Technical Report 32, c4e-Preprint Series, Cambridge

A stochastic approximation scheme and convergence theorem for particle interactions with perfectly reflecting boundaries

ref: Technical Report 32, c4e-Preprint Series, Cambridge

Associated Themes: Numerics, Nanoparticles, and Particle Processes

Abstract

We prove the existence of a solution to an equation governing the number density within a compact domain of a discrete particle system for a prescribed class of particle interactions taking into account the effects of the diffusion and drift of the set of particles. Each particle carries a number of internal coordinates which may evolve continuously in time, determined by what we will refer to as the internal drift, or discretely via the interaction kernels. Perfectly reflecting boundary conditions are imposed on the system and all the processes may be spatially and temporally inhomogeneous. We use a relative compactness argument to construct a sequence of measures that converge weakly to a solution of the governing equation. Since the proof of existence is a constructive one, it provides a stochastic approximation scheme that can be used for the numerical study of molecular dynamics.

Material from this preprint has been published in: Monte Carlo Methods and Applications 12 (3), 291-342, (2006)

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