Geometric Brownian information engine with finite cycle time: Optimisation of output work, power and efficiency

We consider a Geometric Brownian Information Engine to explore the effects of finite cycle time $(\tau)$ on the extractable work, power, and efficiency. We incorporate an error-free feedback controller that converts the information obtained about the state of overdamped Brownian particles, confined within a 2-D monolobal geometry, into extractable work. The performance of the information engine depends on the cycle period $(\tau)$, measurement distance $(x_m)$, and feedback location $(x_f)$ of the controller. Upon increasing the feedback cycle time, the engine transitions from a high non-equilibrium steady state to a completely relaxed state. We set the measurement distance at an optimum position related to a fully relaxed state ($x_m^* \sim 0.6 \sigma$). When the cycle time is finite and short ($\tau <\tau_r$), the best information processing occurs with a shorter distance of the feedback site. While increasing the cycle time towards a fully relaxed state ($\tau \gg \tau_r$), the maximum extractable work that can be achieved with a feedback location is set to be twice that of $x_m^*$, as expected. When the cycle time ($\tau$) is longer than the relaxation time ($\tau_r$), the maximum power is achieved when the scaled feedback location is exactly double the optimum measurement distance ($x_f^{*}=2x_m^*$). In contrast, when $\tau < \tau_r$, the maximum power is achieved when the feedback site is set at a lower value. As the $\tau$ increases, the maximum average power decreases. In the limit of a long $\tau$, the highest efficiency as well extractable work is attained when $x_f$ is located at $2x_m$, regardless of the level of entropic control. As the dominance of entropic control increases, the extractable work and efficiency in the fully relaxed state decrease due to higher information loss during relaxation.