Tuning up Maxwell’s demon

Within just a couple of decades, Maxwell’s demon has gone from being one of the most fundamental and intriguing Gedankenexperiments in physics—but one that appeared impossible to imagine in practice—to being a real setup in several laboratories around the world (1). Maxwell introduced his demon in 1867 to question the validity of the second law of thermodynamics (2). He imagined a “very observant and neat-fingered being” able to monitor the velocity and position of the molecules of a gas and use this information to separate the fast from the slow ones without energy expenditure. This operation is compatible with Newton’s mechanics but appears to defeat the second law, which rules out that temperature differences can develop without energy dissipation somewhere else. The solution to this apparent failure of the second law came when analyses included the physical nature of the demon as well as the thermodynamic consequences of the measurement and the information processing needed by the demon’s actions. Nowadays, we are interested not only in reconciling the Maxwell demon and the second law but also in exploring information as a fuel or thermodynamic resource (3) in artificial microscopic devices as well as in biological motors. In PNAS, Saha et al. (4) tackle an issue that has been barely studied to date: the maximum speed that a Maxwell demon or information engine can reach and the maximum power that it can supply.

Maxwell’s observation was based on his own discovery, a few years earlier, of the distribution of molecular velocities in a gas, a manifestation of the randomness of microscopic magnitudes at finite temperature. In general, an information engine consists of a device that monitors thermal fluctuations, for example, the wiggling position of individual particles, and uses this information to extract energy from a thermal bath, essentially converting heat to work. The fluctuating Brownian motion of particles controlled by optical tweezers has become one of the main experimental tools to explore the physics of information engines (1). A simple version of this setup is used in ref. 4 in order to lift a micrometer-sized particle against the gravitational force (Fig. 1). The particle is immersed in water and trapped by the tweezers, and its time-dependent position x(t) is monitored by microscope optics. The density of the particle is higher than that of the water, so the net force, gravity plus buoyancy, points downward. When the particle undergoes an uphill fluctuation larger than a predefined threshold, the trap is rapidly repositioned upward to give the particle a new equilibrium position from which to fluctuate upward and downward (Fig. 1). The action of the trap can thus be compared to a ratchet that allows fluctuations in one direction but blocks fluctuations in the opposite direction. In order to run the experiment, one must frequently monitor the position of the bead and be able to very quickly respond to fluctuations. Saha et al. manage to check the bead position up to every 20 µs, and can equally quickly enact changes to the setup with nanometer precision.

Fig. 1.

(Left) How the position of the optical trap is adjusted upward each time the bead reaches a given threshold. As an outcome, the random motion of the bead is rectified upward. (Right) An example trajectory x(t) of the bead and the position λ(t) of the center of the trap.

The experiment in ref. 4 has two key features that enable the authors to systematically explore the achievable power and speed of their demon. First, they ensure that the repositioning of the trap—a necessary part of the ratcheting action—is done in such a way that the trap does not push or pull the bead. Hence, the bead gains potential gravitational energy only from the thermal bath via fluctuations. This is achieved by using a trap with a potential energy that is approximately quadratic. If the trap is moved twice the distance between the particle and the center of the trap, the potential energy does not change. In practice, the unavoidable feedback delay slightly distorts this strategy, and the experimenter must calibrate the length of the trap jump. Continuous monitoring of the position of particle and trap allows verification that the jumps indeed do not change the energy of the particle. Therefore, the particle only exchanges energy with the thermal bath, and the motor can be considered a pure information engine.

Second, by working to lift the bead against the gravitational field, the demon stores the gained energy in the same way that a water reservoir can be used to store electric power, such that it can be recovered later as work. The experiment thus represents a true realization of a “useful” demon, in contrast with previous experiments with rotatory Brownian particles (5) or single-electron boxes (6), where the energy is extracted by modifying an external field and is spread out over the many degrees of freedom of the apparatus generating the field.

