Non-Poisson renewal events and memory

We study two different forms of fluctuation-dissipation processes generating anomalous relaxations to equilibrium of an initial out-of-equilibrium condition, the former being based on a stationary although very slow correlation function and the latter characterized by the occurrence of crucial events, namely, non-Poisson renewal events, incompatible with the stationary condition. Both forms of regression to equilibrium have the same nonexponential Mittag-Leffler structure. We analyze the single trajectories of the two processes by recording the time distances between two consecutive origin recrossings and establishing the corresponding waiting time probability density function (PDF), $\psi \left(t\right)$. In the former case, with no crucial events, $\psi \left(t\right)$ is an exponential, and in the latter case, with crucial events, $\psi \left(t\right)$ is an inverse power law PDF with a diverging first moment. We discuss the consequences that this result is expected to have for the correct interpretation of some anomalous relaxation processes.

Figure 1

Time evolution of $x\left(t\right)$ driven by Eq. (5) with the parameters $H=0.25$ and ${\mathrm{\Delta }}^2=0.3$.

Figure 2

The red and the green curves are the correlation functions of FGN (obtained using the algorithm of Ref. [29]) with $H=0.75$ and $H=0.25$, respectively. The blue (diamonds), the purple (circles), and the pink (stars) curves are the correlation functions of the trajectory $x\left(t\right)$ obtained from GLE of Eq. (5) using FGN with $H=0.75$ as the noise $\xi \left(t\right)$ and increasing ${\mathrm{\Delta }}^2$, namely, ${\mathrm{\Delta }}^2=0.09,0.02$, and 0.9, respectively.

Figure 3

Survival probability for the recrossing of the origin of $x\left(t\right)$ of Eq. (5) for $H=0.25$. The black dots refer to the case ${\mathrm{\Delta }}^2=0.08$, and the blue squares refer to ${\mathrm{\Delta }}^2=0.3$. The brown dashed line is the fitting $\approx e^{-0.11t}$, which is the truncation expected in the long time limit. The red dashed line is the fitting $\approx e^{-0.35t}$, which shows the exponential waiting time PDF of Eq. (67) with $H$ of Eq. (68) replaced by $H^{\prime }=1-H$. Inset: The top green dashed line illustrates the slope predicted by Eq. (65) with ${\mu }_R-1=0.5$. The agreement between numerical data and theory limited to only the very short-time region is explained in the text.

Figure 4

Survival probability for the recrossing of the origin of $x\left(t\right)$ of Eq. (5) for $H=0.4$. The red dots refer to the case ${\mathrm{\Delta }}^2=0.009$, the brown stars refer to ${\mathrm{\Delta }}^2=0.05$, and the pink triangles refer to ${\mathrm{\Delta }}^2=0.5$. The blue dashed line is the fitting $\approx e^{-0.009t}$, and the green dashed line is the fitting $\approx e^{-0.05t}$, which are the truncations expected in the long time limit. The black dashed line is the fitting $\approx e^{-0.44t}$, which shows the exponential waiting time PDF of Eq. (67) with $H$ of Eq. (68) replaced by $H^{\prime }=1-H$. Inset: The top green dashed line illustrates the slope predicted by Eq. (66) with ${\mu }_R-1\approx 0.63$.

Figure 5

Survival probability for the recrossing of the origin of $x\left(t\right)$ of Eq. (5) for $H=0.8$. The red triangles refer to ${\mathrm{\Delta }}^2=0.009$, and the green dots refer to ${\mathrm{\Delta }}^2=0.61$. The black dashed line is the fitting $\approx e^{-0.055t}$, which is the exponential truncation expected in the long-time limit. The orange dashed line is the fitting $\approx e^{-0.61t}$, which shows the exponential waiting time PDF of Eq. (67) with $H$ of Eq. (68) replaced by $H^{\prime }=1-H$. Inset: The blue dashed line is the slope corresponding to Eq. (66), ${\mu }_R-1\approx 0.22$.

Figure 6

Survival probability for the recrossing of the origin for different values of the fiction $\omega$. From the top: The blue dots represent the numerical data for $\omega =0.001$, and the red dashed line is the fitting with scaling ${\mu }_R-1\approx 0.35$; the brown triangles represent numerical data for $\omega =1$, and the green dashed line is the fitting with scaling ${\mu }_R-1\approx 0.73$. We use the theoretical predictions of Eq. (69) for $\omega =1$ and of Eq. (70) for $\omega =0.001$.

Figure 7

Correlation function given by Eq. (99) of the permanence times of FGN (obtained from Ref. [29]) with $H=0.2$. The blue upper blue envelope goes down as $t^{-1.59}$.

Figure 8

Correlation function given by Eq. (99) of the permanence times of FGN (obtained from Ref. [29]) with $H=0.75$. The upper red envelope goes down as $t^{-0.5}$, and the lower green envelope goes as $-t^{-0.5}$.