Experimental Realization of Feynman's Ratchet

Experimental realization of Feynman’s ratchet

Jaehoon Bang,1, ∗ Rui Pan,2, ∗ Thai M. Hoang,3, † Jonghoon Ahn,1 Christopher Jarzynski,4 H. T. Quan,2, 5, ‡ and Tongcang Li1, 3, 6, 7, § 1School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA 2School of Physics, Peking University, Beijing 100871, China 3Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA 4Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742 USA 5Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 6Purdue Quantum Center, Purdue University, West Lafayette, IN 47907, USA 7Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USA (Dated: April 24, 2021) Feynman’s ratchet is a microscopic machine in contact with two heat reservoirs, at temperatures TA and TB , that was proposed by Richard Feynman to illustrate the second law of thermodynamics. In equilibrium (TA = TB ), thermal fluctuations prevent the ratchet from generating directed motion. When the ratchet is maintained away from equilibrium by a temperature difference (TA 6= TB ), it can operate as a heat engine, rectifying thermal fluctuations to perform work. While it has attracted much interest, the operation of Feynman’s ratchet as a heat engine has not been realized experimentally, due to technical challenges. In this work, we realize Feynman’s ratchet with a colloidal particle in a one dimensional optical trap in contact with two heat reservoirs: one is the surrounding water, while the effect of the other reservoir is generated by a novel feedback mechanism, using the Metropolis algorithm to impose detailed balance. We verify that the system does not produce work when TA = TB , and that it becomes a microscopic heat engine when TA 6= TB . We analyze work, heat and entropy production as functions of the temperature difference and external load. Our experimental realization of Feynman’s ratchet and the Metropolis algorithm can also be used to study the thermodynamics of feedback control and information processing, the working mechanism of molecular motors, and controllable particle transportation.

In his Lectures on Physics, Richard Feynman intro- been proposed, and experimental demonstration of the duced an ingenious model to illustrate the inviolability directed Brownian motion in asymmetric potentials had of the second law of thermodynamics [1]. A ratchet and been accomplished [15–20]. Recently, a macroscopic (10- pawl are arranged to permit a wheel to turn in only one direction, and the wheel is attached to a windmill immersed in a gas. Random collisions of gas molecules A B LED against the windmill’s panes would then seemingly drive the wheel to rotate systematically in the allowed direc- Water (T ) tion. This rotation could be used to extract useful work B from the thermal fluctuations in the gas, in flagrant vi- Mode 2: engaged olation of the second law of thermodynamics. As dis- Objective Switching probability: ∆푬 cussed by Feynman, this violation does not occur because − AOD 풌 푻 thermal fluctuations of the pawl occasionally allow the 퐦퐢퐧[ퟏ, 풆 푩 푨] ratchet to move in the “forbidden” direction. However, Laser if the pawl is maintained at a temperature that differs from that of the gas, then the device is indeed able to Metropolis rectify thermal fluctuations to produce work – in this Mode 1: disengaged algorithm (T ) case it operates as a microscopic heat engine. Camera A

