Finite-size effect on optimal efficiency of heat engines

The optimal efficiency of quantum (or classical) heat engines whose heat baths are $n$-particle systems is given by the strong large deviation. We give the optimal work extraction process as a concrete energy-preserving unitary time evolution among the heat baths and the work storage. We show that our optimal work extraction turns the disordered energy of the heat baths to the ordered energy of the work storage, by evaluating the ratio of the entropy difference to the energy difference in the heat baths and the work storage, respectively. By comparing the statistical mechanical optimal efficiency with the macroscopic thermodynamic bound, we evaluate the accuracy of the macroscopic thermodynamics with finite-size heat baths from the statistical mechanical viewpoint. We also evaluate the quantum coherence effect on the optimal efficiency of the cycle processes without restricting their cycle time by comparing the classical and quantum optimal efficiencies.

Figure 1

The finite but large region. This region is the intermediate region between the very little region and the infinitely large region. In the very little region, the optimal efficiency can be numerically calculated in existing results. In the infinitely large region, the optimal efficiency can be calculated with the thermodynamic limit or the quasistatic assymption. However, no existing result can solve the optimal efficiency in the finite but large region.

Figure 2

Average work extraction includes any heat engine that satisfies energy conservation law in average. Our upper bound is valid for any average work extraction. CP-instrument work extraction is more narrow (and easy to realize) class than average work extraction. Our upper bound is attained by the CP-instrument work extraction.

Figure 3

The five optimal efficiencies in (C34) have the same asymptotic expansion.

Figure 4

We plot Carnot's bound, the thermodynamic optimal efficiency ${\eta }_T^{\left(n\right)}$, and the results of numerical calculation for ${\eta }_{C\neg }^{\left(n\right)}({\beta }_H,{\beta }_L,{\mathcal{T}}_{Q_n}^{\mathrm{opt}})$, as the red dotted curved line, the blue curved line, and the black dots, respectively.

Figure 5

We plot the difference ${\eta }_T^{\left(n\right)}-{\eta }_{C\neg }^{\left(n\right)}({\beta }_H,{\beta }_L,{\mathcal{T}}_{Q_n}^{\mathrm{opt}})$ as black dots. We also plot the function $d_{{\beta }_H,{\beta }_L}^{\left(1\right)}{Q}_n/n^2-1111Q_n/n^2$ as the blue line. The parameter $d_{{\beta }_H,{\beta }_L}^{\left(1\right)}$ is defined in (50), and 1111 is the fitting parameter. In the present condition, the parameter $d_{{\beta }_H,{\beta }_L}^{\left(1\right)}\approx 14343$.

Figure 6

Structure of $U_{f_n}$. We omit $E$ in this figure just for simplicity.