Non-equilibrium dynamics of a classical tip interacting with a quantum surface

We apply the non-equilibrium statistical operator method within the mixed quantum-classical representation to the non-contact atomic force microscopy system in which a macroscopic tip is made to oscillate above a sample surface. In our treatment the tip is treated classically, whereas a microscopic system comprising atoms in the surface and in the nano-tip is considered quantum mechanically. Using a very general non-Markovian approach, we obtain both the Fokker-Planck equation for the tip and, independently, its equation of motion with friction terms and memory effects completely accounted for. In the classical limit (high temperatures) our equations turn exactly into the classical counterparts derived previously.