Life and Death in a Cage and at the Edge of a Cliff

TL;DR

This paper investigates the survival probabilities of diffusing particles in expanding or receding boundaries, revealing non-universal power-law behaviors dependent on boundary motion rates, with heuristic and exact analyses supporting the findings.

Contribution

It provides a comprehensive analysis of survival probabilities in dynamic boundary conditions, combining heuristic and exact methods to characterize non-universal power-law behaviors.

Findings

Survival probability follows a power-law decay with an exponent depending on boundary motion.

Heuristic approaches yield accurate approximate exponents for slow and fast boundary motions.

Exact analysis confirms the heuristic results across most boundary motion scenarios.

Abstract

The survival probabilities of a ``prisoner'' diffusing in an expanding cage and a ``daredevil'' diffusing at the edge of a receding cliff are investigated. When the diffuser reaches the boundary, he dies. For ``marginal'' boundary motion, i.e., the cage length grows as $\sqrt{At}$ or the cliff location recedes as $x_0(t)=-\sqrt{At}$ and the daredevil diffuses within the domain $x>x_0$, the survival probability of the diffuser exhibits non-universal power-law behavior, $S(t)\sim t^{-\b}$, which depends on the relative rates of boundary and diffuser motion. Heuristic approaches are applied for the cases of ``slow'' and ``fast'' boundary motion which yield approximate expressions for $\b$. An asymptotically exact analysis of these two problems is also performed and the approximate expressions for $\b$ coincide with the exact results for nearly entire range of possible boundary motions.