  This work contributes to the problem of determining effective interaction
between asymmetrically (likely or oppositely) charged objects whose total
charge is neutralized by mobile pointlike counter-ions of the same charge, the
whole system being in thermal equilibrium. The problem is formulated in two
spatial dimensions with logarithmic Coulomb interactions. The charged objects
correspond to two parallel lines at distance d, with fixed line charge
densities. Two versions of the model are considered: the standard
"unconstrained" one with particles moving freely between the lines and the
"constrained" one with particles confined to the lines. We solve exactly both
systems at the free-fermion coupling and compare the results for the pressure
(i.e. the force between the lines per unit length of one of the lines) with the
mean-field Poisson-Boltzmann solution. For the unconstrained model, the large-d
asymptotic behaviour of the free-fermion pressure differs from that predicted
by the mean-field theory. For the constrained model, the asymptotic pressure
coincides with the attractive van der Waals-Casimir fluctuational force. For
both models, there are fundamental differences between the cases of
likely-charged and oppositely-charged lines, the latter case corresponding at
large distances d to a capacitor.
