Brownian particle motion in a system with absorbing boundaries

Abstract

The transition probabilities of brownian particles are found for a system with plane-parallel boundaries and arbitrary absorption coefficients at the boundaries. The connection between the different model boundary conditions (‘radiation boundary conditions’ and Lorentz' model condition) are discussed. The transition probabilities under consideration have been represented as a series of the normal distributions with variable coefficients depending on the absorption coefficients of particles by the boundaries and coordinates. The first and second moments of transition probabilities are calculated. The influence of the absorptive properties of the boundaries on these moments have been investigated in detail. The symmetry properties for both moments are established. Numerical calculations of the mean and mean-squared displacements of brownian particles have been performed in the cases of totally reflecting and perfectly absorbing boundaries. The space and time distributions of brownian particles as well as the number of particles absorbed by the boundaries have been described in the systems with partially reflecting boundaries for homogeneous and local initial distributions of brownian particles. The generalization of the basic results obtained for a plane layer to systems bounded in all directions by plane boundaries is presented. The full analysis of the problem under consideration has also been done for the particular case of a semi-bounded system.