Partial Absorption and ``Virtual'' Traps,
E. Ben-Naim, S. Redner and G. H. Weiss
The spatial probability distribution associated with diffusion
and attenuation in partially absorbing media is studied. An equivalence
is established between a system with free diffusion for x>0 and
partial absorption for x<0, and a semi-infinite system x>0) with a
radiation boundary condition at x=0. By exploiting this equivalence,
it is shown that the effect of a partially absorbing medium in the
long-time limit is equivalent to that of a perfect, ``virtual'' trap
whose size is smaller than the original absorbing medium. For short
times, however, there is substantial penetration of diffusing particles
into the absorber. The virtual trap approach is readily generalized to
higher dimensions. This allows one to obtain the density profile of
diffusing particles around a partially absorbing spherical trap. An
unusual crossover between short-time penetration and long-time trapping
occurs in two dimensions; the size of the virtual trap is
exponentially small in the case of weak absorption, corresponding to an
absorption time which is exponentially large.
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