Emerging cost-time Pareto front for diffusion with stochastic return

Resetting, in which a system is regularly returned to a given state after a fixed or random duration, has become a useful strategy to optimize the search performance of a system. While earlier theoretical frameworks focused on instantaneous resetting, wherein the system is directly teleported to a given state, there is a growing interest in physical resetting mechanisms that involve a finite return time. However employing such a mechanism involves cost and the effect of this cost on the search time remains largely unexplored. Yet answering this is important in order to design cost-efficient resetting strategies. Motivated from this, we present a thermodynamic analysis of a diffusing particle whose position is intermittently reset to a specific site by employing a stochastic return protocol with external confining trap. We show for a family of potentials $U_R(x) \sim |x|^{m}$ with $m>0$, it is possible to find optimal potential shape that minimises the expected first-passage time for a given value of the thermodynamic cost, i.e. mean work. By varying this value, we then obtain the Pareto optimal front, and demonstrate a trade-off relation between the first-passage time and the work done.