Subdiffusion in an external potential: Anomalous effects hiding behind normal behavior

We propose a model of subdiffusion in which an external force is acting on a particle at all times not only at the moment of jump. The implication of this assumption is the dependence of the random trapping time on the force with the dramatic change of particles behavior compared to the standard continuous time random walk model in the long time limit. Constant force leads to the transition from non-ergodic subdiffusion to ergodic diffusive behavior. However, we show this behavior remains anomalous in a sense that the diffusion coefficient depends on the external force and on the anomalous exponent. For quadratic potential we find that the system remains non-ergodic. The anomalous exponent in this case defines not only the speed of convergence but also the stationary distribution which is different from standard Boltzmann equilibrium.

Figure 1

Variance $\sigma \left(t\right)=\langle x^2\rangle -{\langle x\rangle }^2$ of ensemble calculated with $\mu =0.3$ and $p(x,0)=\delta \left(x\right)$. In all simulations we use $\nu =1,a=0.1$, and ${\tau }_0=1$. For $F=0$ (lowest curve) the variance grows as $D_{\mu }{t}^{\mu }$ (dashed-dotted line) in the limit $t\to \infty$. Constant force $F=0.0001,F=0.001$, and $F=0.01$ (curves from bottom to top on the RHS of the figure) leads to the transition from subdiffusive behavior for short times to normal diffusion in the long time limit, $\sigma \to 2D_{F}t$ (dashed lines), with $D_F$ given by Eq. (47). Intermediate asymptotic of the variance is fitted by the power-law (dashed-dotted lines, see the text). The inset shows transition of densities from subdiffusive form for short times to the Gaussian shape for long times caused by the constant force $F=0.0001$. Densities were calculated at $t={10}^3,{10}^4,5\times {10}^4$, and ${10}^5$.

Figure 2

Main figure: The action of the constant force $F=0.001$. Time-averaged variance ${\sigma }_T(\Delta ,T)$ calculated for 30 individual trajectories each of a length $T=3\times {10}^4$ (each curve corresponds to a single trajectory) with $\mu =0.7$. Bold solid curve (red) represents the average over 30 trajectories. The time averaged variance is proportional to the time lag ${\sigma }_T(\Delta ,T)\sim \Delta$ as indicated by dashed-dotted line. The dashed (blue) line represents the long time asymptotic of the ensemble averaged variance and is given by $2D_{F}t$ with $D_F$ defined by Eq. (47) (see the text). The equality between the time and the ensemble averaged variance reflects the ergodic behavior of the system under the action of constant external force. Inset: Contrast this with the behavior in the quadratic potential $U\left(x\right)=\kappa x^2/2$ with $\kappa =0.001$. Each individual curve corresponds to the time averaged variance of a single trajectory with $\mu =0.7$. The bold solid (red) line represents average over 30 trajectories. The dashed (black) curve is the numerically calculated ensemble averaged variance. For small $t$ the variance has power-law behavior $\sigma \left(t\right)\sim t^{\mu }$ and in the long time limit it saturates due to the confined space. The time averaged variance is proportional to the time lag ${\sigma }_T(\Delta ,T)\sim \Delta$ (as indicated by dashed-dotted line) before it also saturates to a different than the ensemble averaged variance value. This indicates that the behavior of the system in the quadratic potential is non-ergodic.

Figure 3

Density $p(x,t)$ in the quadratic potential $U\left(x\right)=k{x}^2,k=0.001$ calculated with the anomalous exponent $\mu =0.5$ at time $t={10}^5$ and $t={10}^6$. Two densities overlap indicating convergence to stationary solution $p_{st}$. Clearly $p_{st}$ is non-Boltzmann and is well described (accept for the central part) by the long-wave asymptotic Eq. (52) shown by the dashed line. To distinguish the form of the stationary solution $exp(-A|x{|}^{1+\mu })$, we show the Boltzmann equilibrium $exp(-B{x}^2)$ and the function $exp(-C|x\left|\right)$ (dashed-dotted curves) to guide the eye $(A,B,C$ are positive constants). Note that the central part of $p_{st}$ has distinct cusp at $x=0$ where the force vanishes.