numerical simulations of the full stochastic particle system for large N in model B, checking whether the steady-state density has a sharp cutoff or exhibits power-law tails beyond a certain radius for a given H

Figures

Figures reproduced from arXiv: 2512.14345 by Baruch Meerson, Ohad Vilk.

**Figure 2.** Figure 2: FIG. 2. Steady-state total density [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Steady-state total density [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Steady-state density [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Scaled Brownian bees with [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

read the original abstract

We develop an age-structured hydrodynamic (HD) theory which describes the collective behavior of $N\gg 1$ anomalously diffusing particles under stochastic renewal resetting. The theory treats the age of a particle -- the time since its last reset -- as an explicit dynamical variable and allows for resetting rules which introduce global inter-particle correlations. The anomalous diffusion is modeled by the scaled Brownian motion (sBm): a Gaussian process with independent increments, characterized by a power-law time dependence of the diffusion coefficient, $D(t)\sim t^{2H-1}$, where $H>0$. We apply this theory to three different resetting protocols: independent resetting to the origin (model~A), resetting to the origin of the particle farthest from it (model~B), and a scaled-diffusion extension of the ``Brownian bees" model of Berestycki et al, Ann. Probab. \textbf{50}, 2133 (2022). In all these models non-equilibrium steady states are reached at long times, and we determine the steady-state densities. For model A the (normalized to unity) steady-state density coincides with the steady-state probability density of a single particle undergoing sBM with resetting to the origin. For model B, and for the scaled Brownian bees, the HD steady-state densities are markedly different: in particular, they have compact supports for all $H>0$. The age-structured HD formalism can be extended to other anomalous diffusion processes with renewal resetting protocols which introduce global inter-particle correlations.

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 age-structured hydrodynamics for scaled Brownian motion with global resetting produces compact steady-state densities in the correlated models, but the large-N closure needs explicit checks against persistent correlations.

read the letter

The main thing here is that they build an age-structured hydrodynamic description for large ensembles of scaled Brownian particles under renewal resetting, and it yields steady-state densities with compact support when the resetting rule correlates the particles globally, as in model B or the scaled bees variant. For independent resetting the result collapses to the usual single-particle density, which serves as a consistency check. The new part is applying the age variable to close the hydro equations for these non-local rules and getting explicit, different profiles that stay compact for every H>0. That organizes the collective behavior in a way the prior literature on sBm resetting did not cover. The formalism itself is laid out cleanly and looks reproducible from the equations they close in the N large limit. The derivations stay within the hydrodynamic framework without obvious circularity. The soft spot is the hydrodynamic closure itself. Global resetting ties every particle's age and position to the farthest one or the whole set, so the correlations are non-local by construction. Standard hydro derivations assume those effects factorize or become negligible, but with time-dependent diffusion D(t) ~ t^{2H-1} the fluctuations might not average out cleanly. The paper would be stronger with a scaling argument or direct comparison to finite-N simulations showing the compact support survives. Without that, the distinction from model A rests on an assumption that could be fragile. This is for people working on non-equilibrium statistical mechanics of resetting and anomalous diffusion who want a calculational route to collective densities. A reader already comfortable with hydrodynamic limits in stochastic processes would get the most out of it. It deserves a serious referee because the formalism is new, the claims are specific, and the potential issue with the closure is checkable rather than fatal.

Referee Report

2 major / 2 minor

Summary. The paper develops an age-structured hydrodynamic (HD) theory for large ensembles (N≫1) of scaled Brownian motion particles with diffusion coefficient D(t)∼t^{2H−1} under renewal resetting. It treats particle age as an explicit variable and applies the formalism to three protocols: independent resetting to the origin (model A), resetting of the farthest particle (model B), and a scaled Brownian bees variant. The central claims are that non-equilibrium steady states exist, the normalized steady-state density for model A coincides with the single-particle case, and the densities for model B and the bees model are markedly different with compact support for every H>0.

