Efficiency Fluctuations in Microscopic Machines

Nanoscale machines are strongly influenced by thermal fluctuations, contrary to their macroscopic counterparts. As a consequence, even the efficiency of such microscopic machines becomes a fluctuating random variable. Using geometric properties and the fluctuation theorem for the total entropy production, a “universal theory of efficiency fluctuations” at long times, for machines with a finite state space, was developed by Verley et al. [Nat. Commun. 5, 4721 (2014); Phys. Rev. E 90, 052145 (2014)]. We extend this theory to machines with an arbitrary state space. Thereby, we work out more detailed prerequisites for the “universal features” and explain under which circumstances deviations can occur. We also illustrate our findings with exact results for two nontrivial models of colloidal engines.

Figure 1

Illustration of the relation (5) between $\varphi ({\lambda }_Q,{\lambda }_W)$ and $J(\eta )$. Left: Contour plot of $\varphi ({\lambda }_Q,{\lambda }_W)$ showing its convexity along with the lines ${\lambda }_Q=\eta {\lambda }_W$ (dashed) for the macroscopic efficiency $\overline{\eta }$ (black), reversible efficiency ${\eta }_{\mathrm{C}}$ (orange), and an intermediate value (gray). The red curve marks the minimizing $\tilde{\mathit{\lambda }}(\eta )$ for all $\eta$. Right: The resulting $J(\eta )=-\varphi \mathbf{(}{\tilde{\lambda }}_Q(\eta ),{\tilde{\lambda }}_W(\eta )\mathbf{)}$.

Figure 2

Contour plots of a typical $\varphi ({\lambda }_Q,{\lambda }_W,{\lambda }_S)$ for ${\lambda }_S=0,\frac{1}{2},1$ along with the domain of convergence $C$ and the domain of convergence $C_0$ for $\varphi$ at ${\lambda }_S=0$. The functional form of $\varphi ({\lambda }_Q,{\lambda }_W,{\lambda }_S)$ is the same in all ${\lambda }_S=\text{const}$ planes, but the limited domain of convergence leads to cutoffs whose location changes as a function of ${\lambda }_S$. The symmetry around the point $({\lambda }_Q^{*}/2,{\lambda }_W^{*}/2,1/2)$ is a consequence of the fluctuation theorem.

Figure 3

$\varphi ({\lambda }_Q,{\lambda }_W)$ (left) and $J(\eta )$ (right) for the isothermal work-to-work converter from Ref. [24]. Exact results are shown for two different parameter sets of the model (see Refs. [24, 37] for details). Dashed black lines mark the corresponding macroscopic efficiencies $\overline{\eta }$ (negative in this case), dashed orange lines the reversible efficiency ${\eta }_{\mathrm{C}}=1$. Top (model parameters $\theta =1$, $\alpha =1/2$): Since the cutoff does not intersect the minimizing curve $\tilde{\mathit{\lambda }}(\eta )$, the emerging shape of the rate function $J(\eta )$ still follows the predictions of the VWVE theory: it is smooth and has a unique maximum at the reversible efficiency ${\eta }_{\mathrm{C}}=1$. Bottom ($\theta =4$, $\alpha =1/2$): The cutoff interferes with the minimizing curve $\tilde{\mathit{\lambda }}(\eta )$ so that the latter runs along the boundary of the domain of convergence for $\eta \in [\frac{1}{2},\frac{7}{2}]$ (purple, dashed lines) and the rate function (right panel, blue curve) becomes distorted from the shape that would be obtained without cutoffs (dotted, gray curve). In particular, it develops kinks that become visible in its first derivative. One of these is displayed in the inset of the bottom-right panel.

Figure 4

$\varphi ({\lambda }_Q,{\lambda }_W)$ (left) and $J(\eta )$ (right) for two configurations of the Brownian gyrator. Dashed black lines mark the macroscopic efficiency $\overline{\eta }$, dashed orange lines the reversible efficiency ${\eta }_{\mathrm{C}}$. The minimizing curve $\tilde{\mathit{\lambda }}(\eta )$ is shown in red in the left panels. The efficiencies at the edges of the plateau region are marked by dotted blue lines in the right panels. Top: $u_1=5$, $u_2=1$, $\alpha =\pi /4$, $f_{\mathrm{ext}}=-1/2$, $k_{\mathrm{B}}{T}_1=1$, $k_{\mathrm{B}}{T}_2=1/3$. Bottom: $u_1=4$, $u_2=2$, $\alpha =\pi /4$, $f_{\mathrm{ext}}=-1/2$, $k_{\mathrm{B}}{T}_1=2$, $k_{\mathrm{B}}{T}_2=1/10$. In all plots, ${\gamma }_1={\gamma }_2=1$. In this case, the kinks occur in $J(\eta )$ itself.