  In this paper, by comparing the time scales associated with the velocity
relaxation and correlation time of the random force due to dust charge
fluctuations, memory effects in the velocity relaxation of an isolated dust
particle exposed to the random force due to dust charge fluctuations are
considered, and the velocity relaxation process of the dust particle is
considered as a non-Markovian stochastic process. Considering memory effects in
the velocity relaxation process of the dust particle yields a retarded friction
force, which is introduced by a memory kernel in the fractional Langevin
equation. The fluctuation-dissipation theorem for the dust grain is derived
from this equation. The mean-square displacement and the velocity
autocorrelation function of the dust particle are obtained, and their
asymptotic behavior, the dust particle temperature due to charge fluctuations,
and the diffusion coefficient are studied in the long-time limit. As an
interesting feature, it is found that by considering memory effects in the
velocity relaxation process of the dust particle, fluctuating force on the dust
particle can cause an anomalous diffusion in a dusty plasma. In this case, the
mean-square displacement of the dust grain increases slower than linearly with
time, and the velocity autocorrelation function decays as a power-law instead
of the exponential decay. Finally, in the Markov limit, these results are in
good agreement with those obtained from previous works for Markov (memoryless)
process of the velocity relaxation.
