Minimal model for short-time diffusion in periodic potentials

We investigate the dynamics of a single, overdamped colloidal particle, which is driven by a constant force through a one-dimensional periodic potential. We focus on systems with large barrier heights where the lowest-order cumulants of the density field, that is, average position and the mean-squared displacement, show nontrivial (nondiffusive) short-time behavior characterized by the appearance of plateaus. We demonstrate that this “cage-like” dynamics can be well described by a discretized master equation model involving two states (related to two positions) within each potential valley. Nontrivial predictions of our approach include analytic expressions for the plateau heights and an estimate of the “de-caging time” obtained from the study of deviations from Gaussian behavior. The simplicity of our approach means that it offers a minimal model to describe the short-time behavior of systems with hindered dynamics.

Figure 1

(a) The potential landscape of a particle in a tilted washboard. The period is $a$ and the minimum peak-to-trough height is $\Delta u$. (b) The discrete two-state-per-well (2SW) model in which the state of the system is described by well index $n$ and $\alpha =L,R$, describing whether the particle is found in the left or right side of the well. The particle therefore assumes discrete positions at $x=na+x_{L/R}$, where $x_{\alpha }$ is the displacement of internal state $\alpha$. Transitions between the states are described by a rate equation with rates ${\gamma }^{\pm }$ and ${\Gamma }^{\pm }$ as shown.

Figure 2

Comparison of diffusion coefficient $D={\lim }_{t\to \infty }\frac{1}{2}\frac{d}{dt}{\langle x^2\left(t\right)\rangle }_c$ of Eq. (22) with the exact result of Ref. [10]. The results of the 2SW model agree well with the exact ones in the regime of deep valleys, i.e., $u_0\ll kT$ and $F\ll F_c$, with $F_c$ the force for which potential valleys disappear.

Figure 3

The mean-squared displacement ${\langle x^2\left(t\right)\rangle }_c$ for different amplitudes $u_0$ with no applied force ($F=0$). The lines show the result from the 2SW model, and the symbols those from numerical integration of the SE. For the parameters shown here, the 2SW model reproduces the essential features of the short-time dynamics. The agreement with the numerically exact results increases with increasing well-depth (increasing $u_0$).

Figure 4

Plateau height ${\langle x^2\rangle }_c^{\mathrm{plat}}$ as a function of driving force, $F$, calculated in three different ways. We have defined the plateau height as the value of ${\langle x^2\rangle }_c$ at the time for which $d\left(\ln {\langle x^2\rangle }_c\right)/d\left(\ln t\right)$ reaches its minimum value. The solid curves show the plateau heights obtained from the 2SW theory, the symbols from integrating the SE, and the dashed lines show the approximation of Eq. (26). The 2SW method underestimates the plateau heights slightly, becoming better as $u_0/kT$ becomes larger. Only when the two 2SW results agree with one another does the cumulant really show a well defined plateau.

Figure 5

The kurtosis $\kappa \left(t\right)$ of Eq. (4) as a function of time for the same parameters as in Fig. 3. The dominant feature is a peak which occurs around the crossover time. As before, good agreement between 2SW model and numerics is obtained for $u_0\gg kT$.

Figure 6

The time of the kurtosis peak as a function of driving force $F$ for different values of $u_0$. Again, the agreement is good for deep valleys.