Geometric characterization for cyclic heat engines far from equilibrium

Considerable attention has been devoted to microscopic heat engines in both theoretical and experimental aspects. Notably, the fundamental limits pertaining to power and efficiency, as well as the trade-off relations between these two quantities, have been intensively studied. This study aims to shed further light on the ultimate limits of heat engines by exploring the relationship between the geometric length along the path of cyclic heat engines operating at arbitrary speeds and their power and efficiency. We establish a trade-off relation between power and efficiency using the geometric length and the timescale of the heat engine. Remarkably, because the geometric quantity comprises experimentally accessible terms in classical cases, this relation is useful for the inference of thermodynamic efficiency. Moreover, we reveal that the power of a heat engine is always upper bounded by the product of its geometric length and the statistics of energy. Our results provide a geometric characterization of the performance of cyclic heat engines, which is universally applicable to both classical and quantum heat engines operating far from equilibrium.

Figure 1

(a) Schematic of cyclic heat engines. The working medium is constantly coupled to a heat bath. The system Hamiltonian $H_t=H\left({\lambda }_t\right)$ and the bath temperature $T_t$ are periodically controlled in order to extract work. (b) For a given control path $\mathcal{C}$, the corresponding curve of engine states ${\left\{{\varrho }_t\right\}}_{0\le t\le \tau }$ is depicted by the solid line, and its geometric length with respect to a metric $d(·,·)$ is denoted by $\ell$. Our aim is to elucidate the relationship between the geometric length $\ell$ and power and efficiency of heat engines.

Figure 2

Numerical illustration for the qubit heat engine. The top panels show contour plots of (a) the geometric length ${\ell }_W$ and (b) the lower bound in Eq. (9) on the $\mathrm{\Omega }\text{−}\gamma$ plane. The bottom panels describe upper bounds of (c) efficiency [Eq. (11)] and (d) power [Eq. (14)]. Parameter $\mathrm{\Omega }$ is varied while other parameters are fixed as $T_c=1, T_h=10, \gamma =0.1$, and $\tau =10$.

Figure 3

Numerical illustration of the geometric quantities $(2T_c/{\overline{a}}_{\tau }^{\infty }){\left({\ell }_W^{\infty }\right)}^2$ and ${\mathcal{L}}^2$ for the qubit heat engine in the adiabatic regime. The protocol is specified as ${\widehat{\beta }}_t=[T_h+(T_c-T_h)sin{\left(\pi t\right)}^2]/\left(T_h{T}_c\right), {\widehat{\varepsilon }}_t=\mathrm{\Omega }[1+0.5sin\left(2\pi t\right)+0.1sin\left(6\pi t\right)]$, and ${\widehat{\theta }}_t=0$. Parameter $\mathrm{\Omega }$ is varied while other parameters are fixed as $T_c=1, T_h=1.1$, and $\gamma =0.1$.