Stochastic resetting with refractory periods: pathway formulation and exact results

Abstract

We look into the problem of stochastic resetting with refractory periods. The model dynamics comprises diffusive and motionless phases. The diffusive phase ends at random time instants, at which the system is reset to a given position -- where the system remains at rest for a random time interval, termed the refractory period. A pathway formulation is introduced to derive exact analytical results for the relevant observables in a broad framework, with the resetting time and the refractory period following arbitrary distributions. For the paradigmatic case of Poissonian distributions of the resetting and refractory times, in general with different characteristic rates, closed-form expressions are obtained that successfully describe the relaxation to the steady state. Finally, we focus on the single-target search problem, in which the survival probability and the mean first passage time to the target can be exactly computed. Therein, we also discuss optimal strategies, which show a non-trivial dependence on the refractory period.

Figures (4)

• Figure 1: Single trajectory for stochastic resetting with refractory periods. The labels $t_i$, $i=1,2,\ldots$ , mark a new reset to $x_0$ where the propagation phase (blue line) ends with an instantaneous reset (dashed black line) and the refractory period begins (red line). Analogously, $\tau_i$ marks the end of the refractory phase after $t_i$, $i=1,2,\ldots$ Herein, $n$ stands for the number of times the system is renewed, $n=i$ for $\tau_i < t < \tau_{i+1}$. The duration $\sigma$ of each propagation (refractory) phase comes from the PDF $f(\sigma)$ ($w(\sigma)$). For the sake of simplicity, the initial condition is equal to the resetting position $x_r=x_0$.
• Figure 2: PDF of the propagation phase. Numerical integration of $p^{\hbox{\scriptsize$\text{(p)}$}}(x,t|x_0)$ (colourful dashed lines), given by equation eq:pdiff_final, at different times and the infinite time NESS eq:Ness_pprop (solid black line) are shown. All the results are shown using $x_0=0$, $D=1$, $r_1=r_2=1$ as parameters.
• Figure 3: Comparison of the numerical and analytical PDFs of the propagation phase. Parameter values are $D=1$, $r_1=1$, $r_2=2$ and $x_0=0$. Symbols stand for numerical simulations for $t=1.5$ (light blue circles) and $t=3$ (red squares), while solid black lines stand for the analytical approximation obtained in \ref{['app:Laplace-method']}, which asymptotically converges to the rough estimation given by equation \ref{['eq:Approximation_NESS']}. Vertical dashed lines at $|x|/t= \sqrt{4 D r_1}$ indicate the separation between the inner region, where the NESS has already been reached, and the outer region, where the transient behaviour is still observed.
• Figure 4: Mean first passage time $T(r_1,r_2)$ as a function of the resetting rate $r_1$ for fixed refractory rate $r_2$. An excellent agreement between simulations (symbols) and theory (solid lines) is found. The optimal resetting rate $r_1^{\hbox{\scriptsize$\text{opt}$}}$, represented by five-pointed stars, monotonically increases with $r_2$---in the limit $r_2\to \infty$, the value $r_1^{(0)}=\left(2-W(-2e^{-2})\right)^2= 2.53964$ is reached. Inset: Optimal resetting rate as a function of the refractory period rate. We show the numerical solution of the implicit equation for $r_1^{\hbox{\scriptsize$\text{opt}$}}$, as given by equation eq:DerivativeMFPT (solid line), and the analytical approximation for small $r_2$, as given by equation eq:expansion_r1(r2) (dashed line).