A Langevin equation that governs the irregular stick-slip nano-scale friction

Friction force at the nanoscale, as measured from the lateral deflection of the tip of an atomic force microscope, usually shows a regular stick-slip behavior superimposed by a stochastic part (fluctuations). Previous studies showed the overall fluctuations to be correlated and multi-fractal, and thus not describable simply by e.g. a white noise. In the present study, we investigate whether one can extract an equation to describe nano-friction fluctuations directly from experimental data. Analysing the raw data acquired by a silicon tip scanning the NaCl(001) surface (of lattice constant 5.6 Å) at room temperature and in ultra-high vacuum, we found that the fluctuations possess a Markovian behavior for length scales greater than 0.7 Å. Above this characteristic length, the Kramers-Moyal approach applies. However, the fourth-order KM coefficient turns out to be negligible compared to the second order coefficients, such that the KM expansion reduces to the Langevin equation. The drift and diffusion terms of the Langevin equation show linear and quadratic trends with respect to the fluctuations, respectively. The slope 0.61 ± 0.02 of the drift term, being identical to the Hurst exponent, expresses a degree of correlation among the fluctuations. Moreover, the quadratic trend in the diffusion term causes the scaling exponents to become nonlinear, which indicates multifractality in the fluctuations. These findings propose the practical way to correct the prior models that consider the fluctuations as a white noise.