Three essential features are needed to produce directed FIG. 1. Schematic drawings of the experiment. A, A saw- arXiv:1711.04968v1 [cond-mat.stat-mech] 14 Nov 2017 motion in Feynman’s model: 1) when the ratchet and tooth potential and a uniform potential correspond to the pawl are engaged, the potential energy profile must be engaged and disengaged modes, respectively, of the ratchet asymmetric; 2) the device must be in contact with ther- and pawl. We switch between the two potential modes fol- mal reservoirs at different temperatures; and 3) the de- lowing the Metropolis algorithm to generate the effects of a heat reservoir at temperature TA. B, A 780-nm-diameter sil- vice must be small enough to undergo Brownian motion ica microsphere is trapped in a 1D optical trap inside a water driven by thermal noise. Extensive theoretical analyses chamber using a 1064 nm laser. The water is at temperature of Feynman’s device [2–9] and related ratchet models [10] TB = 296 K. The location of the particle is recorded by a have been used to gain insight into motor proteins [11– camera for computer feedback control. AOD: acousto-optic 14]. Simpler models with a single heat reservoir have deflector. 2 cm-scale) ratchet driven by mechanical collisions of 4- A B mm-diameter glass beads was demonstrated [21]. How- TA = 30 K ever, an experimental realization of Feynman’s ratchet, which can rectify thermal fluctuations from two heat reservoirs to perform work, has not been reported to date. A major challenge is to devise a microscopic system that is in contact with two heat reservoirs at different temperatures, without side effects such as fluid convection that C D can smear the effects of thermal fluctuations. TA = 296 K TA = 3000 K Here we realize Feynman’s two-temperature ratchet and pawl with a colloidal particle confined in a one- dimensional (1D) optical trap (Fig. 1). The particle’s 1D Brownian motion emulates the collision-driven rotation of the ratchet, and we use optical tweezers to generate both uniform and sawtooth potentials, simulating respectively the disengaged and engaged modes of the pawl (Fig. 1A). The uniform potential is smooth enough FIG. 2. Potential profiles of the 1D optical trap and trajecto- that the colloidal particle moves freely in the disengaged ries of the microsphere. A, Measured mode 1 (red) and mode mode. The water surrounding the colloidal particle pro- 2 (blue) potential profiles at room temperature TB = 296 K. vides a heat reservoir at temperature TB. The other heat Black lines indicate the fitted linear potentials used in numer- reservoir is generated by using feedback control of the ical simulations. The asymmetry in mode 2 is about 1:3. B-D, 50 individual trajectories (thin lines) and the average trajec- optical tweezer array [22] to toggle between the disen- tory (thick blue line) of the microsphere when the second heat gaged and engaged modes of the pawl, implementing the reservoir is at temperature TA = 30 K (B), TA = 296 K (C), Metropolis algorithm [23] as in Ref. [6] to satisfy the and TA = 3000 K (D). The average displacement of the par- detailed balance condition at a chosen temperature TA ticle at 60 s is h∆xi = 2.1 ± 0.6µm in B, h∆xi=−0.1 ± 0.9µm (Eq. (1)). Our setup, inspired by both continuous [3–5] in C, and h∆xi=−4.5 ± 0.9µm in D. and discrete [6] theoretical models, captures the essential features of Feynman’s original model, with the pawl diffusive motion of the Brownian particle in the 1D trap maintained at one temperature (TA) and the ratchet at to the stochastic switching between the engaged and dis- another (TB). As described in detail below, both numerical simulation and experimental data clearly show the engaged potential modes. Letting Pi(x, t) denote the influence of the two heat reservoirs on the operation of joint probability density to find the particle at position the ratchet, in agreement with theoretical analyses [1, 6]. x and the potential in mode i ∈ {1, 2} at time t, we In our experiment, a silica microsphere with a diameter construct the reaction-diffusion equation [9] of 780 nm, immersed in deionized water, undergoes diffu- ∂P (x, t) ∂ U 0(x) ∂2P (x, t) i = i P (x, t) + D i sion in a 1D optical trap created with an array of 19 opti- ∂t ∂x γ i ∂x2 cal tweezers. The array is generated by an acousto-optic +k (x)P (x, t) − k (x)P (x, t), (2) deflector (AOD) controlled by an arbitrary function gen- ji j ij i erator [22]. By tuning the power of each optical tweezer where γ is the Stokes friction coefficient, D = kBTB/γ individually, the trap potential can be modified to be ei- the diffusion constant, and kij (kji)(i 6= j) the switch- ther uniform or sawtooth-shaped. The water chamber ing rate from potential mode i(j) to j(i). This rate is mimics the gas heat reservoir in Feynman’s model; its the product of the constant attempt rate, Γ, and the temperature TB was measured to be around 296 K. We Metropolis acceptance probability, Eq. (1): use computer feedback to generate the effects of an arti- Uj(x) − Ui(x) ficial heat reservoir, at temperature TA, coupled to the kij = Γ min 1, exp − , (3) pawl as it switches between its two modes: (1) disengaged kBTA and (2) engaged. Letting U1(x) and U2(x) denote the po- In our experiment, Γ is chosen to optimize the particle tential energies of these modes, as functions of the parti- velocity. The total particle probability distribution is cle location x, we generate attempted switches between P1(x, t) + P2(x, t). modes at a rate Γ, and each such attempt is accepted with The potentials U1(x) and U2(x) are determined from a probability given by the Metropolis algorithm [6, 23]: the measured equilibrium distribution of the particle in P = min[1, exp(−∆E/k T )], (1) each mode. We fix the 1D trap in a particular mode switch B A and we track the Brownian motion of the silica micro- where kB is Boltzmann’s constant, and ∆E = Uj(x) − sphere in the trap, recording its position every 5 ms. 5 Ui(x) for an attempted switch from mode i to mode j. After more than 5 × 10 data points are collected, the To model the dynamics of our system, we couple the potential profile is extracted using the equation U(x) = 3

