Active particle in one dimension subjected to resetting with memory

The study of diffusion with preferential returns to places visited in the past has attracted increased attention in recent years. In these highly non-Markov processes, a standard diffusive particle intermittently resets at a given rate to previously visited positions. At each reset, a position to be revisited is randomly chosen with a probability proportional to the accumulated amount of time spent by the particle at that position. These preferential revisits typically generate a very slow diffusion, logarithmic in time, but still with a Gaussian position distribution at late times. Here we consider an active version of this model, where between resets the particle is self-propelled with constant speed and switches direction in one dimension according to a telegraphic noise. Hence there are two sources of non-Markovianity in the problem. We exactly derive the position distribution in Fourier space, as well as the variance of the position at all times. The crossover from the short-time ballistic regime, dominated by activity, to the long-time anomalous logarithmic growth induced by memory is studied. We also analytically derive a large deviation principle for the position, which exhibits a logarithmic time scaling instead of the usual algebraic form. Interestingly, at large distances, the large deviations become independent of time and match the nonequilibrium steady state of a particle under resetting to its starting position only.

Figure 1

An active particle (dark blue arrow) moves at constant speed $v_0$ on the line and stochastically switches direction at rate $\gamma$. In addition, the particle can relocate instantaneously to a previously visited location with rate $r$. In this case, a time $t^{\prime }\in [0,t)$ is chosen at random uniformly in the past and the particle adopts the position and velocity it had at that time (light blue arrow).

Figure 2

Variance $V_r\left(t\right)$ of the position $x\left(t\right)$ as a function of $t$ for $v_0=1, \gamma =1$, and $r=0.1$. The blue dashed line corresponds to the exact result in Eq. (13) and the black solid line to the long-time behavior in Eq. (14). The dots are results of Monte Carlo simulations employing the Gillespie algorithm [40].

Figure 3

(a) Analytical rate function ${\mathrm{\Psi }}_r\left(z\right)$ in Eq. (75) plotted against $z$ (solid line) for fixed $r=0.1$ and $\gamma =1$ (or $R=r/\gamma =0.1$); the quadratic behavior at small $z$ in Eq. (82) is displayed with a dashed line. (b) Same rate function at large $z$ (solid line), with the leading asymptotic form in Eq. (82) shown as a black dashed line. The red dashed line represents the leading term plus the first subleading correction.