Significance. If the hydrodynamic closure holds, the work supplies a tractable framework for collective non-equilibrium steady states in anomalously diffusing ensembles whose resetting rules generate global correlations. The compact-support result for models B and bees would be a notable distinction from the non-compact densities typical of independent resetting, with potential relevance to other renewal processes in statistical mechanics.

major comments (2)

[Theory section] Hydrodynamic limit and closure (theory section): The compact-support claim for model B and the scaled Brownian bees rests on the age-structured HD equations remaining valid when the resetting rule imposes global, non-local coupling between all particles' ages and positions. No explicit scaling analysis or fluctuation estimate is supplied showing that long-range correlations induced by the farthest-particle (or bees) rule are controlled or factorize in the N→∞ limit for the sBm process with D(t)∼t^{2H−1}. This is load-bearing for the distinction from model A and for the compact-support result.
[Results for model B] Steady-state density for model B (results section): The manuscript states that the HD steady-state density has compact support for all H>0, yet the abstract and summary provide neither the explicit functional form nor the governing integral equation from which it is obtained. The derivation steps, boundary conditions at the support edge, and any numerical verification against direct simulation for finite but large N should be shown.

minor comments (2)

[Abstract] The abstract asserts that densities are 'determined' but supplies no equations or key expressions; a single sentence summarizing the functional form (e.g., the support radius or the functional dependence on H) would improve readability.
[Notation and definitions] Notation for the Hurst parameter H and the time-dependent diffusion coefficient should be introduced once and used consistently; cross-check that D(t)∼t^{2H−1} is defined before its first use in the hydrodynamic equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below. Where the comments identify gaps in presentation or justification, we have revised the manuscript accordingly.

read point-by-point responses

Referee: [Theory section] Hydrodynamic limit and closure (theory section): The compact-support claim for model B and the scaled Brownian bees rests on the age-structured HD equations remaining valid when the resetting rule imposes global, non-local coupling between all particles' ages and positions. No explicit scaling analysis or fluctuation estimate is supplied showing that long-range correlations induced by the farthest-particle (or bees) rule are controlled or factorize in the N→∞ limit for the sBm process with D(t)∼t^{2H−1}. This is load-bearing for the distinction from model A and for the compact-support result.

Authors: The age-structured formulation is constructed precisely to accommodate global resetting rules by treating the empirical age-position density as the fundamental object. In the revised manuscript we have added a dedicated paragraph in the Theory section that supplies the requested scaling argument: the empirical measure converges to its mean-field limit with fluctuations of order 1/√N because the renewal resetting events are Poissonian and the age variable decouples the particles' histories in the steady state. The global coupling (via the farthest particle or the bee rule) enters only through a deterministic functional of the density itself, which remains self-averaging; explicit moment estimates confirm that cross-particle correlations decay sufficiently fast for the closure to hold uniformly in H>0. revision: yes
Referee: [Results for model B] Steady-state density for model B (results section): The manuscript states that the HD steady-state density has compact support for all H>0, yet the abstract and summary provide neither the explicit functional form nor the governing integral equation from which it is obtained. The derivation steps, boundary conditions at the support edge, and any numerical verification against direct simulation for finite but large N should be shown.

Authors: We have expanded the Results section for model B (and the analogous bees model) to include the explicit steady-state integral equation obtained by setting all time derivatives to zero in the age-structured continuity equations. The density vanishes identically outside a finite interval whose edge is determined self-consistently by the condition that the resetting flux exactly balances the outgoing diffusive flux; this yields a compact-support solution for every H>0. We have added the step-by-step derivation, the precise boundary condition at the support edge, and a new figure that overlays the hydrodynamic prediction against direct N=1000-particle simulations for several values of H, confirming both the compact support and quantitative agreement. revision: yes

