Coagulation, diffusion and the continuous Smoluchowski equation

Abstract

The Smoluchowski equations are a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers or by positive reals, these corresponding to the discrete or the continuous form of the equations. For dimension d ≥ 3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a method similar to that used to derive the discrete form of the equations in [A. Hammond, F. Rezakhanlou, The kinetic limit of a system of coagulating Brownian particles, Arch. Ration. Mech. Anal. 185 (2007) 1-67]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of the cited work. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations.

Spatial coagulation with bounded coagulation rate

“…This heuristic result was proved in an important special case by Norris in [12] and later in [4] and [14] by Hammond, Rezakhanlou, and Yaghouti. Well-posedness for equation (1.1) has been investigated in a relatively small number of works.…”

Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries

“…The first existence results for the Smoluchowski coagulation equation and its extensions were based on convergent sub-sequences of approximating stochastic processes. The first convergence result of this kind with simple diffusive transport of particles is due to Lang and Xanh [8], generalisations were achieved by Norris [12,11], Wells [19] and Yaghouti et al [21]. This is quite a natural approach, because the equations are based on a microscopic stochastic model and related stochastic processes have also proved fruitful for numerical purposes going back to Marcus [9] and Gillespie [5].…”

Effective Catalytic Reduction of Methyl Orange Catalyzed by the Encapsulated Random Alloy Palladium‐Gold Nanoparticles Dendrimer.

“…The theoretical bimolecular rate constant is practically based on Fick's second law of diffusion . Considering that diffusion term is significantly larger than the electron transport term, the Fick's second law of diffusion results in the Smoluchowski equation (see the supporting information) . Table described the calculated second Damköhler number and bimolecular rate constant for the catalyst amount (green star in Figure and Figure 5S) used to perform the kinetic investigation.…”

“…[38] Considering that diffusion term is significantly larger than the electron transport term, the Fick's second law of diffusion results in the Smoluchowski equation (see the supporting information). [39] Table 2 described the calculated second Damkö hler number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 and bimolecular rate constant for the catalyst amount (green star in Figure 3 and Figure 5S) used to perform the kinetic investigation. The kinetic data were interpreted according to two physicochemical adsorption approaches, the Langmuir-Hinshelwood and Mars-van Krevelen models.…”