Optimal refrigerator

We study a refrigerator model which consists of two n -level systems interacting via a pulsed external field. Each system couples to its own thermal bath at temperatures T_h and T_c , respectively (θ≡T_c/T_h<1) . The refrigerator functions in two steps: thermally isolated interaction between the systems driven by the external field and isothermal relaxation back to equilibrium. There is a complementarity between the power of heat transfer from the cold bath and the efficiency: the latter nullifies when the former is maximized and vice versa. A reasonable compromise is achieved by optimizing the product of the heat-power and efficiency over the Hamiltonian of the two systems. The efficiency is then found to be bounded from below by ζ_CA=(1)/(1-θ)-1 (an analog of the Curzon-Ahlborn efficiency), besides being bound from above by the Carnot efficiency ζ_C=(1)/(1-θ)-1 . The lower bound is reached in the equilibrium limit θ→1 . The Carnot bound is reached (for a finite power and a finite amount of heat transferred per cycle) for lnn≫1 . If the above maximization is constrained by assuming homogeneous energy spectra for both systems, the efficiency is bounded from above by ζ_CA and converges to it for n≫1 .