By systematically tuning key parameters of the experiment, and using analytical theory to complement their study, the authors identify some basic performance principles of information engines based on diffusive beads (4). The speed by which an information engine can move a bead is limited by the rate at which fluctuations are generated. The dispersion of the position of the bead in a harmonic potential of stiffness $\kappa$ at temperature $T$ is $\sigma =\sqrt{kT/\kappa }$, $k$ being the Boltzmann constant, and the relaxation time is ${\tau }_r=\gamma /\kappa$, where $\gamma$ is the friction constant. Roughly speaking, in a time ${\tau }_r$, the system develops spatial fluctuations of size $\sigma$; hence a maximum velocity $v_{\text{max}}\approx \sigma /{\tau }_r=\sqrt{kT\kappa }/\gamma$ is expected. Experimental results and analytical calculations reported in the paper confirm this estimation. Moreover, the maximum velocity is reached when the position of the particle is monitored sufficiently frequently, about 10 times per ${\tau }_r$, such that no useful fluctuation is missed, and the trap is moved whenever an uphill fluctuation is detected. Notice also, from the expression for the maximum velocity, that, perhaps unsurprisingly, the fastest speeds are found for smaller beads, which diffuse faster and need to overcome the least gravitational force.

Of key interest is the demon’s power, that is, the amount of work it can produce in a given time—arguably the most important parameter of a motor. It turns out that, for a given optical trap and bead mass density, there is a bead size that will produce the most power: too small, and a bead is too light to gain potential energy, although it may run fast; too large, and it diffuses too slowly.

In total numbers, Saha et al. (4) find that their experimental setup can produce speeds as high as 190 µm/s, and about half that speed at peak power, which can be as high as 1,000 kT/s. Here $kT=4.2\times {10}^{-21}$ J at room temperature is the energy scale of thermal fluctuations that power any diffusive motor. To put this into context, a micrometer-sized bead moving at 100 µm/s moves 100 times its own length per second. Translated into the macroscopic world of cars, this corresponds to supersonic speeds.

Saha et al. tackle an issue that has been barely studied up to date: the maximum speed that a Maxwell demon or information engine can reach and the maximum power that it can supply.

However, the mechanisms for generating force and speed in a Maxwell demon are fundamentally different from those of macroscopic engines, and much more comparable to the way force is generated by molecular motors in biology. Indeed, protein motors, which power our muscles and practically all processes in our cells, are thought to essentially be information engines that make use of fluctuations by recognizing and locking in diffusion in the desired direction. For example, myosin V, which “walks” along actin filaments, is thought to employ a force-triggered mechanism based on allostery to coordinate the stepping of its two “feet” (7, 8), thus using information processing to rectify fluctuations. Thus, one can expect that the performance of biological motors is limited in a way similar to Maxwell’s demon, by the order of size of fluctuations (universally given by $kT$), and the characteristic time scale of motion of the object in question. Indeed, the performance values observed by Saha et al. (4) compare well to biological systems. AMB-1 bacteria, driven by a single flagella motor, and of a size comparable to microbeads, produce a thrust of $F\approx 0.03\text{pN}$ while moving at a speed $v$ of about 20 µm/s (9). This corresponds to a power output of the bacteria’s flagella motor of P = Fv = 1,500 kT/s, very much comparable to the values obtained in the experiment by Saha et al.

A complementary question, not addressed in detail in Saha et al.’s (4) paper, is that of efficiency. It is known that the acquisition of information and/or the subsequent resetting of the memory of the demon has an unavoidable energy cost (2, 10). Monitoring a particle at high frequency maximizes the speed and power but can be expected to yield very low efficiency. Maximizing efficiency requires optimizing the usefulness of the information acquired in each measurement, inspiring very interesting questions about the trade-off between power and efficiency in information engines. Research in this direction will be important, since it is known that information-based ratchets can be more efficient than chemical motors with exactly the same dynamics (11).