Momčilo Gavrilov

Erasure Without Work in an Asymmetric, Double-Well Potential

According to Landauers principle, erasing a memory requires an average work of at least kTln2 per bit. Recent experiments have confirmed this prediction for a one-bit memory represented by a symmetric double-well potential. Here, we present an experimental study of erasure for a memory encoded in an asymmetric double-well potential. Using a feedback trap, we find that the average work to erase can be less than kTln2. Surprisingly, erasure protocols that differ subtly give measurably different values for the asymptotic work, a result we explain by showing that one protocol is symmetric with the respect to time reversal, while the other is not. The differences between the protocols help clarify the distinctions between thermodynamic and logical reversibility.

Brownian Duet: A Novel Tale of Thermodynamic Efficiency

We calculate analytically the stochastic thermodynamic properties of an isothermal Brownian engine driven by a duo of time-periodic forces, including its Onsager coefficients, the stochastic work of each force, and the corresponding stochastic entropy production. We verify the relations between different operational regimes, maximum power, maximum efficiency and minimum dissipation, and reproduce the signature features of the stochastic efficiency. All these results are experimentally tested without adjustable parameters on a colloidal system.

Arbitrarily Slow, Non-quasistatic, Isothermal Transformations

For an overdamped colloidal particle diffusing in a fluid in a controllable, virtual potential, we show that arbitrarily slow transformations, produced by smooth deformations of a double-well potential, need not be reversible. The arbitrarily slow transformations do need to be fast compared to the barrier crossing time, but that time can be extremely long. We consider two types of cyclic, isothermal transformations of a double-well potential. Both start and end in the same equilibrium state, and both use the same basic operations---but in different order. By measuring the work for finite cycle times and extrapolating to infinite times, we found that one transformation required no work, while the other required a finite amount of work, no matter how slowly it was carried out. The difference traces back to the observation that when time is reversed, the two protocols have different outcomes, when carried out arbitrarily slowly. A recently derived formula relating work production to the relative entropy of forward and backward path probabilities predicts the observed work average.

Real-time calibration of a feedback trap

Feedback traps use closed-loop control to trap or manipulate small particles and molecules in solution. They have been applied to the measurement of physical and chemical properties of particles and to explore fundamental questions in the non-equilibrium statistical mechanics of small systems. These applications have been hampered by drifts in the electric forces used to manipulate the particles. Although the drifts are small for measurements on the order of seconds, they dominate on time scales of minutes or slower. Here, we show that a recursive maximum likelihood (RML) algorithm can allow real-time measurement and control of electric and stochastic forces over time scales of hours. Simulations show that the RML algorithm recovers known parameters accurately. Experimental estimates of diffusion coefficients are also consistent with expected physical properties.

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Significance The second law of thermodynamics states that the total entropy of an isolated system is constant or increasing. This constrains the laws of physics, ruling out perpetual-motion machines that convert heat to work without any side effect. At its heart, the second law is a statement about entropy, yet entropy is an elusive concept: To date, it has not been directly measured but is rather inferred from other quantities, such as the integral of the specific heat over temperature. Here, by measuring the work required to erase a fraction of a bit of information, we isolate directly the change in entropy, showing that it is compatible with the functional form proposed by Shannon, demonstrating its physical meaning in this context. Stochastic thermodynamics extends classical thermodynamics to small systems in contact with one or more heat baths. It can account for the effects of thermal fluctuations and describe systems far from thermodynamic equilibrium. A basic assumption is that the expression for Shannon entropy is the appropriate description for the entropy of a nonequilibrium system in such a setting. Here we measure experimentally this function in a system that is in local but not global equilibrium. Our system is a micron-scale colloidal particle in water, in a virtual double-well potential created by a feedback trap. We measure the work to erase a fraction of a bit of information and show that it is bounded by the Shannon entropy for a two-state system. Further, by measuring directly the reversibility of slow protocols, we can distinguish unambiguously between protocols that can and cannot reach the expected thermodynamic bounds.

Feedback traps for virtual potentials

Feedback traps are tools for trapping and manipulating single charged objects, such as molecules in solution. An alternative to optical tweezers and other single-molecule techniques, they use feedback to counteract the Brownian motion of a molecule of interest. The trap first acquires information about a molecules position and then applies an electric feedback force to move the molecule. Since electric forces are stronger than optical forces at small scales, feedback traps are the best way to trap single molecules without ‘touching’ them (e.g. by putting them in a small box or attaching them to a tether). Feedback traps can do more than trap molecules: they can also subject a target object to forces that are calculated to be the gradient of a desired potential function U(x). If the feedback loop is fast enough, it creates a virtual potential whose dynamics will be very close to those of a particle in an actual potential U(x). But because the dynamics are entirely a result of the feedback loop—absent the feedback, there is only an object diffusing in a fluid—we are free to specify and then manipulate in time an arbitrary potential U(x,t). Here, we review recent applications of feedback traps to studies on the fundamental connections between information and thermodynamics, a topic where feedback plays an even more fundamental role. We discuss how recursive maximum-likelihood techniques allow continuous calibration, to compensate for drifts in experiments that last for days. We consider ways to estimate work and heat, using them to measure fluctuating energies to a precision of ±0.03 kT over these long experiments. Finally, we compare work and heat measurements of the costs of information erasure, the Landauer limit of kT ln 2 per bit of information erased. We argue that, when you want to know the average heat transferred to a bath in a long protocol, you should measure instead the average work and then infer the heat using the first law of thermodynamics. This article is part of the themed issue ‘Horizons of cybernetical physics’.

Feedback trap with scattering-based illumination

Feedback traps can explore particle dynamics in virtual potentials. They periodically measure particle position and, after each measurement, apply a force imposed by the potential. Here, a scattering-based illumination improves particle position measurements and bandwidth.

Direct measurement of a nonequilibrium system entropy using a feedback trap

Feedback traps are tools for trapping single charged objects in solution. They periodically measure an object’s position and apply a feedback force to counteract Brownian motion. The feedback force can be calculated as a gradient of a potential function, effectively creating a “virtual potential.” Its flexibility regarding the choice of form of the potential gives an opportunity to explore various fundamental questions in stochastic thermodynamics. Here, we review the theory behind feedback traps and apply it to measuring the average work required to erase a fraction of a bit of information. The results agree with predictions based on the nonequilibrium system entropy. With this example, we also show how a feedback trap can easily implement the complex erasure protocols required to reach ultimate thermodynamic limits.