Diffusion coefficient in periodic and random potentials

Transport and diffusion of particles on modulated surfaces is a nonequilibrium problem which is receiving a great deal of attention due to its technological applications, but analytical calculations are scarce. In earlier work, we developed a perturbative approach to begin to provide an analytic platform for predictions about particle trajectories over such surfaces. In some temperature and forcing regimes, we successfully reproduced results for average particle velocities obtained from numerical simulations. In this paper, we extend the perturbation theory to the calculation of higher moments, in particular the diffusion tensor and the skewness. Numerical simulations are used to check the domain of validity of the perturbative approach.

Figure 1

Approximate (solid), exact (dotted), and numerical (symbols) 1D diffusion results for $D/\mathcal{T}$ as function of external force $F$ at temperatures (a) $\mathcal{T}=0.2$, (b) $\mathcal{T}=0.5$, and (c) $\mathcal{T}=1.0$.

Figure 2

Diffusion coefficient $D/\mathcal{T}$ as a function of force $F$ for a random one-dimensional potential with Gaussian energy spectrum. Parameters: $\epsilon =0.188$, $\gamma =0.15$, and temperature $\mathcal{T}=1.0$. Analytic prediction (line) and numerical simulations (points). Inset: Potential correlation for a periodic case (solid line) and a Gaussian random one (dashed line).

Figure 3

Diffusion coefficients $D_{\parallel }/\mathcal{T}$ (solid and open) and $D_{\perp }/\mathcal{T}$ (dashed and filled) as functions of $F$, at temperature $\mathcal{T}=1.0$, and with the forcing vector at angles (a) $\theta =0$, (b) $\theta ={\text{tan}}^{-1}\sqrt{2}$, and (c) $\theta =\pi /2$. Lines are results using our analytical approximation and points are numerical simulation results.

Figure 4

Diffusion coefficients $D_{\parallel }/\mathcal{T}$ (solid and open) and $D_{\perp }/\mathcal{T}$ (dashed and filled) as functions of angle $\theta$ at temperature $\mathcal{T}=1.0$ and forcing magnitude $F=10$. Lines are approximate analytic results and points are numerical simulation results.

Figure 5

Diffusion coefficients $D_{\parallel }/\mathcal{T}$ (solid and open) and $D_{\perp }/\mathcal{T}$ (dashed and filled) as functions of forcing magnitude $F$ at temperature $\mathcal{T}=0.2$ and angle $\theta ={\text{tan}}^{-1}\sqrt{2}$. Lines are analytic results and points are numerical results.

Figure 6

Diffusion coefficients $D_{\parallel }/\mathcal{T}$ (solid and open) and $D_{\perp }/\mathcal{T}$ (dashed and filled) as functions of angle $\theta$ at temperature $\mathcal{T}=0.2$ and forcing magnitude $F=10$. Lines are approximate analytic results and points are numerical results.

Figure 7

The third moment coefficient $S$ as a function of $F$ for the one-dimensional periodic case at temperature $\mathcal{T}=1.0$. Inset: time evolution of the third moment of the displacement for different forces (a) $F=1$ (black) and $F=5$ (gray), (b) $F=8$, and (c) $F=10$.