Figs. 2B, 2C, and 2D show 60-s trajectories of the particle without external load for different heat reservoir temperatures TA. The light lines display individual trajectories, and the thick blue lines are averages over these trajectories. We interpret positive displacements of the particle as clockwise rotation of the mechanical ratchet, and negative displacements as counter- clockwise rotation. As seen in Fig. 2C, the average final displacement of the particle converges essentially to zero, h∆xi = −0.1 ± 0.9µm, when the temperatures of the two reservoirs are equal (TA = TB = 296K) even though the potential is asymmetric. This experimentally verifies Feynman’s prediction that the ratchet does not produce perpetual motion, as clockwise rotations are cancelled by counter-clockwise rotations, on average. FIG. 3. Average velocity of the microsphere under different When TA = 30K (< TB), the average final position of external loads at different temperatures TA. Blue squares, red circles and black triangles represent results with poten- the microsphere is h∆xi = 2.1 ± 0.6µm, indicating net tial slopes of −0.05, 0 and 0.14 kB TB /µm respectively. The clockwise rotation of the ratchet. When TA = 3000K added slope to the original potential represents the external (> TB), the average final position is observed to be load. Every experimental data point represents the result of h∆xi = −4.5 ± 0.9µm, corresponding to net counter- an average of 50 repetitions. Solid lines are simulation results. clockwise rotation. These results demonstrate that a The vertical dashed line indicates TB = 296 K. temperature difference TB − TA can give rise to unidirectional motion via the rectification of thermal noise [9]. We now investigate whether this motion can be harnesses −kBTB ln[N(x)/Ntotal], where N(x) is the number of to perform work against an external load. count at each point and Ntotal is the total count. The To apply an effective external load to the ratchet, a position x is discretized in bins of 52 nm corresponding slight linear slope f is added to the potentials. Fig. 3 to the pixel size of our camera. The measured uniform shows the observed average particle velocity as a function and sawtooth potentials are shown in Fig. 2A. The depth of pawl temperature TA, for f = −0.05kBTB/µm, f = 0 (from trough to peak) of the sawtooth potential U2(x) is and f = 0.14kBTB/µm, corresponding to positive, zero about 4.8 kBTB. The sawtooth potential is described by and negative external loads, respectively. Each experi- an asymmetry ratio of approximately 1:3 (see Fig. 2A); mental data point is calculated from fifty 60-s trajecto- this is a key parameter for thermal ratchets [24]. The uni- ries. The lines show numerical simulation results, with 4 form potential U1(x) has a standard deviation of about each situation simulated over 5 × 10 times to achieve 0.15 kBTB, which is small enough for the silica micro- high accuracy. In these simulations we use the over- sphere to diffuse freely. After measuring the potentials, 1 ∂U damped Langevin equationx ˙ = − γ ∂x + ξ(t), where we fit U1(x) with a straight line and U2(x) with an ideal ξ(t) is Gaussian random force satisfying hξ(t)i = 0, sawtooth potential (Fig. 2A). These fitted smooth po- 0 2kB TB 0 hξ(t)ξ(t )i = γ δ(t − t ), and γ is the Stokes fric- tentials were used in our numerical simulations to avoid tion coefficient. The simulation time step is 2 ms. The the statistical noise in the measurements. simulations are performed with the same procedure as To implement the ratchet dynamics, we first trap a sil- the experiment, described earlier. The simulation and ica microsphere with a single optical tweezer and position experimental results show good agreement over a wide it near the middle of the trap, which corresponds to the range of TA (Fig. 3). potential minimum at the center of mode 2. The optical By realizing an external load, we can interpret our sys- tweezer array is then turned on at mode 2, and we follow tem as a microscopic heat engine. The work produced by the diffusion of the microsphere as the potential switches the engine is given by the product of the slope and the fi- between modes as described earlier. The position of the nal displacement of the particle W = f∆x. In the case of microsphere is recorded every 5 ms using a complemen- a positive external load (f < 0), data points in Fig. 3 with tary metal-oxide semiconductor (CMOS) camera (Fig. negative average velocity correspond to positive average 1B). Every 200 ms (this time interval is equal to 1/Γ), work performed by the engine. Conversely, in the neg- an attempt to switch modes is made, and is accepted with ative loading case (f > 0), data points in Fig. 3 having the probability given by Eq. (1). In our experiment, the positive average velocity correspond to positive average sawtooth potential has finite length. To mimic the infi- work performed by the engine. nite nature of a rotating ratchet, the particle is dragged The thermodynamic operation of our system as a heat back to the trap center whenever it reaches one of the engine is further illustrated in Fig. 4. In Fig. 4A we plot potential minima located on both ends. the average work as a function of TA, with the points 4

