Optimization in First-Passage Resetting

We investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold. In an infinite domain, this process is nonstationary and its probability distribution exhibits rich features. In a finite domain, we define a nontrivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost.

Figure 1

First-passage resetting on the semi-infinite line. Whenever a diffusing particle, which starts at the origin, reaches $L$, it is instantaneously reset to the origin. Successive first-passage times are denoted by $t_1,t_2,\dots$.

Figure 2

Schematic of diffusion with first-passage resetting. Shown are paths from (0,0) to $(x,t)$ with (a) an odd number or (b) an even number of reset events. We transform either the odd-numbered or the even-numbered pieces of the original path through the reflection $x(t)\to L-x(t)$ to yield free diffusion from either (c) $(L,0)$ to $(x,t)$ or (d) (0,0) to $(x,t)$. Summing over all numbers of reset events gives Eq. (5b).

Figure 3

The objective function of Eq. (9) versus Péclet number Pe for different normalized cost values $C^{\prime }\equiv C/(L^2/D)$. Indicated on each curve is the optimal operating point.