Partial absorption and “virtual” traps

Abstract

The spatial probability distribution associated with diffusion and attenuation in partially absorbing media is studied. An equivalence is established between a system with free diffusion for x>0 and partial absorption for x<0, and a semi-infinite system (x>0) with a radiation boundary condition at x=0. By exploiting this equivalence, it is shown that the effect of a partially absorbing medium in the long-time limit is equivalent to that of a perfect, “virtual” trap whose size is smaller than the original absorbing medium. For short times, however, there is substantial penetration of diffusing particles into the absorber. The virtual trap approach is readily generalized to higher dimensions. This allows one to obtain the density profile of diffusing particles around a partially absorbing spherical trap. An unusual crossover between short-time penetration and long-time trapping occurs in two dimensions; the size of the virtual trap is exponentially small in the case of weak absorption, corresponding to an absorption time which is exponentially large.