Field-Induced Dispersion in Subdiffusion

We discuss the response of continuous-time random walks to an oscillating external field within the generalized master equation approach. We concentrate on the time dependence of the two first moments of the walker’s displacement. We show that for power-law waiting-time distributions with $0<\alpha <1$ corresponding to a semi-Markovian situation showing nonstationarity, the mean particle position tends to a constant; namely, the response to the external perturbation dies out. On the other hand, the oscillating field leads to a new additional contribution to the dispersion of the particle position, proportional to the square of its amplitude and growing with time. These new effects, amenable to experimental observation, result directly from the nonstationary property of the system.

Figure 1

The mean displacement $m_1(t)$ (measured in units of $\mu f_0$) in a CTRW model with $\alpha =1/2$ as a function of time for $\omega =1$. Note that for $t$ large the displacement stagnates: the linear response of the system to external sinusoidal field dies.

Figure 2

Shown is the square of ${\sigma }_1^2(t)$ (measured in units of ${\mu }^2{f}_0^2$) as a function of $t$ for $\alpha =1/2$ and $\omega =1$.