Arrival time distribution for a driven system containing quenched dichotomous disorder

We study the arrival time distribution of overdamped particles driven by a constant force in a piecewise linear random potential which generates the dichotomous random force. Our approach is based on the path integral representation of the probability density of the arrival time. We explicitly calculate the path integral for a special case of dichotomous disorder and use the corresponding characteristic function to derive prominent properties of the arrival time probability density. Specifically, we establish the scaling properties of the central moments, analyze the behavior of the probability density for short, long, and intermediate distances. In order to quantify the deviation of the arrival time distribution from a Gaussian shape, we evaluate the skewness and the kurtosis.

Figure 1

Schematic representation of (a) the piecewise linear random potential $U\left(x\right)$ and (b) the corresponding dichotomous random force $g\left(x\right)=-dU\left(x\right)∕dx$ as functions of the coordinate $x$.

Figure 2

(Color online) Theoretical and simulated probability density of the arrival time for $x=2.4$. The solid line (red online) and histogram represent the analytical result (5.1) and the numerical simulation of the arrival time (2.1), respectively. The parameters of the force field are chosen as $f=1$, $g=0.3$, and $\lambda =1$. The vertical arrow depicts the $\delta$-singular contribution to $P_x\left(t\right)$.

Figure 3

Short-distance behavior of the probability density of the arrival time $P_x\left(t\right)$ at (a) $x={10}^{-3}$ and (b) $x=0.2$. The other parameters are the same as those in Fig. 2.

Figure 4

(Color online) Plots of the probability density of the arrival time for different distances from the origin of the coordinate system. The vertical surfaces (green online) depict the regular part of $P_x\left(t\right)$. In order to visually demonstrate that the intensity of the $\delta$-singular part of $P_x\left(t\right)$ exponentially decreases with $x$, we depicted the length of the vertical arrows (red online) in the form $0.8+2e^{-\lambda x}$. For the convenience of comparison, the force field characteristics are chosen as in Figs. 2, 3.

Figure 5

Plots of the skewness $s\left(x\right)$ (dashed line) and the kurtosis $k\left(x\right)$ (solid line) of the arrival time probability density vs the normalized distance $\lambda x$.