  The generic Berry phase scenario in which a two-level system is coupled to a
second system whose dynamical coordinate is slowly-varying is generalized to
allow for stochastic evolution of the slow system. The stochastic behavior is
produced by coupling the slow system to a heat resevoir which is modeled by a
bath of harmonic oscillators initially in equilibrium at temperature T, and
whose spectral density has a bandwidth which is small compared to the
energy-level spacing of the fast system. The well-known energy-level shifts
produced by Berry's phase in the fast system, in conjunction with the
stochastic motion of the slow system, leads to a broadening of the fast system
energy-levels. In the limit of strong damping and sufficiently low temperature,
we determine the degree of level-broadening analytically, and show that the
slow system dynamics satisfies a Langevin equation in which Lorentz-like and
electric-like forces appear as a consequence of geometrical effects. We also
determine the average energy-level shift produced in the fast system by this
mechanism.
