Fokker-Planck Sample Clauses

Fokker-Planck approach and velocity dependence of friction As we discussed in the previous chapter, thermal activation can make the tip overcome energy barriers even when these are much higher than the thermal energy. This activated process can be formulated in terms of xxx Xxxxxx-Xxxxxx equation [21] which has become a standard approach in this field. Xxxxx et al. proposed an elegant treatment of the thermal activation of the tip, generalizing xxx Xxxxxxxxx model by adding a noise term [22]. Their model takes into account the irregular motion of the tip, as the temperature increases. Although this model only included the effect of forward jumps, they argued that the inclusion of backward jumps would lead to a further reduction of friction at low velocities, when the frequency of such jumps would be substantial. Through this approach, Xxxxx and co-authors conclude that at lower velocities, atomic-friction has a weak logarithmic dependence on the scanning velocity due to thermal fluctuations. At higher velocities, viscous drag dominates and it acts as the major source of dissipation. The stick-slip regime is the highest velocity regime where thermal effects are manifest and temperature acts as a lubricant. At velocities comparable to the phonon velocity of the material, elastic deformations of the tip are so essential that the tip slides smoothly over the surface. Recently, Evstigneev and Xxxxxxx have analyzed the thermal activation of the tip, for a wide range of spring constants, describing the rates of multiple back and forth jumps of the tip [23]. The main message of these approaches is that thermal excitations can result in multiple jumps of the tip, both in the direction of the moving support and against that direction. This implies that the logarithmic velocity dependence of atomic friction, which is the result of only the forward jumps, should be restricted to a small velocity range. Combining all this information, we summarize the different friction regimes as follows: At ultra-low velocities, typically of the order of a few nm/s or lower, thermal activation makes the tip perform multiple backward and forward jumps between neighboring potential wells, thus making friction dramatically low [1]. We will discuss this aspect in detail later in this chapter. At slightly higher velocities of the order of a few tens of nm/s, the tip will perform predominantly pre-critical jumps. These give rise to a logarithmic velocity dependence of the friction force [9].