Pith. sign in

REVIEW 6 major objections 4 minor 25 references

Existence and Non-existence for Exchange-Driven Growth Model

T0 review · 6 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For exchange-driven growth, global solutions stop existing once the interaction kernel grows past a sharp threshold: below $\mu+\nu\le 3$ they exist for all time, while faster kernels cause gelation or instantaneous gelation.

desk verdict A serious EDG paper that likely settles the existence/gelation phase diagram if the imported moment-blow-up-to-gelation lemma is proved for the exchange-driven system; worth refereeing, needs revision. read the letter →

arxiv 2411.14083 v1 pith:6BYFWUMX submitted 2024-11-21 math.AP

Saroj Si , Ankik Kumar Giri This is my paper · ORCID
classification math.AP MSC 34A3534A1246B5034G20
keywords exchange-drivengrowthinteractionkernelsexistencemildsolutionsnon-existencegelationinstantaneousmomentblow-up
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Exchange-driven growth is an infinite system of ordinary differential equations in which clusters exchange monomers; the paper asks when this system has a global solution and when it does not. It claims that symmetric kernels with $K_{j,k}\le C(j^\mu k^\nu+j^\nu k^\mu)$, $\mu,\nu\le 2$, $\mu+\nu\le 3$, admit global classical solutions for initial data in $Y^+_\lambda$ with $\lambda=\max\{\mu,\nu\}>1$, and for data in $Y^+_1$ when $\max\{\mu,\nu\}\le 1$, relaxing earlier requirements of finite higher moments. In the intermediate range $3<\mu+\nu\le 4$, for kernels satisfying $C_1(j^2k^\alpha+j^\alpha k^2)\le K_{j,k}\le Cj^2k^2$ with $1<\alpha\le 2$, it establishes finite-time gelation by blowing up the moment $M_\alpha$, with the exact formula $T_{\mathrm{gel}}=(2M_2(f(0))C)^{-1}$ for $K_{j,k}=Cj^2k^2$. For kernels with $K_{j,k}\ge C(j^\beta+k^\beta)$, $\beta>2$, it proves instantaneous gelation, $T_{\mathrm{gel}}=0$, so no solution exists on any positive time interval. If these results are correct, they settle the existence/non-existence phase diagram that earlier work had conjectured.

What carries the argument

The proof is carried by three mechanisms. Finite truncations of the infinite ODE system produce approximating solutions; uniform moment bounds come from testing the truncated equations with convex functions $G\in\mathcal G_{1,\infty}$ selected by a refined de la Vallée-Poussin theorem, and Arzelà–Ascoli plus the convex-function tail control let the interaction sums pass to the limit. Gelation is detected through the exact evolution of a selected moment: writing $(j+1)^\alpha-2j^\alpha+(j-1)^\alpha$ by the mean value theorem turns the kernel lower bound into a closed differential inequality for $M_\alpha$ whose solution blows up at a computable time. Instantaneous gelation uses the criterion that blow-up of $\sum_j j^m f_j(t_0)$ implies $T_{\mathrm{gel}}\le t_0$, imported from coagulation–fragmentation theory, combined with Jensen's inequality applied to $M_{n-2+\beta}$ to show that for every $n$ the blow-up time is at most $O(1/n)$; letting $n\to\infty$ gives $T_{\mathrm{gel}}=0$.

What would settle it

Find a symmetric kernel with $C_1(j^2k^\alpha+j^\alpha k^2)\le K_{j,k}\le Cj^2k^2$, $1<\alpha\le 2$, and initial data in $Y^+_{2+\alpha}$ with $M_\alpha(f(0))>0$, for which a global mass-conserving solution with all moments finite exists; this would refute Theorem 2.7 and Corollary 2.8. A more direct check is to test the imported lemma on the EDG system by looking for a solution with $\sum_j j^m f_j(t_0)=\infty$ while $\sum_j j f_j(t_0)=\sum_j j f_j(0)$, which would show that moment blow-up need not imply gelation.

