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We evaluate exactly the statistical integral for an inhomogeneousone-dimensional counterion-only Coulomb gas between two charged boundaries andfrom this compute the effective interaction, or disjoining pressure, betweenthe bounding surfaces. Our exact results are compared with the limiting casesof weak and strong coupling which are the same for 1D and 3D systems. Forsystems with a large number of counterions it is found that the weak coupling(mean-field) approximation for the disjoining pressure works perfectly and thatfluctuations around the mean-field in 1D are much smaller than in 3D. In thecase of few counterions it works less well and strong coupling approximationperforms much better as it takes into account properly the discreteness of thecounterion charges.

One-dimensional counterion gas between charged surfaces: Exact results compared with weak- and strong-coupling analysis