Lévy flights and nonhomogenous memory effects: Relaxation to a stationary state

The non-Markovian stochastic dynamics involving Lévy flights and a potential in the form of a harmonic and nonlinear oscillator is discussed. The subordination technique is applied and the memory effects, which are nonhomogeneous, are taken into account by a position-dependent subordinator. In the nonlinear case, the asymptotic stationary states are found. The relaxation pattern to the stationary state is derived for the quadratic potential: the density decays like a linear combination of the Mittag-Leffler functions. It is demonstrated that in the latter case the density distribution satisfies a fractional Fokker-Planck equation. The densities for the nonlinear oscillator reveal a complex picture, qualitatively dependent on the potential strength, and the relaxation pattern is exponential at large time.

Figure 1

Relaxation function $\chi \left(t\right)$ calculated from Eq. (30) for $\alpha =1.5$, $\beta =0.5$, $\lambda =1$ and a few values of $\theta$.

Figure 2

The time evolution of the density distribution for the subordinated process corresponding to two values of $\gamma$ and calculated for $\alpha =1.5$ and $\theta =1$ from Eq. (6). The curves in the upper part correspond to $\tau =0.05$, 0.3, 1, 5, and 20 (from bottom to top). The straight red lines mark the dependence $x^{-1-\alpha }$ and $x^{-\alpha -\gamma }$. Averaging was performed over ${10}^7$ trajectories.

Figure 3

The relaxation function for $\alpha =1.9$, $\beta =0.5$, $\overline{\gamma }=3$ and three values of $\theta$. The solid lines on the right side mark an exponential, $exp(-c_{\chi }t)$, with $c_{\chi }=0.33$, 0.4 and 0.14 (from left to right). The solid line on the left side, fitted to the data for $\theta =1$, marks a function $t^{-0.28}$. Stars correspond to the case $\overline{\gamma }=4$ and $\theta =1$. Averaging was performed over ${10}^4$ trajectories.

Figure 4

Slope of the density $p_0(x,\tau )$ calculated from Eq. (6) for $\alpha =1.5$, $\beta =0.5$, $\gamma =0.3$, and $\theta =1$ (upper part). The lower part presents the slope as a function of the physical time, calculated from Eq. (37). Lines mark the functions ${\tau }^{-0.08}$ and $t^{-0.049}$ for the upper and lower part, respectively. Each point was obtained from the density evaluated by averaging over ${10}^7$ trajectories.