Memory-induced anomalous dynamics in a minimal random walk model

A discrete-time dynamics of a non-Markovian random walker is analyzed using a minimal model where memory of the past drives the present dynamics. In recent work [N. Kumar et al., Phys. Rev. E 82, 021101 (2010)] we proposed a model that exhibits asymptotic superdiffusion, normal diffusion, and subdiffusion with the sweep of a single parameter. Here we propose an even simpler model, with minimal options for the walker: either move forward or stay at rest. We show that this model can also give rise to diffusive, subdiffusive, and superdiffusive dynamics at long times as a single parameter is varied. We show that in order to have subdiffusive dynamics, the memory of the rest states must be perfectly correlated with the present dynamics. We show explicitly that if this condition is not satisfied in a unidirectional walk, the dynamics is only either diffusive or superdiffusive (but not subdiffusive) at long times.

Figure 1

Variation of the asymptotic exponent ($a$) of the variance of the walker's position with $p$ is shown for various values of $q$ (0 to 1 in steps of $0.2$) for the model with variable memory. Here empty and filled points correspond respectively to $\gamma =0.8$ (the walker remembers the recent past more clearly) and $\gamma =-0.3$ (the walker remembers the distant past more clearly). For $\gamma =0.8$ and $q>0$, the dynamics is always essentially diffusive ($a$ is close to 1) while for $\gamma =-0.3$ and $q>0$, the dynamics is either diffusive ($a\sim 1$) or superdiffusive ($a>1$). All three phases of dynamics, diffusive, superdiffusive, and subdiffusive ($a<1$) are accessible only for $q=0$ (filled and empty circles).

Figure 2

Probability distribution of the position of the walker at step $n=2000$ for different values of $\alpha$. Here $q=0$ and $s=1$, and therefore $\alpha =p$. Inset shows results for $\alpha =0.3,0.4$ for which the dynamics is subdiffusive.