Population heterogeneity in the fractional master equation, ensemble self-reinforcement, and strong memory effects

We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find the ensemble-averaged solution of the fractional master equation through subordination involving the fractional Poisson process counting the number of steps at a given time and the underlying discrete random walk with self-reinforcement. We also find the exact solution for the variance which exhibits superdiffusion even as the fractional exponent tends to 1.

Figure 1

Probability distribution of random walkers in continuous time with Mittag-Leffler distributed waiting times $\mu =0.75$, ${\tau }_0=1$, varying values of ${\alpha }_{+}={\alpha }_{-}=\alpha /2$ for the beta distribution (4), $a=1$, $t_{\text{end}}={10}^3$, and $N={10}^4$.

Figure 2

Variance of random walkers in continuous time with Mittag-Leffler distributed waiting times with varying values of $\mu$, ${\tau }_0=1$, $\alpha /2=1/2$, $t_{\text{end}}={10}^3$, and $N={10}^4$. The blue dashed line shows $\text{Var}\left[{\overline{X}}_{\mu }\left(t\right)\right]\propto t^2$. The red dotted line shows $\text{Var}\left[{\overline{X}}_{\mu }\left(t\right)\right]\propto t$.

Figure 3

Variance of random walkers in continuous time with varying values of $\alpha /2$, and Mittag-Leffler distributed waiting times with $\mu =0.75$, ${\tau }_0=1$, $t_{\text{end}}={10}^3$, and $N={10}^4$. The blue dashed line shows $\text{Var}\left[{\overline{X}}_{\mu }\left(t\right)\right]\propto t^2$. The red dotted line shows $\text{Var}\left[{\overline{X}}_{\mu }\left(t\right)\right]\propto t$.