Watch

Extended reading notes

Core claim

The paper's central claim is that the exchange-driven growth system has a kernel-growth threshold with three regimes. For kernels with $\mu+\nu\le 3$ and $\mu,\nu\le 2$, the truncated-system limit is a mild solution that conserves total mass and particle number; the de la Vallée-Poussin machinery upgrades it to a classical solution while removing the earlier need for finite moments of order $p>2$. For kernels between $C_1(j^2k^\alpha+j^\alpha k^2)$ and $Cj^2k^2$ with $1<\alpha\le 2$, the $\alpha$-th moment obeys $M_\alpha(t)\ge (1/M_\alpha(f(0))-C_1\alpha(\alpha-1)2^{\alpha-2}t)^{-1}$, blowing up in finite time and thereby preventing any global mass-conserving solution. For the multiplicative kernel $Cj^2k^2$, the second moment solves $M_2'(t)=2CM_2(t)^2$, so $T_{\mathrm{gel}}=(2M_2(f(0))C)^{-1}$ exactly. Finally, if $K_{j,k}\ge C(j^\beta+k^\beta)$ with $\beta>2$, Jensen's inequality forces every higher moment to blow up at times shrinking to zero, so the gelation time is zero and no $Y^+_2$ solution exists on $[0,T)$ for any $T>0$.

Load-bearing premise

The load-bearing premise is the imported lemma from coagulation–fragmentation theory that if $\sum_j j^m f_j(t_0)=\infty$ for some $m$, then the gelation time is at most $t_0$; the paper does not prove this implication for the exchange-driven system, and both finite-time and instantaneous gelation conclusions depend on it.

Editorial extensions

If this is right

  • If correct, initial data need only lie in $Y^+_\lambda$ (or $Y^+_1$ for slower kernels), not in spaces with finite moments of order $p>2$, so the existence theory covers a strictly larger class of physical cluster distributions.
  • For kernels sandwiched between $C_1(j^2k^\alpha+j^\alpha k^2)$ and $Cj^2k^2$ with $1<\alpha\le 2$, no global mass-conserving solution can exist; the system must lose mass at the finite gelation time.
  • For $K_{j,k}=Cj^2k^2$, the gelation time is exactly $(2M_2(f(0))C)^{-1}$, an explicit, initial-data-dependent formula that can be tested numerically.
  • For kernels growing faster than quadratically in either argument, solutions with finite second moment do not exist on any positive time interval, making the non-existence immediate rather than asymptotic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The exact formula for $K_{j,k}=Cj^2k^2$ suggests a numerical check: integrating the truncated system, $M_2(t)$ should track $(1/M_2(f(0))-2Ct)^{-1}$ up to the blow-up time; any systematic deviation would indicate that finite-size effects change the gelation time.
  • Because the proof of finite-time gelation uses only the lower bound on the kernel and the moment equation, the same mechanism should hold for non-symmetric kernels with comparable growth, extending the non-existence regime beyond symmetric interactions.
  • The instantaneous-gelation theorem implies that for $\beta>2$ the EDG equation is ill-posed in $Y^+_2$ from time zero; any worthwhile regularisation must alter the kernel or impose a cutoff, and it would be informative to compare the resulting limiting solutions as the cutoff is removed.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

6 major / 4 minor

Summary. The paper studies the discrete exchange-driven growth (EDG) model (1.1)--(1.3) and claims a comprehensive existence and gelation phase diagram. The main results are: global classical solutions for symmetric kernels satisfying K_{j,k} \le C(j^\mu k^\nu + j^\nu k^\mu) with \mu,\nu \le 2 and \mu+\nu \le 3, for initial data in Y_\lambda (\lambda=\max\{\mu,\nu\}>1) or in Y_1 in the sublinear case; local existence for K_{j,k} \le C j^2 k^2 with data in Y_2; finite-time gelation for kernels bounded below by C_1(j^2 k^\alpha + j^\alpha k^2), 1<\alpha\le2, with the explicit value T_{\rm gel}=(2M_2(f(0))C)^{-1} for K=C j^2 k^2; and instantaneous gelation, T_{\rm gel}=0, for kernels growing superquadratically, K_{j,k} \ge C(j^\beta+k^\beta), \beta>2. The proofs use truncation, uniform moment estimates based on de la Vall\'ee-Poussin functions, Arzel\`a-Ascoli compactness, and Gronwall-type inequalities.

