Nonlocal electrostatic field theory from microscopic description

The study of electric fields in soft materials converges either to the study of point-like particles (local) in nonlinear theories or to the use of particles with finite sizes in nonlocal linear theories. In this work, we start from the microscopic equations of motion and construct a unified mean-field Fokker-Planck equation that describes the non-equilibrium electrostatics of nonlocal nonlinear systems. We obtain a generalized Poisson-Boltzmann equation for such systems, as well as their electrostatic free energy expression. In the linear approximation, we obtain an anisotropic susceptibility in an isotropic fluid, which allows for local linear response inversion in electrostatics.