Analog of the Carnot engine for fluctuating diffusivity in living cells

Recently, a formal analogy between the fluctuating diffusivity and thermodynamics has been proposed based on phenomena of heterogeneous diffusion observed in living cells. This not only offers the analogs of the quantity of heat and work as well as the internal energy but also achieves that of the Clausius inequality for the entropy concerning diffusivity fluctuations. Here, a discussion is developed about constructing a heat-like engine in terms of the fluctuating diffusivity. The engine constitutes two kinds of processes with the average diffusivity or the average local temperature being kept fixed, along which the fluctuation distribution obeys an exponential law. The efficiency of the engine in a cycle, which quantifies how much the diffusivity change as the analog of work can be extracted, is found to formally coincide with that of Carnot's. During the cycle, the total change of the entropy is also shown to vanish.