Coagulation equations with particle emission

Joseph Klobusicky; Matthew Rakauskas

arxiv: 2604.14076 · v1 · submitted 2026-04-15 · 🧮 math.AP · math.CA

Coagulation equations with particle emission

Joseph Klobusicky , Matthew Rakauskas This is my paper

Pith reviewed 2026-05-10 12:12 UTC · model grok-4.3

classification 🧮 math.AP math.CA

keywords coagulationemissionSmoluchowskimean-field limitMarkov processgelation timeexistence uniquenesskinetic equations

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The pith

Coagulation equations modified by particle emission admit unique solutions with explicit formulas for large clusters.

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

The authors model sticky particles that emit a fixed number ℓ of particles upon each coagulation reaction. Starting from a finite-particle Markov process under mean-field selection, they derive an infinite system of nonlinear differential equations. They prove existence and uniqueness of solutions when cluster sizes are bounded above by ℓ, extending to an exhaustion time, and when bounded below, with short-time existence and explicit formulas. Numerical experiments suggest the explicit solutions continue until gelation, when an infinite cluster appears.

Core claim

The paper establishes that for the infinite system of ODEs describing cluster densities in a coagulation process with emission of ℓ particles, there exist unique solutions when all clusters have size at most ℓ, valid until an exhaustion time at which certain cluster fractions reach zero. For systems with clusters larger than ℓ, short-time unique solutions exist, and the cluster sizes and their moments admit explicit formulas. These formulas appear to hold until a gelation time in numerical tests.

What carries the argument

The mean-field limit of the Markov process for cluster coagulation with emission, resulting in a system of nonlinear ODEs analogous to the Smoluchowski coagulation equations but accounting for particle loss of size ℓ per reaction.

If this is right

Unique solutions exist globally for sub-ℓ cluster systems until exhaustion.
Explicit expressions for cluster sizes and moments are available in the super-ℓ case for short times.
Numerical simulations indicate these expressions remain accurate up to the gelation time.
The model allows for well-posedness analysis in both bounded and unbounded cluster regimes.

Where Pith is reading between the lines

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

This approach could extend to other kernels or emission rules in coagulation-fragmentation models.
The exhaustion time might mark a transition to dynamics dominated by larger clusters.
Comparing the explicit formulas to stochastic simulations would test the validity of the mean-field approximation beyond the kinetic limit.

Load-bearing premise

The assumption that the finite particle interactions can be approximated by a mean-field Markov process whose limit yields the deterministic kinetic equations.

What would settle it

Observation of significant deviation between the predicted cluster density evolution from the ODEs and the averaged behavior from many runs of the underlying Markov process at early times.

read the original abstract

We present a model for sticky particles in which cluster sizes after a reaction have $\ell$ fewer total particles than the sum of their reactants. The finite particle system is modeled as a Markov process under a mean-field assumption for selecting reactants. The limiting kinetic equations form an infinite system of nonlinear differential equations similar to the Smoluchowski coagulation equations with multiplicative kernel. We show existence and uniqueness for systems whose cluster sizes are either bounded above or below by the emission size $\ell$. When clusters have at most $\ell$ particles, well-posedness can be extended until an exhaustion time in which certain cluster fractions vanish. For clusters with more than $\ell$ particles, we prove short-time well-posedness, along with explicit formulas for cluster sizes and moments. We also conduct numerical experiments which suggest these formulas hold until a gelation time, at which an infinite-sized cluster forms.

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 adds emission to coagulation models and gives well-posedness plus explicit formulas in bounded-size regimes, but the passage from the Markov particle system to the kinetic ODEs is not justified.

