Entropy current and efficiency of quantum machines driven by nonequilibrium incoherent reservoirs

Nanotechnology has not only provided us the possibility of developing quantum machines but also noncanonical power sources able to drive them. Here we focus on studying the performance of quantum machines driven by arbitrary combinations of equilibrium reservoirs and a form of engineered reservoirs consisting of noninteracting particles but whose distribution functions are nonthermal. We provide the expressions for calculating the maximum efficiency of those machines without needing any knowledge of how the nonequilibrium reservoirs were actually made. The formulas require the calculation of a quantity that we term entropy current, which we also derive. We illustrate our methodology through a solvable toy model where heat “spontaneously” flows against the temperature gradient.

Figure 1

General scheme of the type of system treated. A local system, connected (or not) to a mechanical device, interchanges particles with equilibrium reservoirs (at temperature $T_i$ and chemical potential ${\mu }_i$) and nonequilibrium incoherent reservoirs.

Figure 2

(a) A local system, consisting of four quantum dots with one narrow resonance each, at energies ${\varepsilon }_i$, is connected to a NIR (the N demon) and two equilibrium reservoirs, denoted by $h$ and $c$, for hot and cold, respectively. Because of the action of the N demon, heat flows “spontaneously” from the cold to the hot reservoir. (b) Coefficient of performance (COP) of the device depicted in Fig. 2. The inset shows the power of the device (${\dot{\varepsilon }}_h$). The parameters used are $k_B{T}_h={\mu }_h={\mu }_c={\varepsilon }_1=1$, ${\varepsilon }_2=2$, and ${\varepsilon }_3=1.5$.