Explicit propagators for a random walker and unidirectionality on linear chains

Explicit propagators are given for a diffusing particle (motor) moving on a linear chain of either infinity or finite length with reflecting ends. Each chain contains a number of thermally accessible barriers and/or potential wells (active sites). All particle interactions with its environment are considered to be short-range and are described by repulsive/attractive delta function potentials. By employing perturbation expansion, closed analytical expressions for the spatio-temporal evolution of the probability density function of the motor are derived, and are valid up to second order with respect to the expansion parameter u, which denotes the strength of interaction between motor and active sites. The mean displacement for two different chains is calculated indicating in both cases that the organization of the motion is done through the interplay of interaction intensities and their positions.