# Optimal refrigerator - Condensed Matter > Statistical Mechanics

Abstract: We study a refrigerator model which consists of two $n$-level systemsinteracting via a pulsed external field. Each system couples to its own thermalbath at temperatures $T h$ and $T c$, respectively $\theta\equiv T c-T h<1$.The refrigerator functions in two steps: thermally isolated interaction betweenthe systems driven by the external field and isothermal relaxation back toequilibrium. There is a complementarity between the power of heat transfer fromthe cold bath and the efficiency: the latter nullifies when the former ismaximized and {\it vice versa}. A reasonable compromise is achieved byoptimizing the product of the heat-power and efficiency over the Hamiltonian ofthe two system. The efficiency is then found to be bounded from below by$\zeta { m CA}=\frac{1}{\sqrt{1-\theta}}-1$ an analogue of the Curzon-Ahlbornefficiency, besides being bound from above by the Carnot efficiency$\zeta { m C} = \frac{1}{1-\theta}-1$. The lower bound is reached in theequilibrium limit $\theta\to 1$. The Carnot bound is reached for a finitepower and a finite amount of heat transferred per cycle for $\ln n\gg 1$. Ifthe above maximization is constrained by assuming homogeneous energy spectrafor both systems, the efficiency is bounded from above by $\zeta { m CA}$ andconverges to it for $n\gg 1$.

Author: Armen E. Allahverdyan, Karen Hovhannisyan, Guenter Mahler

Source: https://arxiv.org/