Piston-Like Information Engine I: Universal Features in Equilibrium

Abstract

The ability to measure the stochastic degrees of freedom of a thermal system enables the extraction of energy from an equilibrium heat bath. This is the underlying principle of Maxwell's demon and subsequent information engines. Here, we experimentally realize a microscopic information engine configured as a compressible piston containing a thermalized colloidal suspension. The particle positions are recorded to identify when a predefined region near the wall is empty, allowing the piston to compress the colloidal suspension without applying work on the system. We find that the stored compression energy is universally set by the probability of a positive measurement outcome, which in turn is controlled by parameters such as density and compression step size. We further demonstrate that mechanical work can be extracted during the decompression of the piston, thereby closing the engine's operating cycle.

Figures (11)

• Figure 1: Schematic view of the experimental setup. A pair of accousto-optic deflectors (AODs) are used to stir a single 1064 nm laser beam at a high rate along a 2D horizontal $16 \times 16 ~\rm{\mu m^2}$ square. Colloidal particles made of silica diffusing in a high refractive index solution ($90\%~\rm{DMSO}$ and $10\%~\rm{H_20}$) are repelled by the laser beam. This setup effectively creates a bidimensional optical box enclosing the particles. The operation of the information engine is sketched on the right: positions of the colloidal particles are monitored in time ; if, at a time $t$, no colloidal particles are measured in an area $\Delta A$ in the vicinity of the left wall of the box, the latter can me shifted inward. Doing so, the box is compressed, increasing the osmotic pressure without applying any direct work onto the particles. To close the cycle, the energy stored in the piston can be converted back to mechanical work by allowing the colloidal suspension to expand against an obstacle, performing work on objects in its path.
• Figure 2: (a) Measured equilibrium probability densities of colloidal particles in the optical box with increasing compression $A = 16 \times 16~$ (1); $11.2 \times 16~$ (2) and $8 \times 16 ~\rm{\mu m^2} ~$(3). The shape of the optical box is superimposed in red. Along the bidimensional heatmap, we show both experimental projections along the x (horizontal) and y (vertical) axis. (b) Osmotic pressure evaluated in three different ways: Experimentally measured mechanical osmotic pressure $\Pi_{\rm osm}$ (red circles), corresponding $\Pi_{\rm EOS}$ (yellow squares) calculated using the scaled particle theory approximation, mechanical osmotic pressure evaluated from numerical simulations (blue triangles), and a the exact expression of $\Pi_{\rm EOS}$ Eq. (\ref{['Eq:EOSPress']}) (black dashed line). The orange numbers (1), (2) and (3) denote the box states corresponding to the densities shown in the top panel.
• Figure 3: (a) Work $W$ stored under the form of osmotic pressure as a function of the area compression $\Delta A = L_y \times \Delta x$ for various initial box sizes (from $A_0 = 16\times16 ~\rm \mu m^2$ in blue, to $8\times16 ~\rm \mu m^2 = 50\% A_0$ in red). (b) Probability $p_1$ of a positive measurement outcome i.e. that the area $\Delta A$ is empty, as a function of the width $\Delta x$ of the probes area. Experimental result (blue to red solid lines and filled area) in close agreement with numerical simulations (gray solid lines); both stay similar to the result of an ideal gas (black dashed line) $p_1^{\rm ideal} = (1 - \Delta_x/L_x)^N$. (c) Normalized mean work per measurement $\overline W= p_1 W$ for each box sizes, plotted against $p_1$. The experimental results (symbols) and numerical simulations (solid lines) bundle around the universal curve Eq. (\ref{['eq:universal']}) (black dash-dotted line). The two most compressed cases depart more significantly.
• Figure 4: Converting osmotic pressure into mechanical work. (a) Initial state: A large diameter $d = 5 ~\rm \mu m$) colloidal particle is brought close to the wall of the compressed box, which contains $8$ colloidal particles in equilibrium. The compressed left optical wall is then removed, letting the colloidal particle suspension to expand by freely diffusing out of the left aperture of the box. (b) Final state: after a time $t \approx 8$ minutes, the small colloidal particles have escaped the confining box, pushing the large colloid along the x-axis. (c) Quantitative measurement of the displacement of the large colloidal particle. The drift $\langle x(t) \rangle$ is the average over 6 repetition of the experiment. The same number of control experiments are realized with the same box geometry (which applies a force on the large colloidal particle) without the enclosed small colloids. The represented drift (red line) is the difference between the experiment and the control.
• Figure 5: Optical pattern corresponding to the full (left) and partially compressed (right) optical box created by AOD beam multiplexing. A single trap in sequentially illuminating each point along a $16\times16~\rm \mu m$ square, with 20 traps per side. The red arrow is indicating the position and direction of movement of the trap illuminated last.