Statistical properties of functionals of the paths of a particle diffusing in a one-dimensional random potential

We present a formalism for obtaining the statistical properties of functionals and inverse functionals of the paths of a particle diffusing in a one-dimensional quenched random potential. We demonstrate the implementation of the formalism in two specific examples: (1) where the functional corresponds to the local time spent by the particle around the origin and (2) where the functional corresponds to the occupation time spent by the particle on the positive side of the origin, within an observation time window of size $t$. We compute the disorder average distributions of the local time, the inverse local time, the occupation time, and the inverse occupation time and show that in many cases disorder modifies the behavior drastically.

Figure 1

Schematic plots of $T$ defined by Eq. (2) as a function of $t$, corresponding to four different realizations of the paths [$\left\{x\left(\tau \right)\right\}$, for $0⩽\tau ⩽t$] denoted by $R_1$, $R_2$, $R_3$, and $R_4$ respectively. For fixed $t$ ($t_1$, $t_2$, $t_3$, or $t_4$, shown by vertical dashed lines), $T$ takes different value for different realizations. On the other hand, for a fixed $T$ (horizontal dashed line) the corresponding $t$ is different for different realizations: $t_1$ for $R_1$, $t_2$ for $R_2$, $t_3$ for $R_3$, and $t_4$ for $R_4$.

Figure 2

A classical particle (represented by •) diffusing in a typical realization of the potential $U\left(x\right)=-\mu \mid x\mid +\sqrt{\sigma }B\left(x\right)$, where $B\left(x\right)$ represents the trajectory of a Brownian motion in space with $B\left(0\right)=0$. The three figures are for $\mu =0$, $\mu >0$, and $\mu <0$, respectively. The dashed lines show the potential for $\sigma =0$.

Figure 3

(Color online) Schematic plot showing two realizations of positions of a particle starting from the initial position $x\left(0\right)=x$ evolve to a position $x+\Delta x$ in time $\Delta t$ and then starting from the position $x+\Delta x$ evolve up to time $t+\Delta t$ (color online).

Figure 4

The scaling function $f_1\left(y\right)$ in Eq. (56). The solid line is plotted by using Eq. (62), and the dashed line is plotted by using the limiting form $f_1\left(y\right)\sim \sqrt{2\pi \sigma }{y}^{-3∕2}{e}^{-2y∕\sigma }$ as $y\to \infty$.

Figure 5

The scaling function $f_2\left(y\right)$ in Eq. (70), plotted by using Eq. (75). $\nu =\mid \mu \mid ∕\sigma$.

Figure 6

(Color online) The PDF’s of the inverse local time for stable $(\mu =-1∕2)$, flat $(\mu =0)$, and unstable $(\mu =1∕2)$ potentials, plotted using Eq. (80) and $T=2$.

Figure 7

The scaling function $g_1\left(y\right)$ in Eq. (86). The solid line is plotted by using Eq. (88), and the dashed line is plotted by using the limiting forms $g_1\left(x\right)\approx \sqrt{2\pi \sigma }{x}^{-1∕2}\exp (-2∕\sigma x)$ for small $x$ and $g_1\left(x\right)\sim 2\ln \left(\sigma x\right)∕x$ for large $x$.

Figure 8

The scaling functions $g_2\left(x\right)$ in Eq. (93) plotted by using Eq. (95). $\nu =\mid \mu \mid ∕\sigma$.

Figure 9

The scaling functions $f_0\left(x\right)$ in Eq. (139) plotted by using Eq. (140).

Figure 10

The PDF of the inverse occupation time for simple Brownian motion.

Figure 11

The scaling functions $g_0\left(x\right)$ in Eq. (172) plotted by using Eq. (175).