In conventional thermodynamics, it is widely acknowledged that the realization of an isothermal process for a system requires a quasistatic controlling protocol. Here we propose and design a strategy to realize a finite-rate isothermal transition from an equilibrium state to another one at the same temperature, which is named the “shortcut to isothermality.” By using shortcuts to isothermality, we derive three nonequilibrium work relations, including an identity between the free-energy difference and the mean work due to the potential of the original system, a Jarzynski-like equality, and the inverse relationship between the dissipated work and the total driving time. We numerically test these three relations by considering the motion of a Brownian particle trapped in a harmonic potential and dragged by a time-dependent force.

The figure of merit for refrigerators performing finite-time Carnot-like cycles between two reservoirs at temperature $T_h$ and $T_c$ ($<T_h$) is optimized. It is found that the coefficient of performance at maximum figure of merit is bounded between 0 and $(\sqrt{9+8{\varepsilon }_c}-3)/2$ for the low-dissipation refrigerators, where ${\varepsilon }_c=T_c/(T_h-T_c)$ is the Carnot coefficient of performance for reversible refrigerators. These bounds can be reached for extremely asymmetric low-dissipation cases when the ratio between the dissipation constants of the processes in contact with the cold and hot reservoirs approaches to zero or infinity, respectively. The observed coefficients of performance for real refrigerators are located in the region between the lower and upper bounds, which is in good agreement with our theoretical estimation.

When a Brownian particle in contact with a heat bath at a constant temperature is controlled by a time-dependent harmonic potential, its distribution function can be rigorously derived from the Kramers equation with the consideration of the inertial effect of the Brownian particle. Based on this rigorous solution and the concept of shortcuts to adiabaticity, we construct a stochastic heat engine by employing the time-dependent harmonic potential to manipulate the Brownian particle to complete a thermodynamic cycle. We find that the efficiency at maximum power of this stochastic heat engine is equal to $1-\sqrt{T_c/T_h}$, where $T_c$ and $T_h$ are the temperatures of the cold bath and the hot one in the thermodynamic cycle, respectively.

The efficiency at maximum power output of linear irreversible Carnot-like heat engines is investigated based on the assumption that the rate of irreversible entropy production of the working substance in each ``isothermal'' process is a quadratic form of the heat exchange rate between the working substance and the reservoir. It is found that the maximum power output corresponds to minimizing the irreversible entropy production in two isothermal processes of the Carnot-like cycle, and that the efficiency at maximum power output has the form $\eta${}mP=$\eta${}C/(2$-${}$\gamma${}$\eta${}C), where $\eta${}C is the Carnot efficiency, while $\gamma${} depends on the heat transfer coefficients between the working substance and two reservoirs. The value of $\eta${}mP is bounded between $\eta${}$-${}$\equiv${}$\eta${}C/2 and $\eta${}+$\equiv${}$\eta${}C/(2$-${}$\eta${}C). These results are consistent with those obtained by Chen and Yan [J. Chem. Phys. 90, 3740 (1989)] based on the endoreversible assumption, those obtained by Esposito et al. [Phys. Rev. Lett. 105, 150603 (2010)] based on the low-dissipation assumption, and those obtained by Schmiedl and Seifert [Europhys. Lett. 81, 20003 (2008)] for stochastic heat engines which in fact also satisfy the low-dissipation assumption. Additionally, we find that the endoreversible assumption happens to hold for Carnot-like heat engines operating at the maximum power output based on our fundamental assumption, and that the Carnot-like heat engines that we focused on do not strictly satisfy the low-dissipation assumption, which implies that the low-dissipation assumption or our fundamental assumption is a sufficient but non-necessary condition for the validity of $\eta${}mP=$\eta${}C/(2$-${}$\gamma${}$\eta${}C) as well as the existence of two bounds, $\eta${}$-${}$\equiv${}$\eta${}C/2 and $\eta${}+$\equiv${}$\eta${}C/(2$-${}$\eta${}C).