Analytically solvable model of a driven system with quenched dichotomous disorder

We perform a time-dependent study of the driven dynamics of overdamped particles that are placed in a one-dimensional, piecewise linear random potential. This setup of spatially quenched disorder then exerts a dichotomous varying random force on the particles. We derive the path integral representation of the resulting probability density function for the position of the particles and transform this quantity of interest into the form of a Fourier integral. In doing so, the evolution of the probability density can be investigated analytically for finite times. It is demonstrated that the probability density contains both a $\delta$-singular contribution and a regular part. While the former part plays a dominant role at short times, the latter rules the behavior at large evolution times. The slow approach of the probability density to a limiting Gaussian form as time tends to infinity is elucidated in detail.

Figure 1

Model potential and force field. (a) Schematic representation of the piecewise varying 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 particle coordinate $x$.

Figure 2

(a) Behavior of the average velocity $⟨v⟩$ and (b) of the effective diffusion coefficient $D_{\mathrm{eff}}$ versus the external force $f$ for the chosen parameter values $\lambda =g=1$.

Figure 3

Time evolution of the probability density. Depicted are the plots of the probability density $P_t\left(x\right)$ derived from Eqs. (3.17, 3.18, 3.19, 3.20) for the evolution time (a) $t=0.75$, (b) $t=2$, and (c) $t=5$. The values of the other parameters are set at $f=1$, $g=0.3$, and $\lambda =1$. The vertical arrows depict the singular parts of decreasing weight of $P_t\left(x\right)$.

Figure 4

(Color online) Theory versus numerics. We show the probability density $P_t\left(x\right)$ as a function of the coordinate $x$ for $f=1$, $g=0.3$, $\lambda =1$, and time $t=4$. The solid line (red online) represents the theoretical result obtained from Eqs. (3.17, 3.18, 3.19, 3.20), while the histogram refers to our numerical simulation of Eq. (2.1).

Figure 5

Deviation from Gaussian behavior. The kurtosis $k\left(t\right)$ of the probability density is depicted on a linear $\left[k\left(t\right)\right]$ versus logarithmic $\left[t\right]$ scale as a function of the evolution time $t$ for $f=1$, $g=0.3$, and $\lambda =1$.