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\title{}{\bf Stochastic simulation of inhomogeneous diffusion-reaction coagulation-fragmentation
processes with annihilation governed by
many species Smoluchowski equations }
\author{}{Sabelfeld K.K.}
{\it Institute of Computational Mathematics and Mathematical Geophysics, \\ SBRAS, NSU, Novosibirsk}
{\it sabelfeld.karl@yahoo.de}
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The coagulation of particles under diffusion controlled conditions has been well studied in many theoretical
and experimental research works and a huge literature exists in this field (e.g., see the references in \cite{1}).
In the simple case of one specie homogeneous Smoluchowski equation without diffusion, there are both deterministic
and stochastic models and algorithm which are able to solve practically interesting problems. If we however turn to
inhomogeneous case, the situation is drastically changed: you can find little theoretical studies and simulation
algorithms to handle the spatial variability, in particular, the diffusion of coagulating particles (e.g., \cite{2}).
In this lecture we present a series of extensions of the nonlinear integro-differential equations (we call them
Smouchowski type equations) governing the kinetics of processes which include: spatially inhomogeneity, diffusion,
many species, fragmentation-coagulation, annihilation of different species, and sink to randomly distributed
capture centers. The main difficulty in these problems is its multiscale character: the diffusion and coagulations
rates, for instance, differ in many orders of magnitudes. Also, the inhomogeneity leads to segregations,
so the spatial correlations are varying with time, which implies, the standard approaches fail.
We suggest new stochastic algorithm which is able to efficiently simulate this kind of processes by introducing
a macroscopic random time step of the diffusion and combining it with the Random Walk on Spheres method.
Practically interesting examples are also given to show the performance of the suggested method. It should be mentioned
that the method is generally enough, in particular, to be able to simulate kinetics of diffusion-reaction processes, annihilation of
electrons and holes in semiconductors, recombination processes on defects and many others. Important issue is the time asymptotics
of the solutions to the Smoluchowski type equations which we study both theoretically and through simulations.
The work has been supported by RFBR under Grant 12-01-00635-a.
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\bibitem{1}
{\it Sabelfeld K.K.} Random Fields and Stochastic Lagrangian Models.
De Gruyter, Berlin/Boston, 2012, 399 pp.
\bibitem{2}
{\it A.A. Kolodko and K.K. Sabelfeld}.
Stochastic Lagrangian model for spatially
inhomogeneous Smoluchowski equation
governing coagulating and diffusing particles.
Monte Carlo Methods and Applications. vol.7 (2001), N3-4,
223-228.
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