Relation between the classical resistance of inhomogeneous networks and diffusion

Relations between diffusion and conduction on arbitrary networks are investigated. It is shown that the (dimensionless) conductance between two points, A and B, on a network of uniform resistors is proportional to the probability of a random walker, leaving point A for good, to reach point B for the first time. A generalization of the Einstein relation is derived for general networks. It is shown that the ‘‘typical time’’ relevant for the diffusion coefficient between two points is the mean first-passage time between them. The ac conductance, Σ(ω), is calculated by connecting the system to time-dependent reservoirs. It is demonstrated for self-similar structures that Σ(ω)∼‖f(ω)${\Vert }^L$, L being the linear size of the system.