Probability density of the fractional Langevin equation with reflecting walls

We investigate anomalous diffusion processes governed by the fractional Langevin equation and confined to a finite or semi-infinite interval by reflecting potential barriers. As the random and damping forces in the fractional Langevin equation fulfill the appropriate fluctuation-dissipation relation, the probability density on a finite interval converges for long times towards the expected uniform distribution prescribed by thermal equilibrium. In contrast, on a semi-infinite interval with a reflecting wall at the origin, the probability density shows pronounced deviations from the Gaussian behavior observed for normal diffusion. If the correlations of the random force are persistent (positive), particles accumulate at the reflecting wall while antipersistent (negative) correlations lead to a depletion of particles near the wall. We compare and contrast these results with the strong accumulation and depletion effects recently observed for nonthermal fractional Brownian motion with reflecting walls, and we discuss broader implications.

Figure 1

Determination of a suitable value of the time step $\mathrm{\Delta }t$. Main panel: probability density $P(x,t)$ of the particle position at time $t=20972$ for correlation exponent $\alpha =1.5$, temperature $T=1$, and several values of the time step $\mathrm{\Delta }t$ (the particles start at the origin at $t=0$). Inset: Mean-square velocity $\langle v^2\rangle$ at long times as a function of time step $\mathrm{\Delta }t$ (the data are averages of $\langle v^2\rangle$ over the time intervals ${10}^4–{10}^5$ or ${10}^4–{10}^6$ depending on $\mathrm{\Delta }t$).

Figure 2

FLE confined to the interval $(-L,L)$ with $L=20$. The particles start from rest at the origin, $x=0$, at time $t=0$. Main panel: Mean-square displacement $\langle x^2\rangle$ vs time $t$ for several values of the correlation exponent $\alpha$. The data are averages over 10 000 trajectories. The solid lines are fits to the power law (5), $\langle x^2\rangle \sim t^{2-\alpha }$. The dashed line marks the value expected for a uniform distribution of the particles over the interval, $L^2/3=400/3$. Inset: Mean-square velocity $\langle v^2\rangle$ vs time $t$ for the same trajectories.

Figure 3

Probability density $P(x,t)$ at different times $t$ for the FLE restricted to the interval $(-L,L)$ with $L=20$ for $\alpha =1.5$. The data are averages over 10 000 trajectories. For improved statistics, the data are also averaged over the time interval from $0.8t$ to $t$. The dashed line corresponds to a uniform distribution $P\left(x\right)=1/\left(2L\right)$.

Figure 4

FLE on the semi-infinite interval $(0,\infty )$. The particles start from rest at the origin, $x=0$, at time $t=0$. Main panel: Mean-square displacement $\langle x^2\rangle$ and mean-square velocity $\langle v^2\rangle$ vs time $t$ for $\alpha =1.6$. The data are averages over ${10}^6$ trajectories. The (red) horizontal line marks the value $\langle v^2\rangle =1$ expected from the fluctuation dissipation relation. The fit line for $\langle x^2\rangle$ represents a fit to the power law (5), $\langle x^2\rangle \sim t^{2-\alpha }$. Inset: Mean-square displacement $\langle x^2\rangle$ vs time $t$ for several $\alpha$. The solid lines are fits to $\langle x^2\rangle \sim t^{2-\alpha }$.

Figure 5

Probability densities $P(x,t)$ for the free FLE and the FLE confined to the interval $(0,\infty )$ at time $t=328$ and $\alpha =1.5$. The particles start at $x=0$ at time $t=0$. The data are averages over ${10}^6$ runs. The solid line shows the behavior naively expected for the confined case, viz., a half-Gaussian of the same width but twice the magnitude as the unconfined probability density.

Figure 6

Left: Probability densities $P(x,t)$ for the FLE confined to the interval $(0,\infty )$ at different times $t$ for $\alpha =1.5$. The particles start at $x=0$ at time $t=0$. The data are averages over ${10}^6$ to $5\times {10}^6$ runs. Right: Scaling plot $\sigma \left(t\right)P(x,t)$ versus $x/\sigma \left(t\right)$ of the same data. The solid line is a fit of the large-$x$ tail of the distribution to a Gaussian.

Figure 7

Probability density data of Fig. 6, plotted as $P$ vs $x^2$ such that a Gaussian distribution gives a straight line. The solid lines are Gaussian fits of the large-$x$ behavior.

Figure 8

Scaled probability density $\sigma \left(t\right)P(x,t)$ vs $x/\sigma$(t) at time $t=167772$ for several values of the correlation exponent $\alpha$. The data represent averages of $3\times {10}^6$ trajectories.

Figure 9

Scaled probability density $\sigma \left(t\right)P(x,t)$ versus $x/\sigma \left(t\right)$ of particles starting at $x=0$ at $t=0$ at several times $t$ for $\alpha =0.5$ and noise amplitude $K_{\alpha }=1$. The data represent averages of ${10}^6$ trajectories. Inset: Time evolution of the mean-square displacement $\langle x^2\rangle$ and mean-square velocity $\langle v^2\rangle$. The solid lines are fits to $\langle x^2\rangle \sim t^{1.5}$ and $\langle v^2\rangle =\mathrm{const}$, respectively.

Figure 10

Stationary probability density $P\left(x\right)$ on the interval $(-L,L)$ with $L=40$ for $\alpha =1.2$. Shown are data for the FLE that fulfills the fluctuation dissipation theorem (3) and a generalized Langevin equation with long-time correlated random forces but instantaneous damping that violates it. The data are averages over 10 000 trajectories.

Figure 11

Repulsive potential $V\left(x\right)$ modeling a reflecting wall at the origin, $x=0$, for several values of the decay constant $\lambda$.

Figure 12

Probability density $P(x,t)$ at different times $t$ for the FLE restricted to the interval $(-L,L)$ with $L=15$ for $\alpha =1.5$. The data are averages over 4000 trajectories. The reflecting walls are implemented via the reflection condition (A1). The time step $\mathrm{\Delta }t=0.01$, as in the main part of the paper.

Figure 13

Probability density $P(x,t)$ at time $t=167772$ for the FLE on the interval $(-15,15)$ for several values of the integration time step $\mathrm{\Delta }t$. The data are averages over 4000 to 10 000 trajectories. The reflecting walls are implemented via the reflection condition (A1). Inset: Extrapolation of $P(0,t)$ and $P(-15,0)$ to time step $\mathrm{\Delta }t=0$.