Significance. If the results are correct, the paper would settle the exchange-driven growth phase diagram conjectured in [10] and meaningfully extend the global and local existence results of [7, 10, 23]. The explicit gelation-time formula for the multiplicative kernel K=C j^2 k^2 is a crisp, falsifiable prediction, and the use of refined de la Vall\'ee-Poussin moments to relax the initial-data assumptions is a promising technique. The paper is ambitious and addresses an active topic. However, several load-bearing technical steps in the gelation and non-existence proofs are currently not rigorous, so the central claims are not yet fully established.

major comments (6)
  1. [This comment concerns Section 4, Lemma 4.1.] The proof of Lemma 4.1 ends with the algebraic inequality X_N(t) \le M_r(0) + \tilde C_* X_N(t), and the authors then invoke Gronwall's lemma to conclude X_N(t) \le M_r(0) e^{\tilde C_* t}. This is not a Gronwall argument: if \tilde C_* > 1 the algebraic inequality yields no bound at all. One needs a differential inequality of the form (d/dt) M_r(t) \le C M_r(t), or an integrated form with the moment on the right inside the time integral. Since Lemma 4.1 is used to justify that the local solution of Theorem 2.5 remains in Y_{2+\alpha}, this gap directly affects the proof of Theorem 2.7.
  2. [This comment concerns Section 1 and the proofs of Theorems 2.7, 2.9, and Corollary 2.10.] The conversion of moment blow-up into a bound on the gelation time is based entirely on the imported statement [3, Lemma 9.2.2], which is proved in the coagulation-fragmentation literature and is not proved for the exchange-driven system. The paper asserts this lemma as 'well-known' in the introduction and then uses it to conclude T_{\rm gel} \le t_0 from divergence of a moment. Finiteness of M_1 does not imply finiteness of higher moments, so divergence of M_\alpha at t_0 does not by itself establish loss of mass. The equality T_{\rm gel}=(2M_2(f(0))C)^{-1} and the conclusion T_{\rm gel}=0 in Theorem 2.9 are therefore not self-contained. The authors should either prove the EDG analogue of the lemma or reformulate the non-existence statements so that they follow directly from failure of Y_2 or Y_r regularity, without invoking mass loss.
  3. [This comment concerns the proof of Corollary 2.10 in Section 5.] The proof supposes a solution on [0,T) and then invokes Lemma 5.3, which requires 0<T<T_{\rm gel}. But Theorem 2.9 gives T_{\rm gel}=0, so the hypothesis of Lemma 5.3 is not satisfied for any positive T. The contradiction is therefore not obtained. The non-existence of any Y_2 solution on an interval—even one that may have undergone gelation—requires a separate argument that does not rely on pre-gelation moment finiteness.
  4. [This comment concerns Theorem 2.2(b) and Lemma 3.5(2).] Theorem 2.2(b) claims global existence whenever max{\mu,\nu} \le 1, which includes the endpoint \mu=\nu=1. However, Lemma 3.5(2) and the proof of the uniform convergence (3.55) explicitly require min{\mu,\nu}<1. The case \mu=\nu=1 (the linearly growing kernel K_{j,k}\le Cjk) is covered neither by part (a), since \lambda=\max\{\mu,\nu\}=1 is excluded by \lambda>1, nor by the proof of part (b). This is a genuine omission in the theorem as stated; the endpoint should either be proved or handled by citing the known results of [10, 23].
  5. [This comment concerns Lemma 5.3, in particular inequality (5.5).] The step from the tail estimate for \sum_{j\ge m} j^2 f_j(\sigma) to the p-th moment bound (5.5) is not justified. The preceding line provides a bound with the exponential factor e^{-2CC_2 m^{\beta-2}(t-\sigma)} depending on the lower cutoff m. In (5.5) the exponential is evaluated with the summation index j, which would require a separate summation-by-parts or layer-cake argument. As written, the inequality does not follow. Since Lemma 5.3 underpins the proof of Theorem 2.9, the instantaneous gelation result needs a corrected tail estimate.
  6. [This comment concerns the proof of Theorem 2.7 in Section 4.] The lower bound on M_\alpha is derived only on the local existence interval of Theorem 2.5, i.e., for t<T_0<1/(2M_2(0)C). If the claimed blow-up time T^* = 1/(C_1\alpha(\alpha-1)2^{\alpha-2}M_\alpha(0)) is larger than 1/(2M_2(0)C), the solution need not exist up to T^*, and the displayed differential inequality does not show that M_\alpha actually reaches infinity. The proof needs a continuation argument or an explicit comparison with the M_2 blow-up time so that T_{\rm gel} is bounded by the first time at which the solution leaves Y_2.
minor comments (4)
  1. [This comment concerns equation (3.17) in Section 3.] The second term on the right-hand side of (3.17) should read j^{\mu+\nu-2}k^\mu (equivalently j^{\lambda+\nu-2}k^\lambda after setting \lambda=\mu), not j^{\mu+\nu-2}k^\nu. The subsequent estimate (3.20) uses the correct k^\lambda, so this appears to be a typo, but it should be corrected for consistency.
  2. [This comment concerns the abstract and the statements of Theorems 2.2 and 2.5.] The abstract says 'classical solutions' are established, whereas the theorems construct mild solutions and Corollary 2.4 upgrades them to continuously differentiable solutions. Please align the terminology so that the claims match the proofs.
  3. [This comment concerns the affiliation on page 1.] The affiliation text contains the typo 'Roor kee'; it should presumably read 'Roorkee'.
  4. [This comment concerns Section 5, around the derivation of (5.4).] The display after (5.4) has several sign and index conventions that are hard to follow (for example, the lower limit of the integral changes from \sigma to t, and the rearrangement of I_j terms is not fully transparent). Please rewrite this part to make the estimates easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: gelation times are solved from moment differential inequalities; the only concern is the external [3, Lemma 9.2.2], which is a correctness gap, not a self-referential reduction.

