  An uniformly accelerated (Rindler) observer will detect particles in the
Minkowski vacuum, known as Unruh effect. The spectrum is thermal and the
temperature is given by that of the Killing horizon, which is proportional to
the acceleration. Considering these particles are kept in a thermal bath with
this temperature, we find that the correlation function of the random force due
to radiation acting on the particles as measured by the accelerated frame,
shows the fluctuation-dissipation relation. It is observed that the
correlations, in both ($1+1$) spacetime and ($1+3$) dimensional spacetimes, are
of Brownian type. We discuss the implications of this new observation at the
end.