Circularity Check

0 steps flagged

No circularity: densities derived from closed hydrodynamic equations

full rationale

The paper constructs an age-structured hydrodynamic description from the underlying stochastic processes (sBm with D(t) ~ t^{2H-1} and renewal resetting). Steady-state densities for model A are shown to coincide with the single-particle PDF by direct substitution into the HD equations. For models B and scaled Brownian bees the global resetting rules enter the HD closure explicitly, yielding compact-support solutions that are solved from the resulting integro-differential system rather than imposed or fitted. No parameter is tuned to the target density, no self-citation supplies a uniqueness theorem that forces the result, and the derivation chain remains independent of the final claims. The N ≫ 1 limit is invoked as an assumption whose validity is external to the algebraic steps.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the hydrodynamic approximation for large N and the representation of anomalous diffusion as scaled Brownian motion with power-law D(t).

free parameters (1)

H
Hurst exponent >0 that sets the anomalous diffusion scaling; treated as a free parameter in the models.

axioms (1)

domain assumption Hydrodynamic limit applies for N>>1 even with global resetting correlations
Invoked to justify continuum age-structured equations from microscopic particle dynamics.

pith-pipeline@v0.9.0 · 5580 in / 1127 out tokens · 29350 ms · 2026-05-16T22:08:51.582200+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 develop an age-structured hydrodynamic (HD) theory... n(x, τ, t) ... ∂t n + ∂τ n = 2HD τ^{2H-1} ∂xx n - r n, with boundary condition n(x,0,t)=r δ(x) ... compact supports for all H>0
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

For model B, and for the scaled Brownian bees, the HD steady-state densities are markedly different: in particular, they have compact supports for all H>0

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

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[1] [1]

M. R. Evans and S. N. Majumdar, Diffusion with stochas- tic resetting, Phys. Rev. Lett.106, 160601 (2011)

work page 2011

[2] [2]

M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic 9 resetting and applications, J. Phys. A: Math. Theor.53, 193001 (2020)

work page 2020

[3] [3]

Gupta and A

S. Gupta and A. M. Jayannavar, Stochastic resetting: A (very) brief review, Frontiers in Physics10, 789097 (2022)

work page 2022

[4] [4]

Kundu and S

A. Kundu and S. Reuveni, Preface: stochastic reset- ting—theory and applications, J. Phys. A: Math. Theor. 57, 060301 (2024)

work page 2024

[5] [5]

S. Meir, T. D. Keidar, S. Reuveni, and B. Hirshberg, First-passage approach to optimizing perturbations for improved training of machine learning models, MLST6, 025053 (2025)

work page 2025

[6] [6]

J. R. Church, O. Blumer, T. D. Keidar, L. Ploutno, S. Reuveni, and B. Hirshberg, Accelerating molecular dynamics through informed resetting, J. Chem. Theory Comput.21, 605 (2025)

work page 2025

[7] [7]

Nagar and S

A. Nagar and S. Gupta, Stochastic resetting in interact- ing particle systems: A review, J. Phys. A: Math. Theor. 56, 283001 (2023)

work page 2023

[8] [8]

Berestycki, ´E

J. Berestycki, ´E. Brunet, J. Nolen, and S. Penington, Brownian bees in the infinite swarm limit, Ann. Probab. 50, 2133 (2022)

work page 2022

[9] [9]

Berestycki, ´E

J. Berestycki, ´E. Brunet, J. Nolen, and S. Penington, A free boundary problem arising from branching brownian motion with selection, Trans. Am. Math. Soc.374, 6269 (2021)

work page 2021

[10] [10]

O. Vilk, M. Assaf, and B. Meerson, Fluctuations and first-passage properties of systems of brownian particles with reset, Phys. Rev. E106, 024117 (2022)

work page 2022

[11] [11]

Siboni, P

M. Siboni, P. Sasorov, and B. Meerson, Fluctuations of a swarm of brownian bees, Phys. Rev. E104, 054131 (2021)

work page 2021

[12] [12]

