encountered along the chosen sliding path such that $N_{\mathrm{max}}=N_{\mathrm{min}}-1$, as we always start and terminate our path in a minimum of the unrelaxed energy curve for both approaches.

Results and discussion {#sec:Res}
======================

With the methods described above we calculate the friction-versus-load behavior of fcc Cu(111). All averages are collected on paths of $\unit[1]{\mu m}$ length to get reliable values for the constitutive system parameters and to make sure that in the case of quasi aperiodic paths the whole unit cell becomes sampled. For both definitions of the mean friction force, $F^{(1)}$ and $F^{(2)}$, we observe an exponential friction law along all path directions, see Fig. \[fig:4\], given by $$\label{eq:exp}
F(L) = F_0 \exp \left( \frac{\mu}{F_0} L \right) \quad .$$ This result can be traced back to the exponential form for the binding energy in transition metals e.g. as given by Pettifor [@book:pettifor]. In a very general way the energy of bond breaking follows the *universal binding energy relation* (UBER) which also shows an exponential dependence on distance [@rose:83] which in our case depends linearly on load.

![Semi-logarithmic plot of the friction force $F^{(1)}$ versus load $L$ for the slip plane of fcc Cu(111). Both periodic paths $0^\circ$ (red circles) 