full rationale

The paper's main results are genuine derivations from its stated hypotheses. Theorem 2.2 constructs global mild solutions by the truncation method (Section 3), deriving uniform bounds on truncated moments via the discrete convexity lemmas 3.2-3.5 and passing to the limit; the higher-moment bounds use the de la Vallee-Poussin theorem and convexity inequalities from [17], neither of which presupposes the existence of solutions. Theorem 2.5 obtains local existence for K≤Cj²k² directly from the truncated differential inequality d/dt M₂ᴺ ≤ 2C(M₂ᴺ)², with the blow-up time appearing as a solved quantity, not as an assumed input. Theorem 2.7 proves finite-time blow-up of M_α from the lower kernel bound C₁(j²k^α+j^αk²) by solving the integral inequality (4.4); the exact gelation time for K=Cj²k² follows by solving dM₂/dt=2CM₂² and combining with the local mass conservation from Theorem 2.5. Theorem 2.9 proves Tgel=0 by showing M_n blows up at times tending to 0, using Jensen's inequality and the lower kernel bound; the calculation is not circular. No parameter is fitted and then renamed as a prediction. The only external input is [3, Lemma 9.2.2], used to convert moment divergence into Tgel≤t0; this lemma is cited from coagulation-fragmentation theory and is not proved for the EDG system, which is a verification/correctness gap rather than a circular reduction. The sole self-citation ([1]) is used only to justify the truncation approach and is not load-bearing for the existence or gelation conclusions. Hence no significant circularity is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters and no invented entities. Constants C, C1, C2 are hypotheses of the theorems, not fitted values. The auxiliary convex functions Gλ and G1 are produced by de la Vallée-Poussin from the initial data, not chosen by hand. The main external input is the standard analytic toolbox plus the imported gelation lemma.

