  We study the velocity dependence of the frictional force of the tip of an
atomic force microscope as it is dragged across a surface, taking into account
memory effects and thermal fluctuations. Memory effects are described by a
coupling of the tip to low frequency excitation modes of the surface in
addition to the coupling to the periodic corrugation potential. We find that
when the excitation mode frequency is comparable to the characteristic
frequency corresponding to the motion of the tip across the surface, the
velocity dependence of the frictional force is non monotonic, displaying a
velocity range where the frictional force can decrease with increasing
velocity. These results provide theoretical support for the interpretation of
recent experiments which find a frictional force that decreases with velocity
on surfaces covered with a monolayer.
