Maximizing power and velocity of an information engine

Abstract

Information-driven engines that rectify thermal fluctuations are a modern realization of the Maxwell-demon thought experiment. We introduce a simple design based on a heavy colloidal particle, held by an optical trap and immersed in water. Using a carefully designed feedback loop, our experimental realization of an "information ratchet" takes advantage of favorable "up" fluctuations to lift a weight against gravity, storing potential energy without doing external work. By optimizing the ratchet design for performance via a simple theory, we find that the rate of work storage and velocity of directed motion are limited only by the physical parameters of the engine: the size of the particle, stiffness of the ratchet spring, friction produced by the motion, and temperature of the surrounding medium. Notably, because performance saturates with increasing frequency of observations, the measurement process is not a limiting factor. The extracted power and velocity are at least an order of magnitude higher than in previously reported engines.

Keywords: feedback trap; information engine; stochastic thermodynamics.

Fig. 1.

Schematic of the information engine. (A) Ratcheted spring-mass system under gravity. (B) Experimental realization using horizontal optical tweezers in a vertical gravitational field. Feedback operations on the right side in A and B are indicated by the small red “swoosh” arrows.

Fig. 2.

Zero-work condition defining a pure information engine. Trap power $P_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ as a function of feedback gain $\alpha$ for fixed threshold $X_{\mathrm{T}}=0$, scaled relaxation frequency ${\tau }_{\mathrm{r}}/t_{\mathrm{s}}^{\prime }=180$, scaled effective mass ${\delta }_{\mathrm{g}}=0.8$, relaxation time ${\tau }_{\mathrm{r}}$ = 3.6 ms, diffusion constant $D=0.16$ $\mathrm{\mu }{\mathrm{m}}^2$/s, trap stiffness $\kappa$ = 7.0 pN/$\mathrm{\mu }$m, and bead diameter of 3 $\mathrm{\mu }$m. (Bottom-Right Inset) Experimental trajectories of the bead [$x\left(t\right)$, gray] and trap [$\lambda \left(t\right)$, black] during continuous ratcheting. (Top-Left Inset) Naive zero-work condition for a harmonic potential is $\alpha =2$, equivalent to $X_{\mathrm{R}}=X_{\mathrm{T}}$. The black curve denotes the trap potential in the current step and gray curve in the previous step. Error bars here and in other figures are the standard error of the mean (SI Appendix, section M).

Fig. 3.

Optimization of ratcheting power. (A) Power as a function of sampling frequency $f_{\mathrm{s}}={\tau }_{\mathrm{r}}/t_{\mathrm{s}}^{\prime }$. The black solid curve denotes the semianalytic results (SI Appendix, section E) for the same material parameters, feedback gain $\alpha =1.8$, and threshold $X_{\mathrm{T}}=0$. The horizontal dotted line indicates the infinite-frequency limit (Eq. 11b multiplied by ${\delta }_{\mathrm{g}}$), and the dashed line denotes the low-frequency limit (SI Appendix, section D). (B) Power as a function of threshold $X_{\mathrm{T}}$ for fixed $\alpha =1.9$ and sampling frequency of 50 kHz. The gray markers show that the input trap power is small. The black curve follows from Eq. 11a. Red markers denote experimental values. For all data, scaled effective mass ${\delta }_{\mathrm{g}}=0.8$, relaxation time ${\tau }_{\mathrm{r}}$ = 3.5 ms, diffusion constant = 0.16 $\mathrm{\mu }{\mathrm{m}}^2$/s, trap stiffness $\kappa$ = 7.3 pN/$\mathrm{\mu }$m, and bead diameter = 3 $\mathrm{\mu }$m.

Fig. 4.

Power and velocity for bead diameters 0.5 (green), 1.5 (black), 3 (red), and 5 (blue) $\mathrm{\mu }$m for threshold $X_{\mathrm{T}}=0$. Markers denote experimental data (see SI Appendix, Table S2 for experimental parameters). (A) Velocity as a function of trap stiffness $\kappa$. (B) Scaled velocity as a function of scaled effective mass ${\delta }_{\mathrm{g}}$. (C) Power as a function of $\kappa$. (D) Scaled power as a function of ${\delta }_{\mathrm{g}}$. Gray markers show that the corresponding $P_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ values remain small. Solid curves in A and C are calculated from Eqs. 12a and 12b. Solid curves in B and D are calculated from Eq. 11b.