Optimal performance of endoreversible quantum refrigerators

The derivation of general performance benchmarks is important in the design of highly optimized heat engines and refrigerators. To obtain them, one may model phenomenologically the leading sources of irreversibility ending up with results that are model independent, but limited in scope. Alternatively, one can take a simple physical system realizing a thermodynamic cycle and assess its optimal operation from a complete microscopic description. We follow this approach in order to derive the coefficient of performance at maximum cooling rate for any endoreversible quantum refrigerator. At striking variance with the universality of the optimal efficiency of heat engines, we find that the cooling performance at maximum power is crucially determined by the details of the specific system-bath interaction mechanism. A closed analytical benchmark is found for endoreversible refrigerators weakly coupled to unstructured bosonic heat baths: an ubiquitous case study in quantum thermodynamics.

Figure 1

Quantum tricycle. A quantum system selectively coupled through frequency filters to three heat baths (with temperatures $T_w>T_h>T_c$) embodies the prototype of any thermal device. Here the direction of the heat currents (depicted by arrows) correspond to a refrigerator. Reversing them realizes a heat transformer or heat engine.

Figure 2

Optimal normalized COP (blue dots) versus ${\varepsilon }_C$ for about $2\times {10}^5$ $n$-stage endoreversible absorption refrigerators [34] with $n\in \{1,...,10\}$ and coupled to unstructured three-dimensional bosonic baths. All three temperatures $T_{\alpha }$, dissipation rates ${\gamma }_{\alpha }$, and hot frequencies ${\omega }_h$ were picked at random and the COP was optimized in ${\omega }_c$ so as to maximize the cooling power ${\dot{\mathcal{Q}}}_c$ in each case. Equation (14) is plotted in solid gray.