Coagulation of Brownian particles

17 January 2005
Dr James Norris
According to the Stokes-Einstein law, microscopic particles subject to intense bombardment by (much smaller) gas molecules perform Brownian motion with a diffusivity inversely proportion to their radius. Smoluchowski, shortly after Einstein's account of Brownian motion, used this model to explain the behaviour of a cloud of such particles when, in addition their diffusive motion, they coagulate on collision. He wrote down a system of evolution equations for the densities of particles of each size, in particular identifying the collision rate as a function of particle size. We give a rigorous derivation of (a spatially inhomogeneous generalization of) Smoluchowski's equations, as the limit of a sequence of Brownian particle systems with coagulation on collision. The equations are shown to have a unique, mass-preserving solution. A detailed limiting picture emerges describing the ancestral spatial tree of particles making up each particle in the current population. The limit is established at the level of these trees.
  • Stochastic Analysis Seminar