Thermal noise reduction of mechanical oscillators by actively controlled external dissipative forces

Universal form of thermodynamic uncertainty relation for Langevin dynamics

The thermodynamic uncertainty relation (TUR) provides a stricter bound for entropy production (EP) than that of the thermodynamic second law. This stricter bound can be utilized to infer the EP and derive other tradeoff relations. Though the validity of the TUR has been verified in various stochastic systems, its application to general Langevin dynamics has not been successfully unified, especially for underdamped Langevin dynamics, where odd parity variables in time-reversal operation such as velocity get involved. Previous TURs for underdamped Langevin dynamics are neither experimentally accessible nor reduced to the original form of the overdamped Langevin dynamics in the zero-mass limit. Here, we find a TUR for underdamped Langevin dynamics with an arbitrary time-dependent protocol, which is operationally accessible when all mechanical forces are controllable. We show that the original TUR is a consequence of our underdamped TUR in the zero-mass limit. This indicates that the TUR formulation presented here can be regarded as the universal form of the TUR for general Langevin dynamics.

Thermodynamic uncertainty relation for underdamped Langevin systems driven by a velocity-dependent force

Recently, it has been shown that there is a trade-off relation between thermodynamic cost and current fluctuations, referred to as the thermodynamic uncertainty relation (TUR). The TUR has been derived for various processes, such as discrete-time Markov jump processes and overdamped Langevin dynamics. For underdamped dynamics, it has recently been reported that some modification is necessary for application of the TUR. However, the previous TUR for underdamped dynamics is not applicable to a system driven by a velocity-dependent force. In this study, we present a TUR, applicable to a system driven by a velocity-dependent force in the context of underdamped Langevin dynamics, by extending the theory of Vu and Hasegawa [Phys. Rev. E 100, 032130 (2019)2470-004510.1103/PhysRevE.100.032130]. We show that our TUR accurately describes the trade-off properties of a molecular refrigerator (cold damping), Brownian dynamics in a magnetic field, and an active particle system.

Self-Calibration and Performance Control of MEMS with Applications for IoT

A systemic problem for microelectromechanical systems (MEMS) has been the large gap between their predicted and actual performances. Due to process variations, no two MEMS have been able to perform identically. In-factory calibration is often required, which can represent as much as three-fourths of the manufacturing costs. Such issues are challenges for microsensors that require higher accuracy and lower cost. Towards addressing these issues, this paper describes how microscale attributes may be used to enable MEMS to accurately calibrate themselves without external references, or enable actual devices to match their predicted performances. Previously, we validated how MEMS with comb drives can be used to autonomously self-measure their change in geometry in going from layout to manufactured, and we verified how MEMS can be made to increase or decrease their effective mass, damping, and or stiffness in real-time to match desired specifications. Here, we present how self-calibration and performance control may be used to accurately sense and extend the capabilities of a variety of sensing applications for the Internet of things (IoT). Discussions of IoT applications include: (1) measuring absolute temperature due to thermally-induced vibrations; (2) measuring the stiffness of atomic force microscope or biosensor cantilevers; (3) MEMS weighing scales; (4) MEMS gravimeters and altimeters; (5) inertial measurement units that can measure all four non-inertial forces; (6) self-calibrating implantable pressure sensors; (7) diagnostic chips for quality control; (8) closing the gap from experiment to simulation; (9) control of the value of resonance frequency to counter drift or to match modes; (10) control of the value of the quality factor; and (11) low-amplitude Duffing nonlinearity for wideband high-Q resonance.

