Weakly anomalous diffusion with non-Gaussian propagators

A poorly understood phenomenon seen in complex systems is diffusion characterized by Hurst exponent $H\approx 1/2$ but with non-Gaussian statistics. Motivated by such empirical findings, we report an exact analytical solution for a non-Markovian random walk model that gives rise to weakly anomalous diffusion with $H=1/2$ but with a non-Gaussian propagator.

Figure 1

The richness of the $f$-$p$ phase diagram. The parabola ($\delta =1/2$) separates the two escape regimes. The dashed lines ($\delta =0$) separate the localized and slow escape regimes. The vertical dotted line ($p=1/2$) separates the log-periodic (L) region on the left and the non-log-periodic region on the right.

Figure 2

Examples of solutions for $\langle x\left(t\right)\rangle$ within the log-periodic region ($p<1/2$). (a) Exact and experimental solutions, normalized for visual aid. The experimental observations were obtained with ${10}^3$ runs with $6\times {10}^6$ total time units each. (b) The escape regime associated with $0<\delta <1/2$. (c) and (d) The localized regimes. The log-periodic solution for $\delta <0$ in (d) fades away in the asymptotic limit.

Figure 3

(a) Gaussian probability plot of the speed of the random walker measured after ${10}^8$ time units, calculated using $v=[x({10}^8+10)-x\left({10}^8\right)]/10$ vs the normal order statistic medians $m$. Three cases are shown: $f=0.1$ (solid circles, $\delta \approx 0.90$), $f=0.7$ (squares, $\delta \approx 0.37$), and $f=0.9$ (open circles, $\delta \approx -1.12$), all for $p=0.1$. They correspond, respectively, to the fast escape, slow escape, and normal localized regimes. The straight line for $\delta <0$ indicates Gaussian behavior. (b) The persistence length distributions corresponding to the same parameters ($f,p$) as above. The inset shows the corresponding Gaussian probability plot for position $\langle x\rangle$ ($f=0.1$) vs the normal order statistics medians $m$. The propagator is visibly non-Gaussian. (c) The difference ${\Delta }_3=3\langle x\rangle [\langle x^2\rangle -{\langle x\rangle }^2]-[\langle x^3\rangle -{\langle x\rangle }^3]$ as a function of $\ln t$. This difference is equal to zero for a Gaussian distribution. The inset shows ${\Delta }_3$ for $\alpha >0$. (d) The difference ${\Delta }_4=3[{\langle x^2\rangle }^2-{\langle x\rangle }^4]-[\langle x^4\rangle -{\langle x\rangle }^4]$ as a function of $t$ in a log-log plot. This also is identically equal to zero for Gaussian distributions. The straight line represents a linear fit. The inset shows the form of the distribution of the position, but any resemblance to a Gaussian distribution is misleading (see text).