assumptions (5)
  • standard math Arzelà-Ascoli theorem is used to extract convergent subsequences of truncated solutions.
    Invoked in the proofs of Theorem 2.2 and Theorem 2.5 after uniform bounds on f^N_j and their time derivatives (Lemma 3.2).
  • standard math Refined de la Vallée-Poussin theorem and convex-function inequalities from [18] and [17].
    Used in Lemma 3.3 and Lemma 3.4 to control higher moments without assuming moments beyond λ or 1.
  • standard math Gronwall's inequality and Jensen's inequality.
    Used throughout Sections 3 to 5 to close moment estimates and to prove blow-up of M_α and M_n.
  • standard math The external fact [3, Lemma 9.2.2]: if ∑ j^m f_j(t0)=∞ for some m∈N then T_gel≤t0.
    Stated in the introduction and used in Theorems 2.7 and 2.9 to convert moment blow-up into gelation time.
  • domain assumption The exchange-driven growth equations (1.1)-(1.3) with non-negative symmetric kernels are the model under study.
    The paper analyzes this system as given; no derivation from particle systems is attempted.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Existence and Non-existence for Exchange-Driven Growth Model." pith.science (2026). https://pith.science/paper/6BYFWUMX

@misc{pith2026241114083,
  author       = {Pith},
  title        = {Pith review of: Existence and Non-existence for Exchange-Driven Growth Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6BYFWUMX}},
  note         = {Machine review of arXiv:2411.14083}
}
abstract

The exchange-driven growth (EDG) model describes the evolution of clusters through the exchange of single monomers between pairs of interacting clusters. The dynamics of this process are primarily influenced by the interaction kernel $K_{j,k}$. In this paper, the global existence of classical solutions to the EDG equations is established for non-negative, symmetric interaction kernels satisfying $K_{j,k} \leq C(j^{\mu}k^{\nu} + j^{\nu}k^{\mu}) $, where $\mu, \nu \leq 2$, $\mu + \nu \leq 3$, and $C>0$, with a broader class of initial data. This result extends the previous existence results obtained by Esenturk [10], Schlichting [23], and Eichenberg \& Schlichting [7]. Furthermore, the local existence of classical solutions to the EDG equations is demonstrated for symmetric interaction kernels that satisfy $K_{j,k} \leq C j^{2} k^{2}$ with $C > 0$, considering a broader class of initial data. In the intermediate regime $3 < \mu + \nu \leq 4$, the occurrence of finite-time gelation is established for symmetric interaction kernels satisfying $C_{1}\left(j^{2}k^{\alpha}+j^{\alpha}k^{2}\right)\leq K_{j,k}\leq Cj^{2}k^{2}$, where $1 < \alpha \leq 2$, $C>0$, and $C_{1} > 0$, as conjectured in [10]. In this case, the non-existence of the global solutions is ensured by the occurrence of finite-time gelation. Finally, the occurrence of instantaneous gelation of the solutions to EDG equations for symmetric interaction kernels satisfying $K_{j,k}\geq C\left(j^{\beta}+k^{\beta}\right)$ ($\beta>2, C>0)$ is shown, which also implies the non-existence of solutions in this case.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 21 canonical work pages

  1. [10]

    Esenturk

    E. Esenturk. Mathematical theory of exchange-driven grow th. Nonlinearity, 31(7):3460, 2018

    work page 2018

  2. [1]

    M. Ali, A. K. Giri, and P. Lauren¸ cot. Well-posedness of the discre te collision-induced breakage equa- tion with unbounded fragment distribution. Nonlinear Analysis: Real World Applications , 75:103967, 2024

    work page 2024

  3. [2]

    J. M. Ball, J. Carr, and O. Penrose. The becker-d¨ oring cluster equations: basic properties and asymptotic behaviour of solutions. Communications in mathematical physics , 104:657–692, 1986

    work page 1986

  4. [3]

    Banasiak, W

    J. Banasiak, W. Lamb, and Ph. Lauren¸ cot. Analytic methods for coagulation-fragmentation models . Monogr. Res. Notes Math. Boca Raton, FL: CRC Press, 2020

    work page 2020

  5. [4]

    Barik, F

    P. Barik, F. da Costa, J. Pinto, and R. Sasportes. The discrete generalized exchange-driven system. arXiv preprint arXiv:2408.00345 , 2024

    arXiv 2024

  6. [5]

    Ben-Naim and P

    E. Ben-Naim and P. Krapivsky. Exchange-driven growth. Physical Review E , 68(3):031104, 2003

    work page 2003

  7. [6]