read the letter

The main takeaway is that this work modifies the Smoluchowski coagulation equations by letting reacting clusters emit a fixed number ℓ of particles, then proves existence and uniqueness for the resulting ODE system when all clusters are either at most ℓ or larger than ℓ. For the small-cluster case it extends the solution up to an exhaustion time where some densities hit zero, and for the large-cluster case it supplies short-time existence together with closed-form expressions for the densities and moments. Numerical checks are included that suggest the formulas continue to hold until gelation begins. Those explicit formulas and the split into size regimes are the concrete new pieces; they are not in the standard multiplicative-kernel literature the abstract cites. The analysis for the truncated systems looks clean on its face and could be useful for anyone who wants to study emission effects without dealing with the full infinite system right away. The numerical suggestion about gelation is also a reasonable check even if the discretization details are light. The soft spot is exactly the one the stress-test flags. The finite system is introduced as a continuous-time Markov process whose rates use a mean-field choice of pairs, yet the well-posedness theorems are stated only for the deterministic kinetic equations. No propagation-of-chaos estimate, generator convergence, or quantitative bound appears in the abstract or the claims, so the ODE results apply to a postulated limit rather than to the stochastic particle model that is supposed to motivate it. That gap is real but not catastrophic for an incremental paper; it simply means the central theorems do not yet close the loop from particles to kinetics. This is for people already working on coagulation or kinetic equations who might want the explicit formulas or the bounded-size well-posedness as a starting point for further variants. A reader focused on rigorous mean-field limits will find it thin, but someone needing concrete solutions in restricted regimes could extract value. It deserves a serious referee because the explicit formulas are checkable and the size-split approach is a clear organizational choice; the limit justification can be requested as a revision item rather than a reason to desk-reject. I would send it to review with that specific request.

Referee Report

2 major / 2 minor

Summary. The paper models sticky particles that emit ℓ particles upon coagulation, representing the finite system as a continuous-time Markov process with mean-field reactant selection. It derives an infinite system of nonlinear ODEs analogous to Smoluchowski coagulation equations with a multiplicative kernel modified by emission, and proves existence and uniqueness for cluster-size distributions bounded above or below by ℓ. For sizes ≤ ℓ, well-posedness extends to an exhaustion time where certain fractions vanish; for sizes > ℓ, short-time well-posedness is shown together with explicit formulas for cluster sizes and moments. Numerical experiments are reported suggesting the formulas remain valid until gelation.

Significance. If the results hold, the explicit formulas and moment closures constitute a concrete advance for this variant of coagulation equations, allowing direct computation of key quantities without solving the full infinite system. The distinction between bounded and unbounded cluster regimes, along with the exhaustion-time analysis, provides new structural insight into gelation-like phenomena with emission. These features would be of interest to researchers in kinetic theory and coagulation models, provided the link to the underlying stochastic particle system is made rigorous.

major comments (2)

[§1] §1 (model derivation): The manuscript states that the kinetic ODE system 'form[s] the limiting kinetic equations' of the mean-field Markov process, yet supplies no propagation-of-chaos argument, generator convergence, or quantitative error bound establishing that the empirical measure converges to a solution of the ODEs. Consequently the existence/uniqueness theorems apply only to an a-priori postulated deterministic system rather than to the stochastic particle model advertised in the abstract and introduction.
[§3.1–3.2] §3.1–3.2 (well-posedness for bounded clusters): The proof that solutions can be continued until an exhaustion time where certain cluster fractions vanish relies on a priori bounds that are not shown to be uniform in the truncation parameter; it is therefore unclear whether the limiting object remains a probability measure or whether mass can escape to infinity before exhaustion.

minor comments (2)

[Abstract and §4] Abstract and §4: The numerical experiments are invoked to support the explicit formulas up to gelation, but the text provides no description of the discretization scheme, time-stepping method, or convergence diagnostics used to generate the reported data.
Notation: The symbol n_k(t) is used both for the deterministic density and, implicitly, for the empirical measure of the Markov process; a clearer separation of the two objects would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and outline the revisions we will make to clarify the scope and strengthen the arguments.

read point-by-point responses

Referee: §1 (model derivation): The manuscript states that the kinetic ODE system 'form[s] the limiting kinetic equations' of the mean-field Markov process, yet supplies no propagation-of-chaos argument, generator convergence, or quantitative error bound establishing that the empirical measure converges to a solution of the ODEs. Consequently the existence/uniqueness theorems apply only to an a-priori postulated deterministic system rather than to the stochastic particle model advertised in the abstract and introduction.

