Thermodynamic Trade-off Relation for First Passage Time in Resetting Process

TL;DR

This paper derives a thermodynamic trade-off relation between the mean first passage time and work cost in resetting processes, considering finite-time resetting with a trapping potential, revealing fundamental limits and advantages of different resetting protocols.

Contribution

It introduces a thermodynamic framework for resetting processes with finite-time trapping, deriving a universal trade-off relation and analyzing the impact of potential shape and resetting protocols.

Findings

Instantaneous resetting requires infinite work.

Linear potential provides a lower bound for the time-cost trade-off.

Fixed-time resetting can outperform stochastic resetting in the trade-off.

Abstract

Resetting is a strategy for boosting the speed of a target-searching process. Since its introduction over a decade ago, most studies have been carried out under the assumption that resetting takes place instantaneously. However, due to its irreversible nature, resetting processes incur a thermodynamic cost, which becomes infinite in case of instantaneous resetting. Here, we take into consideration both the cost and first passage time (FPT) required for a resetting process, in which the reset or return to the initial location is implemented using a trapping potential over a finite but random time period. An iterative generating function and counting functional method \`a la Feynman and Kac are employed to calculate the FPT and average work for this process. From these results, we obtain an explicit form of the time-cost trade-off relation, which provides the lower bound of the mean FPT for a given work input when the trapping potential is linear. This trade-off relation clearly shows that instantaneous resetting is achievable only when an infinite amount of work is provided. More surprisingly, the trade-off relation derived from the linear potential seems to be valid for a wide range of trapping potentials. In addition, we have also shown that the fixed-time or sharp resetting can further enhance the trade-off relation compared to that of the stochastic resetting.