Life and death in an expanding cage and at the edge of a receding cliff

The survival probabilities of a particle diffusing within an expanding ''cage'' and near the edge of a receding ''cliff,'' with death occurring when the diffuser reaches a boundary of the system, are investigated. Especially interesting behavior arises when the position of the boundary recedes from the diffuser as root At. In this case, the recession matches the rms displacement root Dt with which diffusion tends to bring the diffuser to its demise. For both the cage and cliff problems, the survival probability S(t) exhibits a nonuniversal power-law decay in time, S(t) similar to t(-beta), in which the value of beta is dependent on the detailed properties of the boundary motion. Heuristic approaches are applied for the cases of ''slow'' (A/D much less than 1) and ''fast'' (A/D much greater than 1) boundary motion which yield approximate expressions for beta. An asymptotic analysis of the survival probability for the cage and cliff problems is also performed. The approximate expressions for beta are in good agreement with the exact results for nearly the entire range of possible boundary motions. (C) 1996 American Association of Physics Teachers.