We discuss the failure of the Markov approximation in the description of
atom-surface fluctuation-induced interactions, both at equilibrium
(Casimir-Polder forces) and out-of-equilibrium (quantum friction). Using
general theoretical arguments, we show that the Markov approximation can lead
to erroneous predictions of such phenomena with regard to both strength and
functional dependencies on system parameters. Our findings highlight the
importance of non-Markovian effects in dispersion interactions. In particular,
we show that the long-time power-law tails of temporal correlations, and the
corresponding low-frequency behavior, of two-time dipole correlations,
neglected in the Markovian limit, dramatically affect the prediction of the
force.Comment: 17 pages, 6 figure
