Diffusion on a Solid Surface: Anomalous is Normal

We present a numerical study of classical particles diffusing on a solid surface. The particles’ motion is modeled by an underdamped Langevin equation with ordinary thermal noise. The particle-surface interaction is described by a periodic or a random two-dimensional potential. The model leads to a rich variety of different transport regimes, some of which correspond to anomalous diffusion such as has recently been observed in experiments and Monte Carlo simulations. We show that this anomalous behavior is controlled by the friction coefficient and stress that it emerges naturally in a system described by ordinary canonical Maxwell-Boltzmann statistics.

Figure 1

Left: A trajectory for $\gamma =1$ over $t=20000$ time units. Right: A trajectory for $\gamma =0.04$ over $t=15000$ time units. The period of the potential is $\lambda =4$. Note the different scales in the two panels.

Figure 2

$⟨r^2⟩/4\tau$ for a particle in the periodic potential, for $\gamma =0.0004$ (solid line), $0.004$ (dotted line), $0.04$ (dashed line), and $0.4$ (dot-dashed line). The straight-line segment has unit slope as a guide to the eye. Inset: Diffusion coefficient vs $\gamma$. The solid lines correspond to the theoretical calculations [Eqs. (3, 4)].

Figure 3

Log-log plot of $P(r,\tau )$ for $\gamma =0.0004$ and three different values of time intervals: $\tau =20$ (squares), $100$ (open circles), $1000$ (triangles), and for $\gamma =1.0$ at the time interval $\tau =100$ (solid curve).

Figure 4

$⟨r^2⟩/4\tau$ for a particle in the random potential for $\gamma =0.0001$ (solid line), $0.001$ (dotted line), $0.003$ (dashed line), and $0.008$ (dot-dashed line). The straight-line segment has unit slope as a guide to the eye. Inset: Exponents $\alpha$ versus friction coefficient $\gamma$.