Coalescense with arbitrary-parameter kernels and monodisperse initial conditions: A study within combinatorial framework

Abstract

For this work, we studied a finite system of discreet-size aggregating particles for two types of kernels with arbitrary parameters, a condensation (or branched-chain polymerization) kernel, $K(i,j)=(A+i)(A+j)$, and a linear combination of the constant and additive kernels, $K(i,j)=A+i+j$. They were solved under monodisperse initial conditions in the combinatorial approach where discreet time is counted as subsequent states of the system. A generating function method and Lagrange inversion were used for derivations. Expressions for an average number of clusters of a given size and its corresponding standard deviation were obtained and tested against numerical simulation. High precision of the theoretical predictions can be observed for a wide range of $A$ and coagulation stages, excepting post-gel phase in the case of the condensation kernel (a giant cluster presence is preserved). For appropriate $A$, these two kernels reproduced known results of the constant, additive and product kernels. Beside a previously solved linear-chain kernel, they extend the number of arbitrary-parameter kernels solved in the combinatorial approach.

arXiv comment14 pages, 5 figures, original work. In this version, adding journal reference

References in corpus (16)

• Scaling Theory and Exactly Solved Models In the Kinetics of Irreversible Aggregation
• "Explosive Percolation" Transition is Actually Continuous
• Cluster aggregation model for discontinuous percolation transition
• Hybrid Percolation Transition in Cluster Merging Processes: Continuously Varying Exponents
• Statistical Thermodynamics of Clustered Populations
• Irreversible Aggregation and Network Renormalization
• Exact solutions for mass-dependent irreversible aggregations
• Spatial evolution of Hindmarsh-Rose neural network with time delays
• Uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation with inverse power law kernels
• Exact combinatorial approach to finite coagulating systems
• Coagulation kinetics beyond mean field theory using an optimised Poisson representation
• Analytical kinetics of clustering processes with cooperative action of aggregation and fragmentation
• Convergence of Achlioptas processes via differential equations with unique solutions
• Exact combinatorial approach to finite coagulating systems through recursive equations
• Coagulation with product kernel and arbitrary initial conditions: Exact kinetics within the Marcus-Lushnikov framework
• Combinatorial solutions to generalized electrorheological kernel aggregation

Cited by in corpus (2)

• Rate equation limit for a combinatorial solution of a stochastic aggregation model
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