  Following the ideas of Davydov's soliton theory, we study the bio-energy
transport in protein molecules. By using a quantum Brownian motion model for a
phonon dressed vibrational exciton, we calculate the time-dependence on the
mean square distance, diffusion coefficient, and energy of the vibrational
exciton. We find the time-dependence by solving the quantum Langevin equation
and find oscillatory behaviors due to the super-diffusive non-ohmic
dissipation. We find that the vibrational exciton gains an overall energy due
to the coupling to the phonon bath; it also dissipates its energy to the
environment as it propagates. The amount of energy gain and the oscillatory
features depend on both temperature and the phonon-vibron coupling.
