One-dimensional colloidal model with dielectric inhomogeneity

We consider a one-dimensional model allowing analytical derivation of the effective interactions between two charged colloids. evaluate exactly partition function for an electroneutral salt-free suspension with dielectric jumps at colloids' position. derive contact relation pressure that shows there is like-charge attraction, whether or not counterions are confined In contrast to homogeneous case, possibility colloids attract despite number counter-ions ($N$) being even. The results shown recover mean-field prediction in limit $N\to \infty$.