Stationary engines in and beyond the linear response regime at the Carnot efficiency

The condition for stationary engines to attain the Carnot efficiency in and beyond the linear response regime is investigated. We find that this condition for finite-size engines is significantly different from that for macroscopic engines in the thermodynamic limit. For the case of finite-size engines, the tight-coupling condition in the linear response regime directly implies the attainability of the Carnot efficiency beyond the linear response regime. As opposed to this, for the case of macroscopic engines in the thermodynamic limit, there are three types of mechanisms to attain the Carnot efficiency. One mechanism allows engines to attain the Carnot efficiency only in the linear response limit, while the other two mechanisms enable engines to attain the Carnot efficiency beyond the linear response regime. These three mechanisms are classified by introducing a tight-coupling window.

Figure 1

(a) Schematic of the information engine with switch (IES). A site lies in the middle of two particle baths H and L, and a wall at $r$ ($l$) prevents the jump of particles between the site and the bath L (H). The switch modulates the transition rates of the wall. (b) State space of the IES. The dashed transition paths vanish if we set $\varepsilon =0$.

Figure 2

State space of the coarse-grained autonomous Carnot engine (CGACE). The transitions with $B\leftrightarrow C$ and $A\leftrightarrow D$ are isothermal but not attached to particle baths. Work is extracted via the rotational path $A\to B\to C\to D\to A$. Although the engine contains few particles in this figure, it actually contains infinitely many particles.

Figure 3

Schematic picture of the imaginary system for the case of the CGACE. In the imaginary system, the chemical potential of the particle bath attached to the state $D$ is replaced from ${\mu }_{\mathrm{L}}$ to ${\mu }_{\mathrm{H}}$.

Figure 4

State space of the IES with $\varepsilon =0$ (left) and $\varepsilon >0$ (right) and their probability flow. In the case of $\varepsilon >0$, there exists particle flow without extracting work.

Figure 5

The behavior of $\partial h\left(\nu \right)/\partial \nu$ for an engine attaining the CE. The indifferentiability of $h$ allows that two equations $\partial h/\partial \nu =\beta {\mu }_{\mathrm{H}}$ and $\partial h/\partial \nu =\beta {\mu }_{\mathrm{L}}$ have the same solution $\nu ={\nu }^{*}$.

Figure 6

We plot pairs of chemical potentials with which the engine attains the CE on the ${\mu }_1-{\mu }_2$ coordinate with gray. We see three types of tight-coupling window.

Figure 7

Graph of $1/\beta ·\partial h_A\left(\nu \right)/\partial \nu$ and $1/\beta ·\partial h_B\left(\nu \right)/\partial \nu$. The area of gray region “abcd” represents the upper bound of extracted work in one cycle, while the area of bold rectangular “pbqd” represents the consumed chemical potential in one cycle.