Generating functions and efficient numerical simulation of aggregation-fragmentation kinetics
Abstract
In this work, we review and revisit the generating function techniques that provide exact analytical solutions for aggregation and fragmentation equations across several physical regimes including spontaneous and collisional shattering. For discrete coagulation-fragmentation equations with size-independent rates under monodisperse initial conditions, we show the derivation of several explicit closed-form solutions. We also briefly report the exact solutions for continuous, three-particle, D-particle collisions and two-component generalizations. Source-driven aggregation yields steady distributions featuring a universal s−3/2 power-law decay and a cutoff mass scaling s* ∼ t2. We also remind that exploitation of the low-rank representations of the aggregation kernel allows to organize efficient modelling the aggregation kinetics and extend this idea showing a simple way for the fast accounting of the spontaneous framentation terms. We exploit our numerical algorithm and study the numerical solutions for multiple models of spontaneous fragmentation and find a new example of temporal oscillations for the closed aggregation-fragmentation process.
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BibTeX
@misc{matveev2026generating,
title = {{Generating functions and efficient numerical simulation of aggregation-fragmentation kinetics}},
author = {S. A. Matveev},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.10785},
eprint = {2606.10785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.10785},
}
RIS
TY - UNPB
TI - Generating functions and efficient numerical simulation of aggregation-fragmentation kinetics
AU - S. A. Matveev
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2606.10785
UR - https://arxiv.org/abs/2606.10785
AB - In this work, we review and revisit the generating function techniques that provide exact analytical solutions for aggregation and fragmentation equations across several physical regimes including spontaneous and collisional shattering. For discrete coagulation-fragmentation equations with size-independent rates under monodisperse initial conditions, we show the derivation of several explicit closed-form solutions. We also briefly report the exact solutions for continuous, three-particle, D-particle collisions and two-component generalizations. Source-driven aggregation yields steady distributions featuring a universal s^−3/2 power-law decay and a cutoff mass scaling s_* ∼ t^2. We also remind that exploitation of the low-rank representations of the aggregation kernel allows to organize efficient modelling the aggregation kinetics and extend this idea showing a simple way for the fast accounting of the spontaneous framentation terms. We exploit our numerical algorithm and study the numerical solutions for multiple models of spontaneous fragmentation and find a new example of temporal oscillations for the closed aggregation-fragmentation process.
ER -