Solving fluctuation-enhanced Poisson-Boltzmann equations

Electrostatic correlations and fluctuations in ionic systems can be described within an extended Poisson-Boltzmann theory using a Gaussian variational form. The resulting equations are challenging to solve because they require the solution of a high-dimensional nonlinear partial differential equation for the pair correlation function. This has limited existing studies to simple approximations or to one-dimensional geometries. In this paper we show that the numerical solution of the equations is greatly simplified by the use of selective inversion of a finite difference operator which occurs in the theory. This selective inversion preserves the sparse structure of the problem and leads to substantial savings in computer effort. In one and two dimensions further simplifications are made by using a mixture of selective inversion and Fourier techniques. With the help of numerical examples, we validate the accuracy of the numerical method and show that electrostatic correlations and fluctuations could lead to a large deviation from the classical Poisson-Boltzmann equation with the increase of the coupling parameter.