  We investigate the random motion of a mirror in (1 + 1)-dimensions that is
immersed in a thermal bath of massless scalar particles which are interacting
with the mirror through a boundary condition. Imposing the Dirichlet or the
Neumann boundary conditions on the moving mirror, we evaluate the mean
radiation reaction force on the mirror and the correlation function describing
the fluctuations in the force about the mean value. From the correlation
function thus obtained, we explicitly establish the fluctuation-dissipation
theorem governing the moving mirror. Using the fluctuation-dissipation theorem,
we compute the mean-squared displacement of the mirror at finite and zero
temperature. We clarify a few points concerning the various limiting behavior
of the mean-squared displacement of the mirror. While we recover the standard
result at finite temperature, we find that the mirror diffuses logarithmically
at zero temperature, confirming similar conclusions that have been arrived at
earlier in this context. We also comment on a subtlety concerning the
comparison between zero temperature limit of the finite temperature result and
the exact zero temperature result.
