Optimal finite-time heat engines under constrained control

We optimize finite-time stochastic heat engines with a periodically scaled Hamiltonian under experimentally motivated constraints on the bath temperature $T$ and the scaling parameter $\lambda$. We present a general geometric proof that maximum-efficiency protocols for $T$ and $\lambda$ are piecewise constant, alternating between the maximum and minimum allowed values. When $\lambda$ is restricted to a small range and the system is close to equilibrium at the ends of the isotherms, a similar argument shows that this protocol also maximizes output power. These results are valid for arbitrary dynamics. We illustrate them for an overdamped Brownian heat engine, which can experimentally be realized using optical tweezers with stiffness $\lambda$.

Figure 1

The maximum-efficiency protocol (3) under the constraints in Eq. (2) (dashed line) compared to a suboptimal cycle (dotted line).

Figure 2

Numerical optimization of the efficiency of the Brownian heat engine under constrained control verifies that the maximum-efficiency protocol is given by Eq. (3). (a) Maximum efficiency and (b) the corresponding power (in units of the ultimate maximum power $P_{{\lambda }_{\mathrm{pwl}}}^{*}$ for ${\lambda }_{\mathrm{pwl}}$) as functions of ${\lambda }_{-}/{\lambda }_{+}$. All protocols except for ${\lambda }_{\mathrm{S}}$ perfectly overlap. Parameters used are $t_{+}=t_{-}=1$, $k_{\mathrm{B}}{T}_{+}=1$, $k_{\mathrm{B}}{T}_{-}=0.25$ (thus Carnot efficiency ${\eta }_{\mathrm{C}}\equiv 1-T_{-}/T_{+}=0.75$), ${\lambda }_{+}=0.5$, and $\mu =1$.

Figure 3

Numerical optimization of the output power of the Brownian heat engine illustrates that the maximum-efficiency protocol ${\lambda }_{\eta }$ (3) also yields maximum power when the compression ratio ${\lambda }_{-}/{\lambda }_{+}$ is large and the durations $t_{+}=t_{-}=1$ of the two isotherms are comparable to the relaxation times $1/\left(2\mu {\lambda }_{\pm }\right)$ for $\sigma$. (a) Powers (in units of the ultimate maximum power $P_{{\lambda }_{\mathrm{pwl}}}^{*}$ for ${\lambda }_{\mathrm{pwl}}$) and (b) the corresponding efficiencies obtained using ${\lambda }_{\eta }$ (3) and the protocols in Table 1. For ${\lambda }_{-}/{\lambda }_{+}\ge 0.59$ all protocols except for ${\lambda }_{\mathrm{S}}$ coincide. (c) and (d) show the protocols and the resulting response for ${\lambda }_{-}/{\lambda }_{+}=0.4$. (e) The relative differences $\delta X=(X_{{\lambda }_{\mathrm{pwl}}}-X_{{\lambda }_{\mathrm{pwc}}})/X_{{\lambda }_{\mathrm{pwl}}}$ of power ($X=P$) and efficiency ($X=\eta$) for ${\lambda }_{\mathrm{pwl}}$ and ${\lambda }_{\mathrm{pwc}}$. (f) The optimal values of parameters $b$ and $d$ for ${\lambda }_{\mathrm{pwl}}$. We used the same parameters as in Fig. 2.

Figure 4

The relative differences $\delta X=(X_{{\lambda }_{\mathrm{pwl}}}-X_{{\lambda }_{\eta }})/X_{{\lambda }_{\mathrm{pwl}}}$ of (a) maximum power ($X=P$) optimized with respect to ${\lambda }_{-}$ and (b) the corresponding efficiency ($X=\eta$) for the linear protocol ${\lambda }_{\mathrm{pwl}}$ and the maximum-efficiency protocol (3) for different values of $T_{-}/T_{+}$ and $t_{-}/t_{+}$. (c)–(f) show the corresponding values of maximum power and efficiency. The piecewise constant protocol ${\lambda }_{\mathrm{pwc}}$ and the maximum-efficiency protocol ${\lambda }_{\eta }$ (3) are in this case equal. We used the same parameters as in Fig. 2.

Figure 5

Efficiency at maximum output power ($\eta$, red axis) and the corresponding optimal compression ratio (${\lambda }_{-}^{*}/{\lambda }_{+}$, blue axis) for the maximum-efficiency protocol (3) as functions of the temperature ratio $T_{-}/T_{+}$ for six values of cycle duration $t_{\mathrm{p}}$ (colored curves) and $t_{-}=t_{+}$. The cycle time, $t_{\mathrm{p}}$, is measured in units of the system relaxation time during the hot isotherm. The corresponding ratio during the cold isotherm reads $t_{\mathrm{p}}{\lambda }_{-}^{*}/{\lambda }_{+}$. We used the same parameters as in Fig. 2.

Figure 6

Efficiency at maximum output work obtained using the Hamiltonian $H=\lambda \left(t\right)\left(\right|x{|}^n/n-ln\left|x\right|)$ as a function of $T_{-}/T_{+}$. Parameters used are $k_{\mathrm{B}}{T}_{+}=1$ and ${\lambda }_{+}=0.5$.

Figure 7

Optimal performance for fixed boundary values of the response: ${\sigma }_{-}\equiv \sigma \left(0\right)=\sigma \left(t_{\mathrm{p}}\right)$ and ${\sigma }_{+}\equiv \sigma \left(t_{+}\right)=0.5$. (a) Maximum power (in units of the ultimate maximum power $P_{{\lambda }_{\mathrm{pwl}}}^{*}$ for ${\lambda }_{\mathrm{pwl}}$) and (b) maximum efficiency as functions of ${\sigma }_{-}/{\sigma }_{+}$. Lines corresponding to ${\lambda }_{\mathrm{S}}$ (orange solid) and ${\lambda }_{\mathrm{slow}}$ (blue dotted) perfectly overlap. The maximum-efficiency protocol (3) and the piecewise constant protocol ${\lambda }_{\mathrm{pwc}}$ are in this case equal. We used the same parameters as in Fig. 2.