Asymptotic densities of ballistic Lévy walks

We propose an analytical method to determine the shape of density profiles in the asymptotic long-time limit for a broad class of coupled continuous-time random walks which operate in the ballistic regime. In particular, we show that different scenarios of performing a random-walk step, via making an instantaneous jump penalized by a proper waiting time or via moving with a constant speed, dramatically effect the corresponding propagators, despite the fact that the end points of the steps are identical. Furthermore, if the speed during each step of the random walk is itself a random variable, its distribution gets clearly reflected in the asymptotic density of random walkers. These features are in contrast with more standard nonballistic random walks.

Figure 1

Trajectories of the three stochastic models investigated in the text. Lévy walk (blue), wait-first (green), and jump-first (red) models produce trajectories which take different paths but pass through the same end points on the $(x,t)$ plane. Therefore, only the very last, uncompleted step distinguishes these three models. $t_n$ is the last renewal event before the measurement time $t$, and $t-t_n$ is called the backward recurrence time.

Figure 2

Backward and forward recurrence times. Given a measurement time $t$, the backward recurrence time ${\tau }_b$ is the time elapsed since the time $t_n$ the last event took place. The forward recurrence time ${\tau }_f$ is the time span between $t$ and the time $t_{n+1}$ at which the next renewal will take place.

Figure 3

Propagators of two jump models in the ballistic scaling regime. The rescaled propagators for (a) the wait-first and (b) the jump-first models from Eqs. (27) and (26) are shown for different values of the exponent $\gamma$ determining the power-law tail of the waiting-time and jump-length distributions.

Figure 4

One-sided jump models. In this particular case only jumps to the right are allowed. One-sided wait-first (a) and jump-first models (b) [Eqs. (29) and (28)] are shown for different values of the power-law tail exponent $\gamma$ governing the flight and jump-length distributions. These results are related to the statistics of the backward and forward recurrence times, as discussed in the text.

Figure 5

Theory and simulations for the wait-first model [Eq. (27)] (solid line, triangles, red), the jump-first model [Eq. (26)] (dotted line, diamonds, blue) and the two-state velocity model Eq. (38) (dashed line, squares, green). $\gamma =1/2$, scaling variable $y=x/\left(v_{0}t\right)$.

Figure 6

Scaling functions of the random walks with random velocities model. Results for four different velocity pdfs of the walking particles are depicted: two-state velocity $v=\pm v_0$ [Eq. (39)] (dash-dot, circles, gray), uniform on an interval [Eq. (42)] $[-v_0,v_0]$ (dashed, squares, green), Gaussian [Eq. (35)] (dotted, diamonds, blue), and Cauchy [Eq. (45)] (full line, triangles, red); $\gamma =1/2$. The result for the Gaussian velocity pdf was obtained using mathematica. Theory and simulations nicely match without fitting.