Effects of an absorbing boundary on the average volume visited by N spherical Brownian particles

The number of distinct sites visited by a lattice random walk and its continuum analog, the volume swept out by a diffusing spherical particle are used to model different phenomena in physics, chemistry and biology. Therefore the problem of finding statistical properties of these random variables is of importance. There have been several studies of the more general problem of the volume of the region explored by N random walks or Brownian particles in an unbounded space. We here study the effects of a planar absorbing boundary on the average of this volume. The boundary breaks the translational invariance of the space, and introduces an additional spatial parameter, the initial distance of the Brownian particles from the surface. We derive expressions for the average volume visited in three dimensions and the average span in one dimension as functions of the time for given values of the initial distance to the absorbing boundary and N. The results can be transformed to those for N lattice random walks by appropriately choosing the radius and diffusion constant of the spheres.