Stochastic Particle Approximations for Smoluchoski’s Coagualtion Equation

Eibeck, Andreas; Wagner, Wolfgang

doi:10.1214/aoap/1015345398

Cited by 97 publications

(96 citation statements)

References 27 publications

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“…As for the conjecture for gelling kernels (1.9), it was solved rather recently in [27,40]. An intermediate step is the existence of solutions to (1.1)-(1.2) and (1.5) with non-increasing finite mass, that is, satisfying M 1 (f (t)) ≤ M 1 (f (0)) for t ≥ 0, see [24,45,47,52,69,80] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Weak Compactness Techniques and Coagulation Equations

Laurençot

2014

Lecture Notes in Mathematics

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Abstract. Smoluchowski's coagulation equation is a mean-field model describing the growth of clusters by successive mergers. Since its derivation in 1916 it has been studied by several authors, using deterministic and stochastic approaches, with a blossoming of results in the last twenty years. In particular, the use of weak L 1 -compactness techniques led to a mature theory of weak solutions and the purpose of these notes is to describe the results obtained so far in that direction, as well as the mathematical tools used.

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Section: Introductionmentioning

confidence: 99%

“…We conclude the introduction with a few words on related interesting issues: we focus in these notes on the deterministic approach to the modeling of coagulation and leave aside the stochastic approach which has been initiated in [54,55,78,79] and further developed in [2,8,9,14,23,24,30,40,41,69] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Weak Compactness Techniques and Coagulation Equations

Laurençot

2014

Lecture Notes in Mathematics

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“…Ultimately, however, crude models for the apportionment of impurities among the defect clusters should be supplanted by a detailed accounting of multicomponent aggregation and coalescence reactions and their influence on the non-equilibrium cluster size distribution. Such problems are widely addressed in the literature, including gelation, polymerization, and formation of aerosols and precipitates in solid or fluid media [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. The numerical methods developed for such problems may also be fruitfully applied to radiation swelling.…”

Section: Introductionmentioning

confidence: 99%

Void nucleation, growth, and coalescence in irradiated metals

Surh

Sturgeon

Wolfer

2008

Journal of Nuclear Materials

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A novel computational treatment of dense, stiff, coupled reaction rate equations is introduced to study the nucleation, growth, and possible coalescence of cavities during neutron irradiation of metals. Radiation damage is modeled by the creation of Frenkel pair defects and helium impurity atoms. A multi-dimensional cluster size distribution function allows independent evolution of the vacancy and helium content of cavities, distinguishing voids and bubbles. A model with sessile cavities and no cluster-cluster coalescence can result in a bimodal final cavity size distribution with coexistence of small, high-pressure bubbles and large, low-pressure voids. A model that includes unhindered cavity diffusion and coalescence ultimately removes the small helium bubbles from the system, leaving only large voids. The terminal void density is also reduced and the incubation period and terminal swelling rate can be greatly altered by cavity coalescence. Temperature-dependent trapping of voids/bubbles by precipitates and alterations in void surface diffusion from adsorbed impurities and internal gas pressure may give rise to intermediate swelling behavior through their effects on cavity mobility and coalescence. IntroductionIrradiation of metals has long been known to culminate in macroscopic property changes including void swelling [1]. Characteristic stable voids and steady volumetric swelling develop for a range of temperatures and fluxes, independent of whether radiation bombardment damage occurs as disseminated Frenkel pairs or as small defect clusters. This can occur whether or not helium is generated along with atomic displacements. In either case, small, unstable voids, loops, and other defect clusters will develop almost immediately within the irradiated material. Their subsequent evolution determines the fluence required to create stable voids and achieve 1 arXiv:0803.3829v1 [cond-mat.mtrl-sci]

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…In this report we introduce the extension of the MFA [12,9] for the solution of the Smoluchowski coagulation equation with surface growth and a particle source. The algorithm presented in this paper calculates an exponentially distributed time step based on a majorant kernel and introduces fictitious jumps to compensate for the use of the majorant, uses stochastic jumps for all processes and makes use of a binary tree method for the determination of the particle partners.…”

Section: Introductionmentioning

confidence: 99%