Higher-order efficiency bound and its application to nonlinear nanothermoelectrics

Power and efficiency of heat engines are two conflicting objectives. A tight efficiency bound is expected to give insights on the fundamental properties of such a power-efficiency tradeoff. Here, we derive an upper bound on the efficiency of steady-state heat engines, which incorporates higher-order fluctuations of power. In a prototypical model of nonlinear nanostructured thermoelectrics, we show that the obtained bound is tighter than a well-established efficiency bound derived from the thermodynamic uncertainty relation, demonstrating that the higher-order terms have rich information about the thermodynamic efficiency in the nonlinear regime. In particular, we find that the higher-order bound is exactly achieved if the tight coupling condition is satisfied. The obtained bound gives a consistent prediction with an observation that nonlinearity enhances the power-efficiency tradeoff, and would also be useful in a variety of nanoscale engines.

Figure 1

Sketch of the two-level quantum dot: The dot has two energy levels and electrons are flowing from or dissipating into the hot and cold reservoirs. Each level corresponds to a different direction of the electron spin. In the sequential tunneling regime and under the wide band approximation, the transition rate is determined by the same coupling strength and the Fermi-Dirac distribution of the level at each reservoir, reflecting upon the occupancy of electrons. In the usual experimental setup, the chemical potentials (equivalently the voltages) are varied while fixing the temperatures. A load to the reservoirs has to be attached to extract the electrical power. For simplicity, we assume the left-right symmetry of the coupling, that is, the coupling strength is identical for both the reservoirs. The capacitance is not installed between the levels, and we assume that there is no interaction between them.

Figure 2

Numerical results on the two-level quantum dot: The estimation of the entropy production rate is plotted according to the TUR [Eq. (1)] and the higher-order TUR [Eq. (3)] with $j_d={\beta }_{\mathrm{c}}P$ for the degenerate case ($x_{\mathrm{h}}=2.0$) and the nondegenerate case ($x_{0u}^{\mathrm{h}}=x_{d2}^{\mathrm{h}}=2.0,x_{0d}^{\mathrm{h}}=x_{u2}^{\mathrm{h}}=2.2$). The true value, $\sigma$, is obtained by setting $d_{ij}=x_{ij}$. In this example, the calculation is shown outside the heat engine mode $(F_{\mathrm{M},\text{stop}}\le F_{\mathrm{M}}\le 0)$ for illustration. $F_{\mathrm{H}}=4.0$. $F_{M,\text{stop}}=-8.0$ for the degenerate case.

Figure 3

Numerical results on the two-level quantum dot: The EBs are plotted according to Eq. (4). The estimation of the entropy production is obtained by truncating the cumulant series at the $n\mathrm{th}$-order term and maximizing it in terms of $\chi$. The true value, $\eta$, is obtained by dividing the power by the heat current. This heat current is calculated by setting $d$ such that $d_{ij}$ equals $-{\beta }_{\mathrm{h}}^{-1}{x}_{ij}^{\mathrm{h}}$ if the hot reservoir drives the transition, and equals zero otherwise. $F_M$ is varied as long as the power output is positive, while $F_H$ is fixed at some value, $F_{\mathrm{H}}=4.0$. (a) The degenerate case, where $x_{\mathrm{h}}=2.0$. (b) The nondegenerate case, where $x_{0u}^{\mathrm{h}}=x_{d2}^{\mathrm{h}}=2.0,x_{0d}^{\mathrm{h}}=x_{u2}^{\mathrm{h}}=2.2$.

Figure 4

Numerical results on the two-level quantum dot: The efficiency at maximum power and the EBs with the optimal parameters are plotted. These parameters consist of the thermodynamic force $F_{\mathrm{M}}$ and the entropy production in the single transition $x_{ij}^{\mathrm{h}\left(\mathrm{c}\right)}$. The levels are assumed to be degenerate in this optimization because the two channels can be optimized simultaneously under this condition. The EMP ${\eta }^{*}$ agrees with the one obtained in Ref. [34], where nonlinearity enhances the EMP.

Figure 5

(a) The sketch of the one-dimensional random walk. A single particle is hopping among the infinite sequence of sites. The hopping rate is biased and the particle current is driven by the thermodynamic force $F$. (b) Numerical results on the one-dimensional random walk. The estimations of the entropy production based on the TUR [Eq. (1)] and the higher-order TUR [Eq. (3)] are plotted.

Figure 6

Numerical results on the two-level quantum dot in the degenerate case: The denominators of Eq. (4) before the maximization are plotted. The expansion of $\lambda$ is truncated at the $n\mathrm{th}$-order term. The one with the full series ($n=\infty$) has its global maximum at $\chi =F$. $F_{\mathrm{H}}=4.0,x_{\mathrm{h}}=2.0$. The parameters are given as (a) $F_{\mathrm{M}}=-0.20x_{\mathrm{h}}{F}_{\mathrm{H}}$, (b) $F_{\mathrm{M}}=-0.64x_{\mathrm{h}}{F}_{\mathrm{H}}$.