Transient survival probability of a particle diffusing in a disordered system of perfectly absorbing spherical sinks

Abstract

The transient survival probability of a particle diffusing in a disordered system of perfectly absorbing non-overlapping spherical sinks is studied by use of a self-consistent cluster expansion for its Laplace transform. The self-consistent cluster expansion, which in principle is exact, is formulated in two alternative versions, corresponding to different ways of handling the instantaneous absorption of particles created initially inside a sink. In each version of the theory the cluster expansion is truncated at either the singlet or the pair level, so that four different approximations to the exact result are obtained. An approximation on the pair level constitutes an improvement to the corresponding singlet approximation. It is shown that at moderate volume fraction of the sink system the two pair approximations yield almost identical results for the survival probability. All four approximations lead to a band gap in the spectrum of relaxation rates, followed by a continuum extending to infinity.