Figure 1

(a) Schematic representation of a quantum heat engine employing electrons submitted to a magnetic field of intensity $B$ as working agents. Electrons are shown as two-level systems in either their upper or lower state of energy. The lower-level energy is set at zero, and the upper-level energy is denoted $ϵ=B$, in appropriate units. We consider two reservoirs containing $N=5$ electrons each. In the $l$ reservoir there is a single electron in the upper state, while in the $h$ reservoir two electrons are in the upper state. In (b) electrons in their low-energy level are represented by balls of weight 0 (open circles), while electrons in their high-energy levels are represented by balls of weight 1 (black-filled circles) of potential energy $ϵ$ that equals the altitude. In both (a) and (b) a cycle consists of picking up at random an electron (respectively, a ball) from a reservoir and carrying it to the next. This operation preserves the total number of balls in each reservoir, but some work is being performed (heat engine, the case of the figure) or absorbed (heat pump).Reuse & Permissions

Figure 2

This figure represents the $ϵ\left(f\right)$ diagram of a mechanical device involving two high reservoirs ($h$ subscripts) and two low reservoirs ($l$ subscripts). The four reservoirs are shown as dots with $f\equiv n/N$ in the abscissa, where $n$ represents the average number of balls of weight 1 in the reservoir. The work performed per cycle, $W\approx 0.193\text{J}$, is equal to the shaded area, and the efficiency $\eta \approx 0.643$ is the maximum possible value under the present conditions.Reuse & Permissions

Figure 3

Available heat-engine (right-hand side) and heat-pump (left-hand side) efficiencies as a function of the average work performed per cycle, $W$, in the case of $m=1$ (Otto cycle), $m=2$, $m=3$, and $m=\infty$ subreservoirs. Let us emphasize that to construct a Carnot engine out of Otto cycles, every Otto cycle must be infinitesimal. The Carnot cycles thus require an infinite number of subreservoirs. We have set ${\beta }_l=1.38$, ${\beta }_h=0.42$. For reversible engines the coefficient of performance (COP) of heat pumps is the reciprocal of $\eta$.Reuse & Permissions