Diffusion approach to the linear Poisson–Boltzmann equation

The linear Poisson–Boltzmann equation (LPBE) is mapped onto a transient diffusion problem in which the charge density becomes an initial distribution, the dielectric permittivity plays the role of either a diffusion coefficient or a potential of interaction and screening becomes a sink term. This analogy can be useful in two ways. From the analytical point of view, solutions of the LPBE with seemingly different functional forms are unified as Laplace transforms of the fundamental Gaussian solution for diffusion. From the numerical point of view, a first off-grid algorithm for solving the LPBE is constructed by running Brownian trajectories in the presence of scavenging.