The paper deals with a diffusing particle that escapes from a cavity to the outer world through anarrow cylindrical tunnel. We derive expressions for the Laplace transforms of the particle survivalprobability, its lifetime probability density, and the mean lifetime. These results show how thequantities of interest depend on the geometric parameters (the cavity volume and the tunnel lengthand radius) and the particle diffusion coefficients in the cavity and in the tunnel. Earlier suggestedexpressions for the mean lifetime, which correspond to different escape scenarios, are contained inour result as special cases. In contrast to these expressions, our formula predicts correct asymptoticbehavior of the mean lifetime in the absence of the cavity or tunnel. To test the accuracy of ourapproximate theory we compare the mean lifetime, the lifetime probability density, and the survivalprobability (the latter two are obtained by inverting their Laplace transforms numerically) withcorresponding quantities found by solving numerically the three-dimensional diffusion equation,assuming that the cavity is a sphere and that the particle has the same diffusion coefficient in thecavity and in the tunnel. Comparison shows excellent agreement between the analytical andnumerical results over a broad range of the geometric parameters of the problem.
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