Approximative solution of the coagulation–fragmentation equation by stochastic particle systems

On a Class of Continuous Coagulation-Fragmentation Equations

“…A local uniqueness result is also provided in [18] for the pure coagulation equation ( #0). Let us finally mention that under the assumption (1.14) existence of measure-valued solutions to the pure coagulation equations ( #0) and to the full coagulation-fragmentation equations (with suitable assumptions on ) has been obtained, respectively, in [18,9] by a probabilistic approach (see also [2]). The solutions obtained therein taking only their values in the space of measures thus satisfy (1.1) in a weaker sense than the one required by Definition 1.1.…”

Simulation of coalescence and breakage: an assessment of two stochastic methods suitable for simulating liquid–liquid extraction

“…The general method for direct simulation of solutions to the coalescence-breakage equation has been given by Eibeck and Wagner (2000a). Their paper concentrates on proving existence of a solution, and proposes a simulation algorithm for the coalescence-breakage case.…”

“…This particle ensemble is an approximation of the real life system being examined. As the process relies on random number generation to choose the nature and timing of the coalescence and breakage events, many trajectories are generated, and an average (which converges to the solution of the population balance equation (Eibeck and Wagner, 2000a)) is calculated. For the one-dimensional case (size dependency only), direct stochastic simulation (e.g.…”

“…The purpose of this paper is to bridge the gap between mathematical proof and practical solution by examining the feasibility of using a stochastic algorithm, either DSA or MFA, to simulate solutions to the population balance equation, where coalescence and binary breakage both occur. DSA is examined as described by Eibeck and Wagner (2000a), and MFA (formulated for coagulation only by Eibeck and Wagner (2001), and for discrete coagulation-fragmentation by Jourdain (2003)) is extended to include binary breakage in the continuous case. The unchanging number of stochastic particles used by MFA lends itself to using a binary search method of probability distribution generation with an associated reduction in simulation CPU time.…”

A new numerical approach for the simulation of the growth of inorganic nanoparticles

“…In 1972, Gillespie first used a stochastic model to simulate cloud droplet growth [6]. More recently, Eibeck and Wagner applied these ideas to coagulation and fragmentation, deriving both the direct simulation algorithm (DSA) and mass-flow algorithm (MFA) with accompanying convergence proofs and introducing fictitious jumps for the reduction of the complexity of the algorithm [7,8,9]. These methods have since been applied to chemical engineering by Goodson and Kraft who studied the convergence properties of the algorithm [10] and by Grosschmidt et al [4] who applied the algorithm to the production of silica.…”