  The Brownian motion of small particles interacting with a field at a finite
temperature is a well-known and well-understood phenomenon. At zero
temperature, even though the thermal fluctuations are absent, quantum fields
still possess vacuum fluctuations. It is then interesting to ask whether a
small particle that is interacting with a quantum field will exhibit Brownian
motion when the quantum field is assumed to be in the vacuum state. In this
paper, we study the cases of a small charge and an imperfect mirror interacting
with a quantum scalar field in (1+1) dimensions. Treating the quantum field as
a classical stochastic variable, we write down a Langevin equation for the
particles. We show that the results we obtain from such an approach agree with
the results obtained from the fluctuation-dissipation theorem. Unlike the
finite temperature case, there exists no special frame of reference at zero
temperature and hence it is essential that the particles do not break Lorentz
invariance. We find that that the scalar charge breaks Lorentz invariance,
whereas the imperfect mirror does not. We conclude that small particles such as
the imperfect mirror {\it will} exhibit Brownian motion even in the quantum
vacuum, but this effect can be so small that it may prove to be difficult to
observe it experimentally.
