Efficiency estimation for an equilibrium version of Maxwell refrigerator

Maxwell refrigerator as a device that can transfer heat from a cold to hot temperature reservoir making use of information reservoir was introduced by Mandal et al. \cite{Mandal2013a}. The model has a two state demon and a bit stream interacting with two thermal reservoirs simultaneously. We work out a simpler version of the refrigerator where the demon and bit system interact with the reservoirs separately and for a duration long enough to establish equilibrium. The efficiency, $\eta$, of the device when working as an engine as well as the coefficient of performance (COP) when working as a refrigerator are calculated. It is shown that the maximum efficiency matches that of a Carnot engine/refrigerator working between the same temperatures, as expected. The COP at maximum power decreases as $\frac{1}{T_h}$ when $T_h >T_c \gg \Delta E$ ($k_B = 1$), where $T_h$ and $T_c$ are the temperatures of the hot and cold reservoirs respectively and $\Delta E$ is the level spacing of the demon. $\eta$ at maximum power of the device, when working as a heat engine, is found to be $\frac{T_h}{0.779 + T_h}$ when $T_c \ll \Delta E$ and $T_h \gg \Delta E$.