  We consider the reduced dynamics of a molecular chain weakly coupled to a
phonon bath. With a small and constant inhomogeneity in the coupling, the
excitation relaxation rates are obtained in closed form. They are dominated by
transitions between exciton modes lying next to each other in the energy
spectrum. The rates are quadratic in the number of sites in a long chain.
Consequently, the evolution of site occupation numbers exhibits longer
coherence lifetime for short chains only. When external source and sink are
added, the rate equations of exciton occupation numbers are similar to those
obtained earlier by Fr\"{o}hlich to explain energy storage and energy transfer
in biological systems. There is a clear separation of time scale into a faster
one pertaining to internal influence of the chain and phonon bath, and a slower
one determined by external influence, such as the pumping rate of the source,
the absorption rate of the sink and the rate of radiation loss. The energy
transfer efficiency at steady state depends strongly on these external
parameters, and is robust against a change in the internal parameters, such as
temperature and inhomogeneity. Excitations are predicted to concentrate to the
lowest energy mode when the source power is sufficiently high. In the site
basis, this implies that when sustained by a high power source, a sink
positioned at the center of the chain is more efficient in trapping energy than
a sink placed at its end. Analytic expressions of energy transfer efficiency
are obtained in the high power and low power source limit. Parameters of a
photosynthetic system are used as examples to illustrate the results.
