Experimental realization of Feynman's ratchet

In his Lectures on Physics, Richard Feynman introduced an ingenious model to illustrate the inviolability of the second law of thermodynamics [1]. A ratchet and 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 against the windmill's panes would then seemingly drive systematic rotation in the allowed direction, which could be used to deliver useful work in violation of the second law. As discussed by Feynman, this violation does not occur, because thermal fluctuations of the pawl occasionally allow the ratchet to move in the 'forbidden' direction. However, if the pawl is maintained at a temperature that differs from that of the gas, then the device is indeed able to rectify thermal fluctuations to produce work—in this case it operates as a microscopic heat engine. This model elegantly illustrates the idea that thermodynamic laws governing heat and work, originally derived for macroscopic systems, apply equally well at the nanoscale where fluctuations dominate.

Three essential features are needed to produce directed motion in Feynman's model: (1) when the ratchet and pawl are engaged, the potential energy profile must be asymmetric; (2) the device must be in contact with thermal reservoirs at different temperatures; and (3) the device must be small enough to undergo Brownian motion driven by thermal noise. Feynman's ratchet as a paradigm for rectifying thermal noise has inspired extensive theoretical studies [2–8] and related ratchet models [9] have been used to gain insight into motor proteins [10, 11]. Directed Brownian motion in asymmetric potentials, with a single heat reservoir, has been demonstrated experimentally in the presence of non-thermal driving [12–15]. Recently, a macroscopic (10 cm scale) ratchet driven by mechanical collisions of 4 mm diameter glass beads was demonstrated [16]. However, an experimental realization of Feynman's ratchet, which rectifies 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 can smear the effects of thermal fluctuations.

Here we realize Feynman's two-temperature ratchet and pawl with a colloidal particle confined in a one-dimensional (1D) optical trap (figure 1). The particle's 1D Brownian motion emulates the collision-driven rotation of the ratchet, and we use optical tweezers to generate both flat and sawtooth potentials, simulating, respectively, the disengaged and engaged modes of the pawl (figure 1(A)). In the flat potential, the colloidal particle moves freely. The water surrounding the colloidal particle provides a heat reservoir at temperature T_B. The other heat reservoir is generated by using feedback control of the optical tweezer array [17] to toggle between the disengaged and engaged modes of the pawl, implementing the Metropolis algorithm [18] as in reference [6] to satisfy the detailed balance condition at a chosen temperature T_A (equation (1)). Our setup, inspired by theoretical models [3–6], captures the essential features of Feynman's original model, with the pawl maintained at one temperature (T_A) and the ratchet at another (T_B). As described in detail below, both numerical simulation and experimental data clearly show the influence of the two heat reservoirs on the operation of the ratchet, in agreement with theoretical analyses [1, 6].

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In our experiment, a silica microsphere with a diameter of 780 nm, immersed in deionized water, undergoes diffusion in a 1D optical trap created with an array of 19 optical tweezers. The array is generated by an acousto-optic deflector (AOD) controlled by an arbitrary function generator [17]. By tuning the power of each optical tweezer individually, the trap potential can be modified to be either flat or sawtooth-shaped. The water, at temperature T_B = 296 K, plays the role of the gas in Feynman's model, providing thermal noise that drives the motion of the ratchet. We use feedback control to generate the effects of a second heat reservoir, at temperature T_A, that drives the pawl as it switches between its two modes: (1) disengaged and (2) engaged. Letting U₁(x) and U₂(x) denote the potential energies of these modes as functions of the particle location x, we generate attempted switches between modes at a rate Γ, and each such attempt is accepted with a probability given by the Metropolis algorithm [6, 18]:

$$\begin{eqnarray}&&{P}_{\mathrm{switch}}(x)=\min [1,\exp (-{\rm{\Delta }}E(x)/{k}_{{\rm{B}}}{T}_{A})],\end{eqnarray} \tag{ 1 }$$

where k_B is Boltzmann's constant, and ${\rm{\Delta }}E(x)={U}_{j}(x)-{U}_{i}(x)$ for an attempted switch from mode i to mode j.

By enforcing the detailed balance condition, equation (1) ensures that the toggling between the two modes consistently reflects the exchange of energy with a thermal reservoir at temperature T_A. When we set T_A = T_B the system relaxes to equilibrium, but when ${T}_{A}\ne {T}_{B}$ we obtain a non-equilibrium steady state in which there is a net flow of energy between the system, the two reservoirs, and (as described later) an external work load.

