Performance at maximum figure of merit for a Brownian Carnot refrigerator

This paper focuses on the coefficient of performance (COP) at maximum ${\chi }^R$ figure of merit for a Brownian Carnot-like refrigerator, within the context of the low-dissipation approach. Our proposal is based on the Langevin equation for a Brownian particle bounded to a harmonic potential trap, which can perform Carnot-like cycles at finite time. The theoretical approach is related to the equilibrium ensemble average of ${〈x^2〉}_{\mathrm{eq}}$ which plays the role of a statelike equation, $x$ being the Brownian particle position. This statelike equation comes from the macroscopic version of the corresponding Langevin equation for a Brownian particle. We show that under quasistatic conditions the COP has the same expression as the macroscopic Carnot refrigerator, while for irreversible cycles at finite time and under symmetric dissipation the optimal COP is the counterpart of Curzon-Ahlborn efficiency as also obtained for irreversible macroscopic refrigerators.