A mini-review on combinatorial solutions to the Marcus-Lushnikov irreversible aggregation

Abstract

Over the past decade, a combinatorial framework for discrete, finite, and irreversibly aggregating systems has emerged. This work reviews its progress, practical applications, and limitations. We outline the approach's assumptions and foundations, based on direct enumeration of system states, contrasting with classical Smoluchowski and Marcus-Lushnikov methods. Using the constant kernel as an example, we derive combinatorial expressions for the average number of clusters of a given size and their standard deviation, and present the complete probability distribution for cluster counts. The method is then extended to several kernels (additive, product, linear-chain, condensation) by explicitly enumerating ways to form clusters of a given size. For general kernels, approximate solutions are obtained via recursive expressions, enabling predictions without explicit solutions. Applications to aerosol growth and planetesimal formation are demonstrated, with comparisons to numerical results. We summarize issues of validity and precision and propose open problems. The appendix includes partial Bell polynomials, generating functions, Lagrange inversion, potential applications, and links between combinatorial and scaling solutions of the Smoluchowski equation.

42 pages, 15 figures, a mini-review; in this version, editorial correction in Eq. (14)

References in corpus (12)

• Scaling Theory and Exactly Solved Models In the Kinetics of Irreversible Aggregation
• Growing the gas-giant planets by the gradual accumulation of pebbles
• Planetesimal rings as the cause of the Solar System's planetary architecture
• Nonequilibrium continuous phase transition in colloidal gelation with short-range attraction
• Population-balance description of shear-induced clustering, gelation and suspension viscosity in sheared DLVO colloids
• A Monte Carlo algorithm to measure probabilities of rare events in cluster-cluster aggregation
• Exact combinatorial approach to finite coagulating systems
• Exact combinatorial approach to finite coagulating systems through recursive equations
• Exact asymptotic solution of an aggregation model with a bell-shaped distribution
• Rate equation limit for a combinatorial solution of a stochastic aggregation model
• Exact solutions of temperature-dependent Smoluchowski equations
• Coalescense with arbitrary-parameter kernels and monodisperse initial conditions: A study within combinatorial framework