To gain insight, we consider a simple model that couples the diffusive motion of the Brownian particle in the 1D trap to the stochastic switching between the engaged and disengaged potential modes. Letting P_i(x, t) denote the joint probability density to find the particle at position x and the potential in mode $i\in \{1,2\}$ at time t, we construct the reaction-diffusion equation

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial {{P}}_{i}(x,t)}{\partial t} & = & \displaystyle \frac{\partial }{\partial x}\left[\displaystyle \frac{{U}_{i}^{\prime} (x)}{\gamma }{P}_{i}(x,t)\right]+D\displaystyle \frac{{\partial }^{2}{P}_{i}(x,t)}{\partial {x}^{2}}\\ & & +{k}_{{ji}}(x){P}_{j}(x,t)-{k}_{{ij}}(x){P}_{i}(x,t),\end{array}\end{eqnarray} \tag{ 2 }$$

where γ is the Stokes friction coefficient, $D={k}_{{\rm{B}}}{T}_{B}/\gamma$ the diffusion constant, and k_ij (k_ji)($i\ne j$) the switching rate from potential mode i(j) to j(i). This rate is the product of a constant attempt rate, Γ, and the Metropolis acceptance probability, equation (1):

$$\begin{eqnarray}&&{k}_{{ij}}(x)={\rm{\Gamma }}\min \left[1,\exp \left(-\displaystyle \frac{{U}_{j}(x)-{U}_{i}(x)}{{k}_{{\rm{B}}}{T}_{A}}\right)\right].\end{eqnarray} \tag{ 3 }$$

The total particle probability distribution is ${P}_{1}(x,t)+{P}_{2}(x,t)$.

To implement the ratchet dynamics, we first trap a silica microsphere with a single optical tweezer and position it near the middle of the trap, which corresponds to the potential minimum at the center of mode 2. The optical tweezer array is then turned on at mode 2, and we follow the diffusion of the microsphere as the potential switches between modes as described earlier (and in the method section at the end of this article).

Figures 2(B)–(D) show 60 s trajectories of the particle without external load for different heat reservoir temperatures T_A. 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 figure 2(C), the average final displacement of the particle converges essentially to zero, $\langle {\rm{\Delta }}x\rangle =-0.1\pm 0.9\,\mu {\rm{m}}$, when the temperatures of the two reservoirs are equal (T_A = T_B = 296 K) even though the potential is asymmetric. This experimentally verifies Feynman's prediction that the ratchet does not produce perpetual motion, as clockwise rotations are canceled by counter-clockwise rotations, on average. When T_A = 30 K (<T_B), the average final position of the microsphere is $\langle {\rm{\Delta }}x\rangle =2.1\pm 0.6\,\mu {\rm{m}}$, indicating net clockwise rotation of the ratchet. When T_A = 3000 K (>T_B), the average final position is observed to be $\langle {\rm{\Delta }}x\rangle =-4.5\pm 0.9\,\mu {\rm{m}}$, corresponding to net counter-clockwise rotation. These results demonstrate that a temperature difference T_B − T_A can give rise to unidirectional motion via the rectification of thermal noise [7]. We now investigate whether this motion can be harnessed to perform work against an external load.

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To apply an effective external load to the ratchet, a slight linear slope f is added to the potentials. Figure 3 shows the observed average particle velocity as a function of pawl temperature T_A, for $f=-0.05{k}_{{\rm{B}}}{T}_{B}/\mu {\rm{m}}$, f = 0 and $f=0.14{k}_{{\rm{B}}}{T}_{B}/\mu {\rm{m}}$, corresponding to positive, zero and negative external loads, respectively. Each experimental data point is calculated from fifty 60 s trajectories. The lines show numerical simulation results, with each situation simulated over 5 × 10⁴ times to achieve high accuracy. In these simulations we use the overdamped Langevin equation $\dot{x}=-\tfrac{1}{\gamma }\tfrac{\partial U}{\partial x}+\xi (t)$, where ξ (t) is Gaussian random force satisfying $\langle \xi (t)\rangle =0$, $\langle \xi (t)\xi (t^{\prime} )\rangle =\tfrac{2{k}_{{\rm{B}}}{T}_{B}}{\gamma }\delta (t-t^{\prime} )$. The simulation time step is 2 ms. The simulations are performed with the same procedure as the experiment, described earlier. The simulation and experimental results show good agreement over a wide range of T_A (figure 3).