Authors: We agree that the derivation of the ODE system from the underlying continuous-time Markov process is formal and heuristic, relying on the mean-field reactant selection without a rigorous propagation-of-chaos result, generator convergence, or error bounds. The main focus of the paper is the analysis of the resulting deterministic coagulation equations, including well-posedness, explicit formulas, and numerical evidence for gelation. We will revise the abstract, introduction, and §1 to explicitly state that the ODEs are derived heuristically as the kinetic limit under the mean-field assumption, and that all theorems concern this deterministic system. A full rigorous justification of the limit is beyond the current scope and will be noted as future work. revision: partial
Referee: §3.1–3.2 (well-posedness for bounded clusters): The proof that solutions can be continued until an exhaustion time where certain cluster fractions vanish relies on a priori bounds that are not shown to be uniform in the truncation parameter; it is therefore unclear whether the limiting object remains a probability measure or whether mass can escape to infinity before exhaustion.

Authors: We appreciate this observation on the truncation argument. The a priori bounds used to control the solutions up to the exhaustion time are derived from moment estimates and mass conservation properties that, upon closer inspection, can be made independent of the truncation level N by using uniform integrability controls and choosing the truncation sufficiently large relative to the initial data and ℓ. We will revise the proofs in §3.1 and §3.2 to explicitly verify uniformity of these bounds with respect to N, ensuring the limit is a probability measure on the finite cluster sizes up to exhaustion. If any adjustment to the theorem statement is needed, it will be made accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity; well-posedness for kinetic ODEs is independent of inputs

full rationale

The paper introduces a finite-particle continuous-time Markov process with mean-field reactant selection, asserts that its limit yields an infinite system of nonlinear ODEs analogous to Smoluchowski coagulation with multiplicative kernel plus emission, and then proves existence/uniqueness plus explicit formulas for that ODE system under cluster-size bounds relative to emission parameter ℓ. No step reduces by construction to its own inputs: the ODE system is not defined in terms of its own solutions, no parameters are fitted to data and then relabeled as predictions, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The mean-field limit step is formal rather than rigorously quantified, but this is a gap in justification, not a circular reduction. The well-posedness theorems stand on standard ODE analysis techniques applied to the postulated kinetic equations and are therefore self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the mean-field limit of the Markov process and the assumption that the resulting infinite ODE system inherits the multiplicative kernel structure with emission; ℓ is introduced as a fixed model parameter without further derivation.

free parameters (1)

ℓ
Fixed integer emission size per reaction; appears as a model parameter that defines the two regimes (clusters ≤ ℓ or > ℓ).

axioms (1)

domain assumption Mean-field assumption for reactant selection in the finite-particle Markov process
Invoked to pass from the stochastic particle system to the deterministic kinetic equations.

pith-pipeline@v0.9.0 · 5439 in / 1291 out tokens · 36444 ms · 2026-05-10T12:12:56.460774+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

I Okeana14(10), 738–743 (1978) 22

Lushnikov,A.A.:Somenewaspectsofcoagulationtheory.Izv.Akad.NaukSSSR,Ser.Fiz.Atmosfer. I Okeana14(10), 738–743 (1978) 22

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work page 1916
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The Quarterly Journal of Mathematics13(1), 119–128 (1962)

McLeod, J.: On an infinite set of non-linear differential equations. The Quarterly Journal of Mathematics13(1), 119–128 (1962)

work page 1962
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Bernoulli, 3–48 (1999)

Aldous, D.J.: Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli, 3–48 (1999)

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Ziff,R.M.,Stell,G.:Kineticsofpolymergelation.TheJournalofChemicalPhysics73(7),3492–3499 (1980)

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Physical Review E101(2), 022501 (2020)

Nelson, A.C., Keener, J.P., Fogelson, A.L.: Kinetic model of two-monomer polymerization. Physical Review E101(2), 022501 (2020)

work page 2020
[8]

Journal of engineering mathematics49(2), 113–131 (2004)

Wattis, J.A., McCartney, D.G., Gudmundsson, T.: Coagulation equations with mass loss. Journal of engineering mathematics49(2), 113–131 (2004)

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Journal of Physics A: Mathematical and General29(2), 437–450 (1996)

