Transport and diffusion of overdamped Brownian particles in random potentials

We present a numerical study of the anomalies in transport and diffusion of overdamped Brownian particles in totally disordered potential landscapes in one and in two dimensions. We characterize and analyze the effects of three different disordered potentials. The anomalous regimes are characterized by the time exponents that exhibit the statistical moments of the ensemble of particle trajectories. The anomaly in the transport is always of the subtransport type, but diffusion presents a greater variety of anomalies: Both subdiffusion and superdiffusion are possible. In two dimensions we present a mixed anomaly: subdiffusion in the direction perpendicular to the force and superdiffusion in the parallel direction.

Figure 1

Temporal evolutions of the velocity (top) and diffusion (bottom). Simulation of the Langevin equation were performed for 100 particles and 100 realizations of the potential with Gaussian correlation.

Figure 2

Transport (top) and diffusion (bottom) exponents for different distributions of disorder. The parameters are the same as in Fig. 1, with $\varepsilon =1/3$ and $2/3$ for the power-law case. Wide horizontal colored bars (shown from bottom to top are Exp, Ga, Pl 1/3, and Pl 2/3 disorder) indicate each dynamical regime (as described in the text).

Figure 3

The top shows correlation functions $g\left(\right|z\left|\right)$ for the Gaussian (Ga), double-sided exponential with discretization $\Delta =0.1$ (Exp), and power law with $\varepsilon =1/3$ (Pl $1/3$) and $\varepsilon =2/3$ (Pl $2/3$). Colored circles at $g\left(\right|z\left|\right)=0.25$ provide a measure of the effective correlation lengths. The bottom shows potential landscapes for each case; $\lambda =1$ for all cases. A vertical shift is implemented for better visualization (2, 1, and $-$2 potential units for Pl $2/3$, Pl $1/3$, and Exp, respectively).

Figure 4

Displacement distribution of the particles $P\left(\Delta z\right)$ for different combinations of displacements and diffusions. $\mathcal{F}=0.6$ (a), $1.8$ (b), $2.6$ (c), and $4.0$ (d) at $\tau =20000$. It is important to pay attention to the horizontal scale in each plot. Plot (d) includes the representation of a Gaussian function of zero mean ($\mu =0$) and variance ${\sigma }^2=2D\tau$ ($D=\mathcal{T}=0.1$, $\tau =20000$).

Figure 5

Parallel (top) and perpendicular (bottom) mean velocities of 400 particles, averaged over 20 realizations of the random potential surface, with $\theta =\arctan (100/N\Delta )$.

Figure 6

Parallel (top) and perpendicular (bottom) diffusion coefficients. The parameters are the same as in Fig. 5.

Figure 7

The top shows particle positions with respect to initial conditions at different times for $\mathcal{F}=0.6$ and, from bottom to top, $\tau =250,1000,4000,16000$. The bottom shows the particle displacement distribution at $\tau =1000$ along the parallel (top) and perpendicular (bottom) directions relative to the force.

Figure 8

The top shows particle positions relative to the initial condition at different times for $\mathcal{F}=1.5$ and, from bottom to top, $\tau =250,1000,2000,4000$. The bottom shows the particle displacement distribution at $\tau =1000$ along the parallel (top) and perpendicular (bottom) directions relative to the direction of the force.

Figure 9

The top shows particle positions relative to the initial conditions at different times for $\mathcal{F}=2.0$ and, from bottom to top, $\tau =250,1000,2000,4000$. The bottom shows the particle displacement distribution at $\tau =1000$ along the parallel (top) and perpendicular (bottom) directions relative to the direction of the force.

Figure 10

The top shows the particle cloud at different times for $\mathcal{F}=3.0$ and, from bottom to top, $\tau =250,400,550,700$. The bottom shows the particle displacement distribution at $\tau =1000$ along the parallel (top) and perpendicular (bottom) directions relative to the direction of the force.