    Carr and F

    J. Carr and F. P. da Costa. Instantaneous gelation in coagulatio n dynamics. Zeitschrift f¨ ur ange- wandte Mathematik und Physik ZAMP , 43(6):974–983, 1992

    work page 1992

  8. [7]

    Eichenberg and A

    C. Eichenberg and A. Schlichting. Self-similar behavior of the exch ange-driven growth model with product kernel. Communications in Partial Differential Equations , 46(3):498–546, 2021

    work page 2021

Show all 25 references
  1. [8]

    M. H. Ernst, R. M. Ziff, and E. Hendriks. Coagulation processes w ith a phase transition. Journal of Colloid and Interface Science , 97(1):266–277, 1984. 27

    1984

  2. [9]

    Escobedo, S

    M. Escobedo, S. Mischler, and B. Perthame. Gelation in coagulatio n and fragmentation models. Communications in Mathematical Physics , 231:157–188, 2002

    2002

  3. [11]

    Esenturk and J

    E. Esenturk and J. Velazquez. Large time behaviour of exchan ge-driven growth. Discrete and Con- tinuous Dynamical Systems. Series A , 41(2):747–775, 2021

    2021

  4. [12]

    Grosskinsky and W

    S. Grosskinsky and W. Jatuviriyapornchai. Derivation of mean- field equations for stochastic particle systems. Stochastic Processes and their Applications , 129(4):1455–1475, 2019

    2019

  5. [13]

    Ispolatov, P

    S. Ispolatov, P. Krapivsky, and S. Redner. Wealth distribution s in models of capital exchange. arXiv preprint cond-mat/9708018, 1997

    1997 arXiv

  6. [14]

    Ke and Z

    J. Ke and Z. Lin. Kinetics of migration-driven aggregation proce sses. Physical Review E , 66(5):050102, 2002

    2002

  7. [15]

    C. Y. Lam and A. Schlichting. Variational convergence of excha nge-driven stochastic particle systems in the thermodynamic limit. arXiv preprint arXiv:2401.06696 , 2024

    2024 arXiv

  8. [16]

    Lauren¸ cot

    P. Lauren¸ cot. The lifshitz–slyozov equation with encounters . Mathematical Models and Methods in Applied Sciences, 11(04):731–748, 2001

    2001

  9. [17]

    Lauren¸ cot

    P. Lauren¸ cot. Weak compactness techniques and coagulatio n equations. In Evolutionary equations with applications in natural sciences , pages 199–253. Springer, 2014

    2014

  10. [18]

    C. H. Lˆ e. Etude de la classe des op´ erateurs m-accr´ etifs deL1(Ω) et accr´ etifs dans L∞ (Ω). PhD thesis, Universit´ e de Paris VI, 1977. Th` ese de 3 ` eme cycle

    1977

  11. [19]

    F. Leyvraz. Scaling theory and exactly solved models in the kinet ics of irreversible aggregation. Physics Reports, 383(2-3):95–212, 2003

    2003

  12. [20]

    J. McLeod. On an infinite set of non-linear differential equations . The Quarterly Journal of Mathe- matics, 13(1):119–128, 1962

    1962

  13. [21]

    J. McLeod. On an infinite set of non-linear differential equations (ii). The Quarterly Journal of Mathematics, 13(1):193–205, 1962

    1962

  14. [22]

    Menon and R

    G. Menon and R. L. Pego. Approach to self-similarity in smoluchow ski’s coagulation equations. Communications on Pure and Applied Mathematics: A Journal I ssued by the Courant Institute of Mathematical Sciences, 57(9):1197–1232, 2004

    2004

  15. [23]

    Schlichting

    A. Schlichting. The exchange-driven growth model: basic prope rties and longtime behavior. Journal of Nonlinear Science , 30(3):793–830, 2020

    2020

  16. [24]

    Van Dongen

    P. Van Dongen. On the possible occurrence of instantaneous g elation in smoluchowski’s coagulation equation. Journal of Physics A: Mathematical and General , 20(7):1889, 1987

    1987

  17. [25]

    R. M. Ziff. Kinetics of polymerization. Journal of Statistical Physics , 23(2):241–263, 1980

    1980

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.