In this article a stochastic particle system approximation to the parametric sensitivity in the Smoluchowski coagulation equation is introduced. The parametric sensitivity is the derivative of the solution to the equation with respect to some parameter, where the coagulation kernel depends on this parameter. It is proved that the particle system converges weakly to the sensitivity as the number of particles N increases. A Monte Carlo algorithm is developed and variance reduction techniques are applied. Numerical experiments are conducted for two kernels: the additive kernel and one which has been used for studying soot formation in a free molecular regime. It is shown empirically that the techniques for variance reduction are indeed very effective and that the order of convergence is O(1/N). The algorithm is then compared to an algorithm based on a finite difference approximation to the sensitivity and it is found that the variance of the sensitivity estimators are considerably lower than that for the finite difference approach. Furthermore, two methods of establishing `efficiency' are considered and the new algorithm is found to be significantly more efficient.
The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [11]. It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique. The survey's p…
The basis of this paper is the elementary observation that the $n$-step descendant distribution of any Galton-Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we obtain necessary and sufficient criteria for the convergence of scaling limits of Galton-Watson processes that are simpler (and equivalent) to the classical criteria obtained by Grimvall in 1974. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the Lévy jump measure of certain CSBPs satisfies a generalized Smoluchowski equation. …
Turbulence in growth of rain droplets and rain formation is studied under an approximating particle system representing aggregation at the level of individuals, depending on their volume and distance in space, of the Smoluchowski Coagulation equation. A random field is introduced to model the air flow interaction with the particles and it is proved that the empirical density of the individual converges to solutions of limiting equation under different setting for the random field of interaction. A brief numerical study for the continuous density is proposed using the particles approach, to ana…
In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of $β$-amyloid peptide (A$β$) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial di…
We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form $π(y)\cdot Aπ(x)$ for a vector of conserved quantities $π$, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time $t_\mathrm{g}\in (0,\infty)$, which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process b…
A Smoluchowski type model of coagulation in a turbulent fluid is given, first expressed by means of a stochastic model, then in a suitable scaling limit as a deterministic model with enhanced diffusion in the velocity component. A precise link between mean intensity of the turbulent velocity field and coagulation enhancement is obtained by numerical simulations, and a formula for the mean velocity difference, in agreement with the gas-kinetic model, is proved by a new method.
We introduce and analyze a novel type of coalescent processes called cross-multiplicative coalescent that models a system with two types of particles, $A$ and $B$. The bonds are formed only between the pairs of particles of opposite types with the same rate for each bond, producing connected components made of particles of both types. We analyze and solve the Smoluchowski coagulation system of equations obtained as a hydrodynamic limit of the corresponding Marcus-Lushnikov process. We establish that the cross-multiplicative kernel is a gelling kernel, and find the gelation time. As an applicat…
