Abrupt transitions in the optimization of diffusion with distributed resetting

TL;DR

The paper investigates Brownian diffusion with resetting to random positions drawn from compact-support PDFs, focusing on how the averaged MFPT $\mathcal{T}_r$ can exhibit abrupt, first-order transitions in the optimal resetting rate $r^*$. By analyzing truncated Gaussian (TGD) and truncated exponential (TED) distributions, it shows that two local minima in $\mathcal{T}_r$ can coexist and merge at a critical point, which is captured by a Ginzburg-Landau-type expansion with $\mathcal{T}_r\simeq T_c+\frac{a_4}{4}(r-r_c)^4$ at criticality and $\delta=3$, $\beta=1/2$ scaling near the transition. The authors also introduce the last-resetting-point density $\rho_l(z)$, revealing how the efficient resetting zones balance entropic and energetic factors, and extend the analysis to a two-point resetting scenario, where $r^*$ can jump between qualitatively different strategies. Together, these results provide a general framework for understanding and predicting discontinuous resetting-optimality in bounded-resetting landscapes and highlight potential hysteresis effects in experiments or simulations.

Abstract

Brownian diffusion subject to stochastic resetting to a fixed position has been widely studied for applications to random search processes. In an unbounded domain, the mean first-passage time at a target site can be minimized for a convenient choice of the resetting rate. Here we study this optimization problem in one dimension when resetting occurs to random positions, chosen from a probability density function with compact support that does not include the target. Depending on the shape of this distribution, the optimal resetting rate either varies smoothly with the mean distance to the target, as in single-site resetting, or exhibits a discontinuity caused by the presence of a second local minimum in the mean first-passage time. These two regimes are separated by a critical line containing a singular point that we characterize through a Ginzburg-Landau theory. To quantify how useful is a given resetting point for the search, we calculate the probability density function of the last resetting position before absorption. The discontinuous transition separates two markedly different optimal strategies: one with a small resetting rate where the last path before absorption starts from a rather distant but likely position, while the other strategy has a large resetting rate, favoring last paths starting from not-so-likely points but which are closer to the target.

Figures (5)

• Figure 1: (a) Sample path of a $1d$ diffusive particle subject to resetting at rate $r$ to random positions $z$, with an absorbing wall located at $x=0$. At each reset, the particle is set to an independent position $z$ distributed with the PDF $\rho(z)$. In gray filling, two examples of distribution for $z$: (b) truncated Gaussian distribution (TGD), and (c) truncated exponential distribution (TED), both with mean $\mu$. In these cases, the support of $\rho(z)$ is $z\in[1,2\mu-1]$, while $\rho(z)=0$ outside.
• Figure 2: Left column: averaged mean first-passage time ${\cal T}_r$ given by Eq. \ref{['pdf-6']} as a function of $r$, when the resetting distribution is a truncated Gaussian with $\sigma=8.8$ and: (a) $\mu=19.117$, (b) $\mu=\mu_{t}=19.1179...$, (c) $\mu=19.119$. Right column: Same quantity in the truncated exponential case, see Eq. \ref{['pdf-8']}, fixing $\sigma=8.26$ and: (d) $\mu=13.05$, (e) $\mu=\mu_{t}=13.0557...$, (f) $\mu=13.07$. The insets in panels (b) and (e) show ${\cal T}_r$ at the critical parameters $\sigma_{c}$ and $\mu_c$ where the two minima merge into a single one, located at $r=r_c$. The critical parameter values are obtained further in Section \ref{['sec:GL']} by solving Eqs. \ref{['pdf-9']}. One obtains $(\mu_c,\sigma_c,r_c)\simeq(19.0105,8.7610,0.1382)$ and $(12.9373,8.2355,0.2028)$ for the TGD and TED, respectively. For $\sigma < \sigma_{c}$, ${\cal T}_r$ admits a single minimum (not shown).
• Figure 3: Optimal resetting rate $r^{*}$ minimizing Eq. \ref{['pdf-6']} in the TGD case [panel (a)], and Eq. \ref{['pdf-8']} in the TED case [panel (b)], as a function of $\mu$ for various $\sigma$. The critical point with diverging derivative is shown in black in each case. For any value of $\sigma$ greater than $\sigma_c$, i.e., to the right of the critical curve in red, $r^*$ exhibits a discontinuity at $\mu=\mu_t(\sigma)$. The labels 1 and 2 denote the minima in Figs. \ref{['fig:mfpt_gauss_expo']}b and \ref{['fig:mfpt_gauss_expo']}e at the transition. The inset of (b) is a zoom of the small $\mu$ region that illustrates the non-monotonous character of $r^*$ when $\sigma>\sigma_c$.
• Figure 4: Last resetting point PDF given by Eq. \ref{['pdf-22']} at the two equivalent optimal rates $r_1^*(\sigma)$ and $r_2^*(\sigma)$, for different values of $\sigma > \sigma_{c}$. The corresponding transition values $\mu_t(\sigma)$ are $19.0352$, $19.3941$, $20.7817$, and $28.1604$, in panels (a), (b), (c) and (d) respectively.
• Figure 5: (a) Transition in the global minimum of the AMFPT for the two-site resetting problem ($m=12.28$). The red markers are the result of numerical simulations. (b) Probability that the last diffusive path has started from the site closer to the target, given by Eq. \ref{['pdf-21']}, for several values of $m$, and evaluated at the optimal rate $r^{\ast}$.