Thermodynamic Bound on Heat-to-Power Conversion

In systems described by the scattering theory, there is an upper bound, lower than Carnot, on the efficiency of steady-state heat-to-work conversion at a given output power. We show that interacting systems can overcome such bound and saturate, in the thermodynamic limit, the much more favorable linear-response bound. This result is rooted in the possibility for interacting systems to achieve the Carnot efficiency at the thermodynamic limit without delta-energy filtering, so that large efficiencies can be obtained without greatly reducing power.

Figure 1

Electrical conductivity $\sigma$ (a), thermopower $S$ (b), thermal conductivity $\kappa$ (c), and figure of merit $ZT$ (d) as a function of the mean number $N$ of particles inside the system, for the one-dimensional, diatomic hard-point gas. Here and in the other figures, the data are obtained for $M=3$, $T=1$, and $\mu =0$.

Figure 2

Relative efficiency $\eta /{\eta }_C$ versus normalized power $P/P_{\max }$ for $\mathrm{\Delta }T=0.2$ ($T_L=1.1$, $T_R=0.9$) and different system sizes. The dotted, dashed, and dot-dashed curves show the expectation from linear response, Eq. (9), at the $ZT(N)$ value corresponding to the given system size $N$. The solid line is Eq. (9) for $ZT=\infty$, corresponding to $N=\infty$ in our model. The upper branch of this curve sets the linear-response upper bound on efficiency for a given power.

Figure 3

Maximum efficiency ${\eta }_{\max }$ versus the corresponding power $P({\eta }_{\max })$, from linear response with the values of $ZT$ obtained numerically (red circles) and directly from numerical computation of power and efficiency (black and white circles) for various system sizes with $T_L=1.1$ and $T_R=0.9$. The dot-dashed line is for the analytical expectation from linear response at large $ZT$, Eq. (11). We also show the bound for classical noninteracting systems (solid line) and its approximation at the low power limit given by Eq. (5) (dashed line) as a comparison. Data from the stochastic model described in the text, with $p$ crossing probability each time two particles meet, are also reported (for further details on this model, see Sec. A in the Supplemental Material [18]).