Hyperaccurate thermoelectric currents

Thermodynamic currents, such as energy, heat, and entropy production, can fluctuate significantly at the nanoscale. However, some fluctuate less than others. Hyperaccurate currents are defined as those which fluctuate the least, in the sense that they maximize the signal-to-noise ratio (precision). In this Letter we analytically determine what are the hyperaccurate currents in quantum thermoelectrics, modeled by coherent transport in the Landauer-Büttiker formalism. Our results yield a tight and general bound on precision, which replace the classical thermodynamic uncertainty relations, that can be violated in quantum thermoelectrics. They also allow us to address the question of how close to hyperaccurate is a given current. We illustrate our findings for smooth boxcar functions, and for a double quantum dot operating as a thermal machine. In the latter, we use our results to establish the parameter ranges for which the output power of an autonomous engine can become hyperaccurate arbitrarily far from equilibrium.

Figure 1

(a) Average particle current $J_N$, and particle current fluctuations ${\mathrm{\Delta }}_N^2$, for the smooth boxcar (7), as a function of the chemical potential bias $\beta {\delta }_{\mu }$, with constant ${\beta }_L={\beta }_R=\beta$. (b) Signal-to-noise ratio ${\mathcal{S}}_N=J_N^2/{\mathrm{\Delta }}_N^2$. The region for which the curve is blue represents violations of the classical TUR (2). Also shown is our new hyperaccurate bound, ${\mathcal{S}}_{\mathrm{hyp}}$ [Eq. (9)]. In this case ${\mathcal{S}}_N={\mathcal{S}}_{\mathrm{thyp}}$, so that the particle current is the thermal hyperaccurate current. Parameters: $\beta a=2.5, \beta \gamma =0.5$.

Figure 2

Thermodynamic properties of a double quantum dot engine [Eq. (13)] as a function of the temperature gradient ${\delta }_T/T=(T_L-T_R)/T$, with fixed $T=(T_L+T_R)/2$. (a) Average heat currents $J_Q^{L\left(R\right)}$, and average output power $P$. The system operates as an engine when $P,J_Q^L,J_Q^R>0$. (b) Corresponding SNRs. The black dot dashed curve is the linear hyperaccurate bound (11). For low ${\delta }_T$ the output power is nearly hyperaccurate. But for high ${\delta }_T$, the situation reverses. Parameters: $\beta \mathrm{\Gamma }=\beta \mathrm{\Omega }=\beta \epsilon /2=1, \beta {\mu }_R=-\beta {\mu }_L=0.35$.

Figure 3

How accurate is the output power and entropy production? The curves represent the ratio (a) ${\mathcal{S}}_P/{\mathcal{S}}_{\mathrm{thyp}}$ and (b) ${\mathcal{S}}_{Q_R}/{\mathcal{S}}_{\mathrm{thyp}}$, with respect to the thermal hyperaccurate bound (11), as a function of $\delta \mu /T=({\mu }_L-{\mu }_R)/T$. Each curve is for a different value of ${\delta }_T/T=(T_L-T_R)/T$, where $T=(T_L+T_R)/2$. From top to bottom: ${\delta }_T/T=0.05,0.2,0.6$.

Figure 4

Factor $({\mathrm{\Theta }}_4+6{\mathrm{\Psi }}_4)/{\delta }_{\beta }^4$ (assuming ${\delta }_{\beta \mu }=0$) in Eq. (16), as a function of $T$ and $\mu$, for the boxcar transmission function (7). Negative (blue) means the classical TUR can be violated. (a) Sharp boxcar, $a=1/2, \gamma =0$. (b) Smoothed boxcar, $a=1/2, \gamma =0.1$.