Anomalous diffusion in non-Markovian walks having amnestically induced persistence

We report numerically and analytically estimated values for the Hurst exponent for a recently proposed non-Markovian walk characterized by amnestically induced persistence. These results are consistent with earlier studies showing that log-periodic oscillations arise only for large memory losses of the recent past. We also report numerical estimates of the Hurst exponent for non-Markovian walks with diluted memory. Finally, we study walks with a fractal memory of the past for a Thue-Morse and Fibonacci memory patterns. These results are interpreted and discussed in the context of the necessary and sufficient conditions for the central limit theorem to hold.

Figure 1

(Color online) (a) Complete phase diagram showing the four phases [9], plotted according to the exact solutions given by Eqs. (11, 21). The dashed line $f_0\left(p\right)$ delineates the threshold for log-periodicity and cleaves the nonpersistent regime into two [Eq. (7)]. The dark region represents numerical results with $H\gtrsim 0.85$, having strong log-periodicity, because $a_0$ and ${\kappa }_0$ are small in this region. (b), (c), and (d) show first, second moments and correlation, respectively, for $p=0.1$ and $f=0.15$. We normalized the first moment by $t^{0.5}$ and the correlation by $t^{2H}$, while was taken the logarithm of the second moment to the base two. Points represent the simulations and continuous lines fittings with the functions shown. The total number of steps (maximum time) was $16777216$, and the number of runs was 5000 to obtain the averages. We can easily verify the relations (16, 17) in the text. Note the consistency between the parameters ${\text{B}}_2=B\text{ln}\left(2\right)$, $C$ and $H$, because the fittings were accomplished independently; also, we can verify that $H=\delta$ considering the precision of the numbers.

Figure 2

Much loss of memory (except $f=0$) and small values of $p$, drives to large oscillations to the second moment. For small values of $f$, the walker tends to go far away, but his (or her) walk is dampened for small values of $p$, that does the walker prefer to make the opposite of what he (or she) did in a past time, causing oscillations to the moments. Observe that oscillations do not exist for $f=0.1$ and $p=0.9$, the walker will always want to do what he (or she) did in the past, without so much indecision. Averages were accomplished with 1000 runs reaching $3276800$ total time units each.

Figure 3

(a) a semilog plot of the displacement $⟨x_t⟩$ as function of time for $p=0.1$, and several values of $f$. (b) Shows with details that big losses of recent memories (small values of $f$) drive to large amplitudes of variation of position and large log-periodic oscillations.

Figure 4

This figure shows a plot of $H$ versus $f$ (fraction of the old memory), for several values of $p$. Here analytical results are also shown (full lines) and numeric (symbols). For $p<0.75$ two regimes are possible, according to $f$: normal diffusion $\left(H=0.5\right)$ and superdiffusive $\left(H>0.5\right)$. For $p\ge 0.75$ exists only superdiffusive behavior (for $p=0.75$ and $f=1$ is marginally superdiffusive), for any value of $f$. Averages were accomplished with 1000 runs and $3276800$ total time units each.

Figure 5

This figure shows the Hurst exponent $H$ versus $p$, the probability of the walker to accept the decision taken at time $t^{\prime }$ $\left(0<t^{\prime }<t\right)$. The symbols represent the results of the simulations, while the full lines represent the analytical results; 1000 runs were accomplished for several values of $f$ for the walker that forgot $\left(1-f\right)t$ of the recent past, for a total time of $3276800$ steps. For the case $f=0$, the walker does not remember anything, except the first step; his (or her) behavior is superdiffusive, except to $p=0.5$, where the exponent $H$ drops abruptly to 0.5, according to the exact result. For small values of $f$ (0.1 and 0.2), even for $p<0.5$, we can see persistence $\left(H>0.5\right)$. For $f=1.0$ (full memory) we have the well known analytical case showing a good correlation with the simulations; the persistent region just appears for $p>3/4$. Overall, we can see a good correlation between the analytical results and the numerical ones.

Figure 6

$H$ versus $p$ plot. The addition of vacancies (dilution) in the total memory of the walker did not provide significant changes in the Hurst exponent. For a dilution of $95\mathrm{\%}\left(d=0.95\right)$ and several $f$ values, the results obtained are almost equivalent to the zero dilution. The critical case happens for $d=1.0$ (totally diluted memory), in which the walker just reminds the first step, equivalent to $f=0.0$ (no dilution case).

Figure 7

This figure shows Hurst exponent $H$ against $p$ for several $f$: (a) $f=0.1$, (b) $f=0.4$, (c) $f=1.0$, and several random dilutions. We can compare the case without dilution $\left(d=0.0\right)$ with $d=0.2$, 0.6 and 0.8 to conclude that larger dilutions just may cause a small increase in $H$ when $p>0.5$; however, the results might be biased by finite size effects, so the curves might be collapsed. Averages were accomplished with 1000 runs and $3276800$ total time units each.

Figure 8

This figure shows the inclusion of predefined memory to the walker, (a) satisfying the sequence of Thue-Morse that is equivalent to a dilution of 50%. (b) The Fibonacci sequence, where each time given by the number of the sequence corresponds to a point of the memory accessible to be remembered. For the total time of $3276800$ steps, using the sequence of Fibonacci, only 33 more localized positions of memory exist in the initial instants, what is equal to a dilution close to 100%. We can see that the position of the memory seems not to have any relevance in the Hurst exponent. Averages were accomplished with 1000 runs and $3276800$ total time units each.

Figure 9

This figure shows that the distribution of persistence lengths (normalized by the average), typically follows an exponential distribution. In (a), $f=1.0$, the data collapses in a single straight line, showing no appreciable variation in the inclination with different values of $p$. In (b), $f=0.1$, we note a significant deviation from exponential distribution to $p=0.0$. In (c) and (d), we see that the inclusion of a dilution of 95% did not affect the distributions significantly. Averages were accomplished with 1000 runs and $1000000$ total time units each.