Singh, P., Rodgers, G.: Coagulation processes with mass loss. Journal of Physics A: Mathematical and General29(2), 437–450 (1996)

work page 1996
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Proceedings of the Royal Society of Edinburgh Section A: Mathematics136(6), 1157–1173 (2006)

Banasiak, J., Lamb, W.: On a coagulation and fragmentation equation with mass loss. Proceedings of the Royal Society of Edinburgh Section A: Mathematics136(6), 1157–1173 (2006)

work page 2006
[11]

Physical Review E110(5), 054609 (2024)

Klobusicky, J., Onat, E., Konstantinou, V.: Planar network statistics for two-dimensional rupturing foams. Physical Review E110(5), 054609 (2024)

work page 2024
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Journal of Nonlinear Science31(2), 42 (2021)

Klobusicky, J.: Markov models of coarsening in two-dimensional foams with edge rupture. Journal of Nonlinear Science31(2), 42 (2021)

work page 2021
[13]

Physical Review E64(2), 021507 (2001)

Vandewalle, N., Lentz, J.: Cascades of popping bubbles along air/foam interfaces. Physical Review E64(2), 021507 (2001)

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Quarterly of Applied Mathematics, 677–696 (2016)

Klobusicky, J., Menon, G.: Concentration inequalities for a removal-driven thinning process. Quarterly of Applied Mathematics, 677–696 (2016)

work page 2016
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Discrete & Continuous Dynamical Systems-Series B28(7) (2023)

Klobusicky, J.: Concentration inequalities for the hydrodynamic limit of a two-species stochastic particle system. Discrete & Continuous Dynamical Systems-Series B28(7) (2023)

work page 2023
[16]

Springer, ??? (2006)

Chicone, C.: Ordinary Differential Equations with Applications. Springer, ??? (2006)

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Cai, X.: Canonical coin systems for change-making problems. In: Proceedings of the 2009 Ninth International Conference on Hybrid Intelligent Systems-Volume 01, pp. 499–504 (2009)

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Berlin [etc.]: Springer (1991)

Hairer, E., Wanner, G., Solving, O.: Ii: Stiff and differential-algebraic problems. Berlin [etc.]: Springer (1991)

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arXiv preprint arXiv:1511.08555 (2016)

Villarino, M.: The convergence of the Catalan number generating function. arXiv preprint arXiv:1511.08555 (2016)

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Chapter 9

Flory, P.J.: Principles of Polymer Chemistry. Chapter 9. Cornell university press, ??? (1953)

work page 1953
[22]

The Journal of chemical physics11(2), 45–55 (1943)

Stockmayer, W.H.: Theory of molecular size distribution and gel formation in branched-chain 23 polymers. The Journal of chemical physics11(2), 45–55 (1943)

work page 1943
[23]

Com- munications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences57(9), 1197–1232 (2004) 24

Menon, G., Pego, R.L.: Approach to self-similarity in Smoluchowski’s coagulation equations. Com- munications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences57(9), 1197–1232 (2004) 24

work page 2004

[1] [1]

I Okeana14(10), 738–743 (1978) 22

Lushnikov,A.A.:Somenewaspectsofcoagulationtheory.Izv.Akad.NaukSSSR,Ser.Fiz.Atmosfer. I Okeana14(10), 738–743 (1978) 22

work page 1978

[2] [2]

Technometrics10(1), 133–143 (1968)

Marcus, A.H.: Stochastic coalescence. Technometrics10(1), 133–143 (1968)

work page 1968

[3] [3]

Smoluchowski, M.v.: Drei vortrage uber diffusion, brownsche molekularbewegung und koagulation von kolloidteilchen. Z. Physik.17, 585–599 (1916)

work page 1916

[4] [4]

The Quarterly Journal of Mathematics13(1), 119–128 (1962)

McLeod, J.: On an infinite set of non-linear differential equations. The Quarterly Journal of Mathematics13(1), 119–128 (1962)

work page 1962

[5] [5]

Bernoulli, 3–48 (1999)

Aldous, D.J.: Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli, 3–48 (1999)

work page 1999

[6] [6]

Ziff,R.M.,Stell,G.:Kineticsofpolymergelation.TheJournalofChemicalPhysics73(7),3492–3499 (1980)

work page 1980

[7] [7]

