To reset or not to reset in a finite domain: that is the question

Abstract

We investigate the search of a target with a given spatial distribution in a finite one-dimensional domain. The searcher follows Brownian dynamics and is always reset to its initial position when reaching the boundaries of the domain (boundary resetting). In addition, the searcher may be reset to its initial position from any internal point of the domain (bulk resetting). Specifically, we look for the optimal strategy for bulk resetting, i.e., the spatially dependent bulk resetting rate that minimizes the average search time. The best search strategy exhibits a second-order transition from vanishing to non-vanishing bulk resetting when varying the target distribution. The obtained mathematical criteria are further analyzed for a monoparametric family of distributions, to shed light on the properties that control the optimal strategy for bulk resetting. Our work paves new research lines in the study of search processes, emphasizing the relevance of the target distribution for the optimal search strategy, and identifies a successful framework to address these questions.

Figures (3)

• Figure 1: Sketch of a single trajectory for the Brownian search of a spatially distributed target. Left: One trajectory starting from $x=0$. The blue solid trace stands for diffusive motion, with instances of a boundary reset (vertical green stroke) and bulk resets (vertical yellow strokes). The target position $x_T$ (horizontal dotted red line) has been drawn from its distribution $P(x_T)$. Domain boundaries are at $\pm \ell$ (horizontal dashed green lines). Right: Spatial distribution of the target.
• Figure 2: Derivative controlling the stability of the no-resetting strategy. Positive (negative) values, represented as warm (cold) colors, thereof contribute to zero (nonzero) bulk resetting as the best search strategy. Top: Standard derivative $M(x_T)$ defined in Eq. \ref{['eq:m']} for the homogeneous case. The change of stability is reached at $x_T^c$ (black diamond). Bottom: Symmetrized functional derivative $M_s(x_T,x)$ defined in Eq. \ref{['eq:R']} for the inhomogeneous case. White color indicates saturated values. The contour line at which $M_s=0$ is marked (dashed line), as well as the value $x_T^{(0)}$ (black diamond) below which $M_s(x_T,x)<0$, $\forall x$.
• Figure 3: Optimal heterogeneous resetting profiles in the bulk. Specifically, they have been obtained numerically for the target distributions of the family \ref{['eq:PTbeta']} corresponding to $\beta=0.5,2,3$. For $\beta\leq1$, the optimal strategy consistently involves no resetting in the bulk. For $\beta>1$, for which nonzero bulk resetting is the optimal strategy, the features displayed---central region with no resetting, strongly heterogeneous behavior close to the wall---are robust. Symbols and lines respectively stand for meshes with $N=501$ and $N=2001$ nodes. The peaks become consistently higher and narrower when increasing the number of nodes in the mesh.