Nearest-neighbour distances of diffusing particles from a single trap

Abstract

We present a simple method for obtaining the typical smallest distance from an absorbing trap to the nearest particle in a system of Brownian particles. In terms of the overall density profile, we also obtain the distribution function for the minimum distance. By using a quasistatic approximation for the diffusion equation, we derive new results for two dimensions: the characteristic distance to the nearest particle increases asymptotically as m. The technique is useful in diffusion-reaction systems. In the Smoluchowski model for bimolecular reactions, an ideal spherical trap centred at the origin is surrounded by a cloud of Brownian particles which are captured upon contact with the trap. The reaction rate in this formulation depends on the spatial distribution of particles about the trap. Recently, interest has focused on finer details of this distribution such as the distance of the nearest surviving particle from the trap (l-31. The density distribution function of that distance is the key to several exact results concerning some diffusion-reaction systems in one dimension (4). While the spatial density of the particles surrounding the trap can be easily obtained by solving a diffusion equation with the appropriate boundary conditions, the distance to the nearest particle from the trap seems to require a more sophisticated analysis ( 13. In this paper, we suggest a simple derivation for the typical distance from the trap to the nearest particle and of the distribution of those distances, based upon the more readily attainable spatial density of all particles and simple facts from the statistics of extremes (5,6). To demonstrate our method, we rederive the known results for one and three dimensions. The simplicity of our approach, together with using a quasistatic approximation (7) for solving the diffusion equation, allows us to solve the two- dimensional case for the first time. Consider first the one-dimensional problem: a perfect trap is located at the origin and is initially surrounded by a homogeneous density, co, of Brownian particles which diffuse with a diffusion constant D. The density c(x, t) of the surviving particles (to the right of the trap) is given by the diffusion equation