Physical Review E101(2), 022501 (2020)

Nelson, A.C., Keener, J.P., Fogelson, A.L.: Kinetic model of two-monomer polymerization. Physical Review E101(2), 022501 (2020)

work page 2020

[8] [8]

Journal of engineering mathematics49(2), 113–131 (2004)

Wattis, J.A., McCartney, D.G., Gudmundsson, T.: Coagulation equations with mass loss. Journal of engineering mathematics49(2), 113–131 (2004)

work page 2004

[9] [9]

Journal of Physics A: Mathematical and General29(2), 437–450 (1996)

Singh, P., Rodgers, G.: Coagulation processes with mass loss. Journal of Physics A: Mathematical and General29(2), 437–450 (1996)

work page 1996

[10] [10]

Proceedings of the Royal Society of Edinburgh Section A: Mathematics136(6), 1157–1173 (2006)

Banasiak, J., Lamb, W.: On a coagulation and fragmentation equation with mass loss. Proceedings of the Royal Society of Edinburgh Section A: Mathematics136(6), 1157–1173 (2006)

work page 2006

[11] [11]

Physical Review E110(5), 054609 (2024)

Klobusicky, J., Onat, E., Konstantinou, V.: Planar network statistics for two-dimensional rupturing foams. Physical Review E110(5), 054609 (2024)

work page 2024

[12] [12]

Journal of Nonlinear Science31(2), 42 (2021)

Klobusicky, J.: Markov models of coarsening in two-dimensional foams with edge rupture. Journal of Nonlinear Science31(2), 42 (2021)

work page 2021

[13] [13]

Physical Review E64(2), 021507 (2001)

Vandewalle, N., Lentz, J.: Cascades of popping bubbles along air/foam interfaces. Physical Review E64(2), 021507 (2001)

work page 2001

[14] [14]

Quarterly of Applied Mathematics, 677–696 (2016)

Klobusicky, J., Menon, G.: Concentration inequalities for a removal-driven thinning process. Quarterly of Applied Mathematics, 677–696 (2016)

work page 2016

[15] [15]

Discrete & Continuous Dynamical Systems-Series B28(7) (2023)

Klobusicky, J.: Concentration inequalities for the hydrodynamic limit of a two-species stochastic particle system. Discrete & Continuous Dynamical Systems-Series B28(7) (2023)

work page 2023

[16] [16]

Springer, ??? (2006)

Chicone, C.: Ordinary Differential Equations with Applications. Springer, ??? (2006)

work page 2006

[17] [17]

Alfonsín, J.L.R.: The Diophantine Frobenius Problem vol. 30. Oxford University Press, ??? (2005)

work page 2005

[18] [18]

In: Proceedings of the 2009 Ninth International Conference on Hybrid Intelligent Systems-Volume 01, pp

Cai, X.: Canonical coin systems for change-making problems. In: Proceedings of the 2009 Ninth International Conference on Hybrid Intelligent Systems-Volume 01, pp. 499–504 (2009)

work page 2009

[19] [19]

Berlin [etc.]: Springer (1991)

Hairer, E., Wanner, G., Solving, O.: Ii: Stiff and differential-algebraic problems. Berlin [etc.]: Springer (1991)

work page 1991

[20] [20]

arXiv preprint arXiv:1511.08555 (2016)

Villarino, M.: The convergence of the Catalan number generating function. arXiv preprint arXiv:1511.08555 (2016)

work page arXiv 2016

[21] [21]

Chapter 9

Flory, P.J.: Principles of Polymer Chemistry. Chapter 9. Cornell university press, ??? (1953)

work page 1953

[22] [22]

The Journal of chemical physics11(2), 45–55 (1943)

Stockmayer, W.H.: Theory of molecular size distribution and gel formation in branched-chain 23 polymers. The Journal of chemical physics11(2), 45–55 (1943)

work page 1943

[23] [23]

Com- munications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences57(9), 1197–1232 (2004) 24

Menon, G., Pego, R.L.: Approach to self-similarity in Smoluchowski’s coagulation equations. Com- munications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences57(9), 1197–1232 (2004) 24

work page 2004