 
Summary: 20 February 1998
Z .Chemical Physics Letters 284 1998 7886
Diffusion approach to the linear PoissonBoltzmann equation
Veaceslav Zaloj 1
, Noam Agmon
Department of Physical Chemistry and The Fritz Haber Research Center, The Hebrew UniÕersity, Jerusalem 91904, Israel
Received 16 September 1997; in final form 19 November 1997
Abstract
Z .The linear PoissonBoltzmann 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 offgrid algorithm for solving the LPBE is
constructed by running Brownian trajectories in the presence of scavenging. q 1998 Elsevier Science B.V.
1. Introduction
w x w xTwo classical fields of continuum physics are electrostatics 1 on the one hand and heat conduction 2 and
w xdiffusion 3,4 on the other. In these fields one studies the solution of elliptic partial differential equations of
Z . Z .similar form. Specifically, one considers the Poisson equation PE or PoissonBoltzmann equation PBE on
Z .the one hand, and the heat equation, Fick or DebyeSmoluchowski equations DSE on the other. The analogy
between diffusion and electrostatics is quite evident, and forms the basis for the socalled ``probabilistic
