Ion diffusion in a Coulombic field

An analytic solution of counter-ion diffusion in a semi-infinite domain near a uniformly charged surface is obtained within the Smoluchowski-Poisson-Boltzmann treatment. The long-ranged Coulombic interaction results in a finite first-passage time to the charged surface, although higher moments of the first-passage time are infinite. This problem is directly related to an exactly solvable model of a lattice random walk with a position-dependent bias.