Numerically exact diffusion coefficients for lattice systems with periodicboundary conditions. I. Theory

The standard method to study the diffusion of a particle in a system with i mmobile obstacles is to use Monte Carlo simulations on finite-size lattices with periodic boundary conditions. For example, the diffusion of proteins on the surface of biomembranes in the presence of fractal and random aggreg ates of obstacles has been studied extensively by M. J. Saxton. In this art icle, we derive two algebraically exact methods to calculate the diffusion coefficient D for such systems. The first method reduces the problem to tha t of a first passage problem. The second one uses the Nernst-Einstein relat ion to transform the problem into a field-driven drift problem where D is r elated to the zero-field mobility. Systems with closed volumes and multiple independent pathways are discussed. In the second part [Mercier and Slater , J. Chem. Phys. 110, 6057 (1999), following paper], a numerical implementa tion will be described and tested, and several examples of applications wil l be given. (C) 1999 American Institute of Physics. [S0021-9606(99)51812-6] .