Aging ballistic Lévy walks

Aging can be observed for numerous physical systems. In such systems statistical properties [like probability distribution, mean square displacement (MSD), first-passage time] depend on a time span $t_a$ between the initialization and the beginning of observations. In this paper we study aging properties of ballistic Lévy walks and two closely related jump models: wait-first and jump-first. We calculate explicitly their probability distributions and MSDs. It turns out that despite similarities these models react very differently to the delay $t_a$. Aging weakly affects the shape of probability density function and MSD of standard Lévy walks. For the jump models the shape of the probability density function is changed drastically. Moreover for the wait-first jump model we observe a different behavior of MSD when $t_a\ll t$ and $t_a\gg t$.

Figure 1

Sample trajectories of Lévy walks. Forward renewal time - $W_{t_a}$, backward renewal time - $V_{t_a}$.

Figure 2

Lines—PDFs of different Lévy walks, $t_a=1, t=1$, and $\alpha =0.4$. Pluses and crosses—PDFs estimated using Monte Carlo methods from ${10}^6$ trajectories of LW and wait-first LW, respectively.

Figure 3

Lines—ensemble averaged MSD $(t_a,t), t_a=1$, and $\alpha =0.4$. Pluses and crosses—ensemble averaged MSDs estimated using Monte Carlo methods from ${10}^4$ trajectories of LW and wait-first model, respectively.

Figure 4

Lines—densities of aging $Z_{t_a}\left(1\right)$ and non-aging $Z\left(1\right)$ jump-first models. Aging time $t_a=1, \alpha =0.4$. Pluses—the density estimated using MC methods from ${10}^6$ trajectories.