Dynamical transition in the temporal relaxation of stochastic processes under resetting

A stochastic process, when subject to resetting to its initial condition at a constant rate, generically reaches a nonequilibrium steady state. We study analytically how the steady state is approached in time and find an unusual relaxation mechanism in these systems. We show that as time progresses an inner core region around the resetting point reaches the steady state, while the region outside the core is still transient. The boundaries of the core region grow with time as power laws at late times with new exponents. Alternatively, at a fixed spatial point, the system undergoes a dynamical transition from the transient to the steady state at a characteristic space-dependent timescale $t^{*}\left(x\right)$. We calculate analytically in several examples the large deviation function associated with this spatiotemporal fluctuation and show that, generically, it has a second-order discontinuity at a pair of critical points characterizing the edges of the inner core. These singularities act as separatrices between typical and atypical trajectories. Our results are verified in the numerical simulations of several models, such as simple diffusion and fluctuating one-dimensional interfaces.

Figure 1

NESS gets established in a core region around the resetting center $O$ whose right and left frontiers ${\xi }_{\pm }\left(t\right)$ grow with time. Outside the core region, the system is transient.

Figure 2

PDFs of the position of a particle diffusing in one dimension with a diffusion constant $D=1/2$, whose position is stochastically reset to the origin at a constant rate $r=1/2$. (a) The points are simulation data for the PDFs at $t=10$ (blue circles) and $t=15$ (green squares). The (magenta) dashed line is the infinite time NESS given in Eq. (3). The vertical dashed lines mark the positions $\pm y^{*}t$. (b) Same data as in (a) compared with large deviation result [Eq. (5)] of the PDF (normalized numerically), denoted by the (magenta) dashed lines. The dashed vertical lines at $y=\pm 1$ mark the $y^{*}$ at which the LDF has a second order discontinuity.

Figure 3

(a) The PDFs of the relative heights of a periodic interface of size $L=2^{15}$ at various times, generated from a TASEP (shown by points) are collapsed by choosing $\mathrm{\Gamma }=2.4$ to the TW GOE PDF with zero mean, shown by the (magenta) dashed line. (b) The points are simulation data for the PDF of the relative heights at $t=50$ (blue circles) and $t=100$ (green squares), with resetting with rate $r=0.05$. The (magenta) dashed line is the infinite time NESS given in Eq. (A8). The vertical dashed lines mark the positions $x_{\pm }^{*}=\pm y_{\pm }^{*}{t}^{1/{\nu }_{\pm }}$. (c) Same data as in (b) compared with Eq. (A7) denoted by the (magenta) dashed lines. The vertical dashed lines mark the positions $\pm y_{\pm }^{*}$. (d) The LDFs computed numerically from (A7) for various $t$ are compared with (A10b).

Figure 4

Schematic plot of the function $\mathrm{\Phi }(w,y)$ as a function of $w$ (but fixed $y$), which has a unique minimum at $w^{*}\left(y\right)$, determined from Eq. (A18) for the two cases (a) $w^{*}<1$ and (b) $w^{*}>1$. For the case (a), the integral in Eq. (A16) is dominated by the minimum $w^{*}$, whereas for the case (b) the integral is dominated by the upper limit $w=1$ of the integral. The minimum values of $\mathrm{\Phi }(w,y)$ in the interval $w\in [0,1]$ in both cases, are marked by the red circles.