  We analyze the friction force exerted on a small probe particle sliding over
an atomic-scale surface by means of a Green-Kubo relation and classical
Molecular Dynamics simulations. We find that, on the atomic scale, the friction
tensor can drastically vary as a function of position and sliding direction.
The Green-Kubo relation yields this positional and directional dependence from
equilibrium simulations of the time dependent covariance of force acting on the
probe. We find, unexpectedly, that the positional and directional dependence of
energy dissipation is related to the (much simpler) static force covariance,
especially in the limit where the probe only mildly perturbs the surface
particles. In contrast, the (free) energy landscape experienced by the probe is
in general not a good indicator of local dissipation. We also discuss
optimization strategies making use of the locally and directionally resolved
friction tensor. This enables us to find optimal sliding paths and velocity
protocols, e.g., minimizing energy dissipation, between two points on the
surface in a given time.