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By realizing an external load, we can interpret our system as a microscopic heat engine. The work produced by the engine is given by the product of the slope and the final displacement of the particle W = fΔx. In the case of a positive external load (f < 0), data points in figure 3 with negative average velocity correspond to positive average work performed by the engine. Conversely, in the negative loading case (f > 0), data points in figure 3 having positive average velocity correspond to positive average work performed by the engine. Overall, our experimental results agree well with simulation results. There is one blue data point in figure 3 (near the vertical dashed line) that deviates from the simulation result. This data point was taken at the end of the experiment, making it susceptible to thermal and mechanical drift of the system. The deviation of this data point may also be due to the small difference between the real potential and the potential used in the simulation (figure 2(A)).

The thermodynamic operation of our system as a heat engine is further illustrated in figure 4. In figure 4(A) we plot the average work as a function of T_A, with the points labeled by circled integers indicating the parameters for which the system generates positive work. For our parameter choices, the greatest observed amount of work extracted in 60 s is 0.14 k_BT_B when $f=-0.05{k}_{{\rm{B}}}{T}_{B}/\mu {\rm{m}}$, and 0.16 ${k}_{{\rm{B}}}{T}_{B}$ when $f=0.14{k}_{{\rm{B}}}{T}_{B}/\mu {\rm{m}}$. Recall that in the absence of external load, our system generates positive velocities when T_A < T_B and negative ones when T_A > T_B (figure 2), suggesting that in these situations it might be able to perform work when f > 0 and f < 0, respectively. The points corresponding to $\langle W\rangle \gt 0$ in figure 4(A) confirm this expectation. (There is a minor exception when $f=-0.05{k}_{{\rm{B}}}{T}_{B}/\mu {\rm{m}}$ and T_A = 500 K, where the experimental uncertainty is too large to observe the very small predicted positive work.) Similarly, in the phase diagram shown in figure 4(C), we see that the system acts as a heat engine (i.e. generates positive work) only when the sign of T_A − T_B is opposite to the sign of f.

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Changes in the potential energy of the particle due to diffusion are associated with heat exchange with reservoir B, and changes in its potential energy during switches between the two modes are associated with heat exchange with reservoir A. For each trajectory, we can thus keep track of the heat absorbed by the system from the two reservoirs, Q_B and Q_A. For the case of positive load, figure 4(B) plots the average values of these quantities. These data show that the system absorbs energy (Q > 0) from the hotter reservoir and releases energy (Q < 0) into the colder reservoir, in agreement with expectations. We have also computed the average entropy production $\langle {\rm{d}}{S}\rangle =-\tfrac{\langle {Q}_{A}\rangle }{{T}_{A}}-\tfrac{\langle {Q}_{B}\rangle }{{T}_{B}}$, shown in figure 4(D). As expected, $\langle {\rm{d}}{S}\rangle =0$ when T_A = T_B as the system is then in equilibrium, but $\langle {\rm{d}}{S}\rangle \gt 0$ when ${T}_{A}\ne {T}_{B}$, in agreement with the second law. In our model, most of the energy absorbed from the hot reservoir is delivered to the cold reservoir, with only a very small portion converted to work—this explains the observation that the entropy production is largely independent of external load, as seen in figure 4(D).

Figure 4 shows agreement between experiments (points) and simulations (lines), and demonstrates the operation of Feynman's ratchet as an engine that rectifies thermal fluctuations to perform work. It is interesting to consider the thermodynamic efficiency of the engine, $\eta =\tfrac{\langle W\rangle }{\langle {Q}_{{\rm{high}}}\rangle }$, where Q_high denotes the heat from the reservoir with a higher temperature. Although Feynman suggested that his ratchet could achieve Carnot efficiency [1], later authors argued that his analysis was incorrect [3–5]. The efficiency that we measured experimentally is η = 0.0015 for the data point labeled by a circled '5' in figure 4(A), which is much lower than the corresponding Carnot efficiency η_C = 0.9, in agreement with the conclusions of references [3–5]. Feynman's ratchet cannot achieve Carnot efficiency, as it is in contact with two heat reservoirs simultaneously and generates work by continuously rectifying fluctuations in a non-equilibrium steady state [3]. This mode of operation is fundamentally different [4, 19] from the thermodynamic cycle of reversible expansion and compression that characterizes a Carnot engine. We note that micron-sized heat engines that operate in cycles have recently been implemented in experiments using colloidal particles [19–22].

