Diffusion on a Solid Surface: Anomalous is Normal

Abstract

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 particlesurface 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.

Transport and diffusion on crystalline surfaces under external forces

“…This problem is effectively one dimensional. Although we could have posed a one-dimensional equation for this paper from the outset, this work is part of a broader program in which the bi-dimensionality is essential in the absence of forces [12] and also when the forces are not along a symmetry direction and/or parallel to one another [32]. In particular, we focus on the role of the friction parameter.…”

“…Consider, for example, our system with a 'dc' external force F 0 along the x-direction and with no oscillating component. Throughout this paper we set λ = 4, the value also used in our earlier work [12]. We set the dimensionless temperature at ε ≡ k B T/V 0 = 0.2, a value that we often use in our simulations (although later we also study the effects on our results by varying the temperature).…”

“…This observation has elicited considerable theoretical interest [8]- [10], because it is at odds with the traditional models that rely on the notion that molecules move by making an occasional jump from one potential minimum to a nearest neighbouring one via a thermal activation process [11]. In a recent work [12], we have suggested that these long, even Lévy-like, motions can be described by ordinary Langevin dynamics in the low-friction regime and do not require the input of any extraordinary assumptions about the driving fluctuations. Since most of the theoretical work in the past has focused (usually for mathematical simplicity) on the overdamped problem, this possibility had not been fully explored.…”

Overdamped thermal ratchets in one and more dimensions. Kinesin transport and protein folding

“…Thus, for those times when it is outside the filament, even though appears statistically to be dragged by the filament as mentioned above, the particle lags behind the centroid, hence effectively producing a wider distribution and a larger second moment. It is interesting to point out that the superdiffusive behavior is a consequence of "standard" noises coupled to a highly nonlinear process and does not arise from an exotic stochastic process [30].…”

A multiscale approach to Brownian motors

“…11], [30] and the references therein. There are various physical systems where Brownian motion in periodic potentials plays a prominent role, such as Josephson junctions [1], surface diffusion [18,34] and superionic conductors [11]. While the system of a Brownian particle in a periodic potential is kept away from equilibrium by an external, deterministic or random, force, detailed balance does not hold.…”