Efficiency Statistics and Bounds for Systems with Broken Time-Reversal Symmetry

Universal properties of the statistics of stochastic efficiency for mesoscopic time-reversal symmetry broken energy transducers are revealed in the Gaussian approximation. We also discuss how the second law of thermodynamics restricts the statistics of stochastic efficiency. The tight-coupling limit becomes unfavorable, characterized by an infinitely broad distribution of efficiency at all times, when time-reversal symmetry breaking leads to an asymmetric Onsager response matrix. The underlying physics is demonstrated through the quantum Hall effect and further elaborated in a triple-quantum-dot three-terminal thermoelectric engine.

Figure 1

Efficiency statistics for TRS and TRB systems. (a) An example of the LDF $J(\eta )$ for TRS (blue, $r=1$) and TRB (green, $r=1.6$) cases with $\alpha =-0.3$ and $q=0.7$. (b) Least probable efficiency ${\eta }^{\star }$ for these two systems at different $\alpha$. (c) LDF $J(\eta )$ as a function of $\alpha$ and $\eta$ for a TRB system with $r=1.6$ and $q=0.7$. Note that the least probable efficiency (the maximum of the LDF) ${\eta }^{\star }$ shifts away from the Carnot efficiency [${\eta }_C\equiv 1$, dashed line in (c)] for TRB systems as demonstrated in panels (a), (b), and (c). In panel (c) the minimum of $J(\eta )$ for $\eta \gg 1$ with large negative $\alpha$ is associated with the reversed machine of which the actual efficiency is $1/\eta$, instead of $\eta$. (d) Our analysis is exemplified on a triple-QD thermoelectric device; see Fig. 4 and text.

Figure 2

Efficiency statistics under the maximum average efficiency condition. (a) Maximum average efficiency ${\overline{\eta }}_{\max }$ (blue curve), thermodynamic upper bound of the average efficiency ${\overline{\eta }}_{\text{bound}}$ (red curve), and the least probable efficiency ${\eta }^{\star }$ (green curve) for $q=0.5$. The blue dot represents the TRS point, $r=1$. (b) Width of efficiency distribution ${\sigma }_{\eta }$. The dashed line represents the TRS limit, $r=1$. The white region is forbidden by the thermodynamic bound, Eq. (4).

Figure 3

Efficiency statistics under the maximum average power condition. (a) Least likely efficiency ${\eta }^{\star }$ as a function of $r$ and $q$. The dashed line represents the TRS limit, $r=1$. The white region outside is forbidden by the thermodynamic bound, Eq. (4). In contrast, the white region inside the color graphics region is due to divergences of ${\eta }^{\star }$ at special $r-q$ lines given by $r=(-1\pm \sqrt{1+8/q^2})/2$. (b) Least probable efficiency ${\eta }^{\star }$, average efficiency $\overline{\eta }(W_{\max })$, and its upper bound ${\overline{\eta }}_{\text{bound}}(W_{\max })$ at $q=0.5$ as a function of $r$. The dots identify values in the TRS limit, $r=1$.

Figure 4

Triple-QD thermoelectric system at maximum average output power condition: (a) Least probable efficiency ${\eta }^{\star }$ and (b) width of efficiency distribution ${\sigma }_{\eta }$. QD “3,” connected to the probe terminal, is set at $E_3=2$. $\varphi =\pi /2$, $\mathrm{\Gamma }=0.5$, and $t=-0.2$. The white region in (a) depicts very large or very small (negative) ${\eta }^{\star }$ values which are not properly displayed.