  Taking an open quantum systems approach, we derive a collective equation of
motion for the dynamics of a matter-wave bright soliton moving through a
thermal cloud of a distinct atomic species. The reservoir interaction involves
energy transfer without particle transfer between the soliton and thermal
cloud, thus damping the soliton motion without altering its stability against
collapse. We derive a Langevin equation for the soliton centre of mass velocity
in the form of an Ornstein-Uhlenbeck process with analytical drift and
diffusion coefficients. This collective motion is confirmed by simulations of
the full stochastic projected Gross-Pitaevskii equation for the matter-wave
field. The system offers a pathway for experimentally observing the elusive
energy-damping reservoir interaction, and a clear realization of collective
Brownian motion for a mesoscopic superfluid droplet.
