We consider charge transport in nanopores where the dielectric constantinside the nanopore is much greater than in the surrounding material, so thatthe flux of the electric fields due to the charges is almost entirely confinedto the nanopore. That means that we may model the electric fields due to chargedensities in the nanopore in terms of average properties across the nanopore assolutions of one dimensional Poisson equations. We develop basic equations foran M component system using equations of continuity to relate concentrations tocurrents, and flux equations relating currents to concentration gradients andconductivities. We then derive simplified scaled versions of the equations. Wedevelop exact solutions for the one component case in a variety of boundaryconditions using a Hopf-Cole transformation, Fourier series, and periodicsolutions of the Burgers equation. These are compared with a simpler model inwhich the scaled diffusivity is zero so that all charge motion is driven by theelectric field. In this non-dissipative case, recourse to an admissibilitycondition is utilised to obtain the physically relevant weak solution of aRiemann problem concerning the electric field. It is shown that theadmissibility condition is Poynting's theorem.
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