In conclusion, by combining the Metropolis algorithm with feedback control, we have realized Feynman's two-temperature ratchet-and-pawl model using a silica microsphere confined in a feedback controlled 1D optical trap. We study the behavior of the ratchet over a range of temperatures and external loads, demonstrating that it rectifies thermal fluctuations to generate work while obeying the thermodynamic laws originally derived for macroscopic heat engines operating in cycles. When the temperatures of the two heat reservoirs are equal, we find no unidirectional average drift of the particle, in agreement with the second law of thermodynamics. When the temperatures differ, we demonstrate that thermal fluctuations can be rectified by the ratchet to generate work. Our system provides a versatile testbed for studying the non-equilibrium thermodynamics of microscopic heat engines and molecular motors [7, 11, 23, 24]. For instance, multiple heat reservoirs can be mimicked by using a position-dependent T_A(x) in the Metropolis algorithm. Moreover, although we have used feedback control to implement an effective heat bath, in an alternative scenario feedback control could be used to mimic the operation of Maxwell's demon [25], and thus to investigate issues related to the thermodynamics of information processing [25–29]. Our study may also have potential applications in particle transportation [7] and separation [30] induced by Brownian motion in asymmetric potential.

Method

The flat and sawtooth trapping potentials are created by 19 optical tweezers (figure 5). To create them using a laser passing through an AOD, we drive the AOD with a combination of 19 radio frequency (RF) components at different frequencies. The power and position of each optical tweezer is controlled by adjusting the amplitude and frequency of each RF component. As shown in figure 5, each optical tweezer (gray curve) has a Gaussian shape. The spacing between neighboring tweezers is smaller than the waist of each optical tweezer. Thus they overlap and create a continuous 1D trap. By carefully changing the power of each RF component, we can create a flat potential or a sawtooth potential. We only used the center-parts (inside dashed green boxes in figure 5) of the potentials in the experiment. To create an effective periodic potential with an infinite length, we use an optical tweezer to drag the microsphere back to the center well every time when it reaches the bottom of the left well or the bottom of the right well of the sawtooth potential. The locations of the edges are chosen carefully to avoid bias.

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The potentials U₁(x) and U₂(x) are determined from the measured equilibrium distribution of the particle in each mode. We fix the 1D trap in a particular mode and track the Brownian motion of the silica microsphere in the trap, recording its position every 5 ms. After more than 5 × 10⁵ data points are collected, the potential profile is extracted using the equation $U(x)=-{k}_{{\rm{B}}}{T}_{B}\mathrm{ln}[N(x)/{N}_{{\rm{total}}}]$, where N(x) is the number of count at each point and N_total is the total count. The position x is discretized in bins of 52 nm corresponding to the pixel size of our camera. The measured flat and sawtooth potentials are shown in figure 2(A) (in the main text). The depth (from trough to peak) of the sawtooth potential U₂(x) is about 4.8 k_BT_B. The sawtooth potential is described by an asymmetry ratio of approximately 1:3 (see figure 2(A)); this is a key parameter for thermal ratchets. The flat potential U₁(x) has a standard deviation of about 0.15 k_BT_B, which is small enough for the silica microsphere to diffuse freely. After measuring the potentials, we fit U₁(x) with a straight line and U₂(x) with an ideal sawtooth potential (figure 2(A)). These fitted smooth potentials were used in our numerical simulations to avoid the statistical noise in the measurements.

To implement the ratchet dynamics, the position of the microsphere is recorded every 5 ms using a complementary metal-oxide semiconductor camera (figure 1(B)). Every 200 ms (this time interval is equal to 1/Γ), an attempt to switch modes is made, and is accepted with the probability given by equation (1) in the main text. In our experiment, the sawtooth potential has finite length. To mimic the infinite nature of a rotating ratchet, the particle is dragged back to the trap center whenever it reaches one of the potential minima located on both ends. The trajectories of the microsphere are then connected end-to-end to create 60 s trajectories. The average velocities are independent of the length of each trajectory.

Acknowledgments

We recently gave an oral presentation about this work in Conference on Lasers and Electro-Optics (CLEO) in May 2018 [31].