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Poisson-Boltzmann theory allows one to study soft matter and biophysicalsystems involving point-like charges of low valencies. The inclusion offluctuation corrections beyond the mean-field approach typically requires theapplication of loop expansions around a mean-field solution for theelectrostatic potential \(\phi({\bf r})\), or sophisticated variationalapproaches. Recently, Poisson-Boltzmann theory has been recast, via a Legendretransform, as a mean-field theory involving the dielectric displacement field\({\bf D}({\bf r})\). In this paper we consider the path integral formulationof the dual theory. Exploiting the transformation between \(\phi\) and \({\bfD}\), we formulate a dual Sine-Gordon field theory in terms of the displacementfield and provide a strategy for precise numerical computations of freeenergies beyond the leading order.

Dual free energies in Poisson-Boltzmann theory