Reappraisal of four different approaches for finding the mean reaction time in the multi-trap variant of the Adam-Delbrück problem.

Adam and Delbrück argued that the dimensionality of the diffusion space determines the average lifetime of a diffusing particle confined to a region with a central trap. Doubts have often been aired as to whether their calculation is relevant to real biological systems, where the number of traps is usually much larger than unity; or whether the rate enhancement is merely a manifestation of an increase in the concentration of the traps; or whether the diverse multi-trap versions of their expression for the mean lifetime in two dimensions are trustworthy. These issues are addressed, and the long-standing problem of finding the low-density limit of trapping time in two dimensions solved, by examining previous treatments of the problem, and by carrying out simulations of two-dimensional systems in which the particles undergo a Pearsonian random walk, and the traps are distributed randomly or on a square lattice. The mean lifetimes are found to be different in the two situations, and it is concluded that the neglect of this aspect lies at the root of the conflict between some of the existing expressions for the mean lifetime. Relations expressing the mean lifetime as a function of the concentration of the traps are presented together with a discussion of their applicability.