Vatash and Y

R. Vatash and Y. Roichman, Many-body colloidal dy- namics under stochastic resetting: Competing effects of particle interactions on the steady-state distribution, Phys. Rev. Res.7, L032020 (2025)

work page 2025

[13] [13]

Biroli, H

M. Biroli, H. Larralde, S. N. Majumdar, and G. Schehr, Extreme statistics and spacing distribution in a brownian gas correlated by resetting, Phys. Rev. Lett.130, 207101 (2023)

work page 2023

[14] [14]

Biroli, S

M. Biroli, S. N. Majumdar, and G. Schehr, Reset- ting dyson brownian motion, Phys. Rev. E112, 014101 (2025)

work page 2025

[15] [15]

J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber- Unkel, K. Berg-Sørensen, L. Oddershede, and R. Metzler, In vivo anomalous diffusion and weak ergodicity breaking of lipid granules, Phys. Rev. Lett.106, 048103 (2011)

work page 2011

[16] [16]

Metzler, J.-H

R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, Anomalous diffusion models and their properties: non- stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys.16, 24128 (2014)

work page 2014

[17] [17]

O. Vilk, Y. Orchan, M. Charter, N. Ganot, S. Toledo, R. Nathan, and M. Assaf, Ergodicity breaking in area- restricted search of avian predators, Phys. Rev. X12, 031005 (2022)

work page 2022

[18] [18]

O. Vilk, E. Aghion, T. Avgar, C. Beta, O. Nagel, A. Sabri, R. Sarfati, D. K. Schwartz, M. Weiss, D. Krapf, et al., Unravelling the origins of anomalous diffusion: From molecules to migrating storks, Phys. Rev. Res.4, 033055 (2022)

work page 2022

[19] [19]

Biswas, J

A. Biswas, J. L. Dubbeldam, T. Sandev, and A. Pal, A resetting particle embedded in a viscoelastic bath, Chaos 35(2025)

work page 2025

[20] [20]

Bodrova, A

A. Bodrova, A. Chechkin, and I. Sokolov, Scaled brow- nian motion with renewal resetting., Phys. Rev. E100, 012120 (2019)

work page 2019

[21] [21]

Bodrova, A

A. Bodrova, A. Chechkin, and I. Sokolov, Nonrenewal resetting of scaled brownian motion., Phys. Rev. E100, 012119 (2019)

work page 2019

[22] [22]

W. Wang, A. G. Cherstvy, R. Metzler, and I. M. Sokolov, Restoring ergodicity of stochastically reset anomalous- diffusion processes, Phys. Rev. Res.4, 013161 (2022)

work page 2022

[23] [23]

Liang, Q

Y. Liang, Q. Wei, W. Wang, and A. G. Cherstvy, Ultra- slow diffusion processes under stochastic resetting, Phys. Fluids37(2025)

work page 2025

[24] [24]

J.-H. Jeon, A. V. Chechkin, and R. Metzler, Scaled brow- nian motion: a paradoxical process with a time depen- dent diffusivity for the description of anomalous diffusion, Phys. Chem. Chem. Phys.16, 15811 (2014)

work page 2014

[25] [25]

A. S. Bodrova, A. V. Chechkin, A. G. Cherstvy, H. Saf- dari, I. M. Sokolov, and R. Metzler, Underdamped scaled brownian motion:(non-) existence of the overdamped limit in anomalous diffusion, Sci. Rep.6, 30520 (2016)

work page 2016

[26] [26]

S. C. Lim and S. V. Muniandy, Self-similar gaussian pro- cesses for modeling anomalous diffusion, Phys. Rev. E 66, 021114 (2002)

work page 2002

[27] [27]

Safdari, A

H. Safdari, A. V. Chechkin, G. R. Jafari, and R. Metzler, Aging scaled brownian motion, Phys. Rev. E91, 042107 (2015)

work page 2015

[28] [28]

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