  A new master equation to mimic the dynamics of a collection of interacting
random walkers in an open system is proposed and solved numerically.In this
model, the random walkers interact through excluded volume interaction
(single-file system); and the total number of walkers in the lattice can
fluctuate because of exchange with a bath.In addition, the movement of the
random walkers is biased by an external perturbation. Two models for the latter
are considered: (1) an inverse potential (V $\propto$ 1/r), where r is the
distance between the center of the perturbation and the random walker and (2)
an inverse of sixth power potential ($V \propto 1/r^6 $). The calculated
density of the walkers and the total energy show interesting dynamics. When the
size of the system is comparable to the range of the perturbing field, the
energy relaxation is found to be highly non-exponential. In this range, the
system can show stretched exponential ($ e^{-{(t/\tau_s)}^{\beta}} $) and even
logarithmic time dependence of energy relaxation over a limited range of time.
Introduction of density exchange in the lattice markedly weakens this
non-exponentiality of the relaxation function, irrespective of the nature of
perturbation.