A B ⑤ ① ② ③ ④

C D ⑤ W (kBTB) ④ 0.16 ③ 0.12

0.08 ② 0.04 ① 0

FIG. 4. Work, heat, and entropy production of Feynman’s ratchet. A, Extracted work values when the added potential slope is −0.05 kB TB /µm (blue squares), 0 kB TB /µm (red circles) and 0.14 kB TB /µm (black triangles). Error bars show the uncertainty of the average. B, Heat absorbed from reservoir A (red circles) and B (blue squares) in the presence of −0.05 kB TB /µm slope is calculated for each trajectory. Each data point shows the average heat calculated from 50 experimental trajectories. Error bars indicating the uncertainty are smaller than the symbols. Solid lines are simulation results. Dashed vertical line indicates TB = 296K. C, The numerically calculated work map of Feynman’s ratchet with representative points corresponding to Fig. 3 and Fig. 4A. The color shaded area is the heat engine regime where the simulated work is positive. D, average entropy production in 60 s as a function of TA.

labeled by circled integers indicating the parameters for reservoirs, QB and QA. For the case of positive load, Fig. which the system generates positive work. For our pa- 4B plots the average values of these quantities. Note that rameter choices, the greatest observed amount of work these data show that the system absorbs energy (Q > 0) extracted in 60 s is 0.14 kBTB when f = −0.05kBTB/µm, from the hotter reservoir and releases energy (Q < 0) and 0.16 kBTB when f = 0.14kBTB/µm. Recall that in into the colder reservoir, in agreement with expectations. the absence of external load, our system generates pos- We have also computed the average entropy production hQAi hQB i itive velocities when TA < TB and negative ones when hdSi = − − , shown in Fig. 4D. As expected, TA TB TA > TB (Fig. 2), suggesting that in these situations it hdSi = 0 when TA = TB as the system is then in equilib- might be able to perform work when f > 0 and f < 0, re- rium, but hdSi > 0 when TA 6= TB, in agreement with the spectively. The points corresponding to hW i > 0 in Fig. second law. In our model, most of the energy absorbed 4A confirm this expectation. (There is a minor exception from the hot reservoir is delivered to the cold reservoir, when f = −0.05kBTB/µm and TA = 500K, where the with only a very small portion converted to work – this experimental uncertainty is too large to observe the very explains the observation that the entropy production is small predicted positive work.) Similarly, in the phase largely independent of external load, as seen in Fig. 4D. diagram shown in Fig. 4C, we see that the system acts Fig. 4 shows agreement between experiments (points) as a heat engine (i.e. generates positive work) only when and simulations (lines), and demonstrates the operation the sign of TA − TB is opposite to the sign of f. of Feynman’s ratchet as an engine that rectifies thermal Changes in the potential energy of the particle due to fluctuations to perform work. It is interesting to consider the thermodynamic efficiency of the engine, η = hW i , diffusion are associated with heat exchange with reservoir hQhighi B, and changes in its potential energy during switches be- where Qhigh denotes the heat from the reservoir with a tween the two modes are associated with heat exchange higher temperature. Although Feynman suggested that with reservoir A. 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