Sub-nanometer Resolution Imaging with Amplitude-modulation Atomic Force Microscopy in Liquid

Atomic force microscopy (AFM) has become a well-established technique for nanoscale imaging of samples in air and in liquid. Recent studies have shown that when operated in amplitude-modulation (tapping) mode, atomic or molecular-level resolution images can be achieved over a wide range of soft and hard samples in liquid. In these situations, small oscillation amplitudes (SAM-AFM) enhance the resolution by exploiting the solvated liquid at the surface of the sample. Although the technique has been successfully applied across fields as diverse as materials science, biology and biophysics and surface chemistry, obtaining high-resolution images in liquid can still remain challenging for novice users. This is partly due to the large number of variables to control and optimize such as the choice of cantilever, the sample preparation, and the correct manipulation of the imaging parameters. Here, we present a protocol for achieving high-resolution images of hard and soft samples in fluid using SAM-AFM on a commercial instrument. Our goal is to provide a step-by-step practical guide to achieving high-resolution images, including the cleaning and preparation of the apparatus and the sample, the choice of cantilever and optimization of the imaging parameters. For each step, we explain the scientific rationale behind our choices to facilitate the adaptation of the methodology to every user's specific system.

Stochastic thermodynamics of Langevin systems under time-delayed feedback control: Second-law-like inequalities

Response lags are generic to almost any physical system and often play a crucial role in the feedback loops present in artificial nanodevices and biological molecular machines. In this paper, we perform a comprehensive study of small stochastic systems governed by an underdamped Langevin equation and driven out of equilibrium by a time-delayed continuous feedback control. In their normal operating regime, these systems settle in a nonequilibrium steady state in which work is permanently extracted from the surrounding heat bath. By using the Fokker-Planck representation of the dynamics, we derive a set of second-law-like inequalities that provide bounds to the rate of extracted work. These inequalities involve additional contributions characterizing the reduction of entropy production due to the continuous measurement process. We also show that the non-Markovian nature of the dynamics requires a modification of the basic relation linking dissipation to the breaking of time-reversal symmetry at the level of trajectories. The modified relation includes a contribution arising from the acausal character of the reverse process. This, in turn, leads to another second-law-like inequality. We illustrate the general formalism with a detailed analytical and numerical study of a harmonic oscillator driven by a linear feedback, which describes actual experimental setups.

Time reversibility and nonequilibrium thermodynamics of second-order stochastic processes

Nonequilibrium thermodynamics of a general second-order stochastic system is investigated. We prove that at steady state, under inversion of velocities, the condition of time reversibility over the phase space is equivalent to the antisymmetry of spatial flux and the symmetry of velocity flux. Then we show that the condition of time reversibility alone cannot always guarantee the Maxwell-Boltzmann distribution. Comparing the two conditions together, we find that the frictional force naturally emerges as the unique odd term of the total force at thermodynamic equilibrium, and is followed by the Einstein relation. The two conditions respectively correspond to two previously reported different entropy production rates. In the case where the external force is only position dependent, the two entropy production rates become one. We prove that such an entropy production rate can be decomposed into two non-negative terms, expressed respectively by the conditional mean and variance of the thermodynamic force associated with the irreversible velocity flux at any given spatial coordinate. In the small inertia limit, the former term becomes the entropy production rate of the corresponding overdamped dynamics, while the anomalous entropy production rate originates from the latter term. Furthermore, regarding the connection between the first law and second law, we find that in the steady state of such a limit, the anomalous entropy production rate is also the leading order of the Boltzmann-factor weighted difference between the spatial heat dissipation densities of the underdamped and overdamped dynamics, while their unweighted difference always tends to vanish.

Electromechanical coupling in the membranes of Shaker-transfected HEK cells

Membranes flex with changes in transmembrane potential as a result of changes in interfacial tension, the Lippman effect. We studied the membrane electromotility of Shaker K(+)-transfected HEK-293 cells in real time by using combined patch-clamp atomic force microscopy. In the voltage range where the channels were closed, Shaker expression had little effect on electromotility relative to wild-type cells. Depolarization between -120 and -40 mV resulted in a linear upward cantilever deflection equivalent to an increase in membrane tension. However, when depolarized sufficiently for channel opening, the electromotility saturated and only recovered over 10 s of milliseconds. This remarkable loss of motility was associated with channel opening, not ionic flux or movement of the voltage sensors. The IL mutant of Shaker, in which the voltage dependence of channel opening but not sensor movement is shifted to more positive potentials, caused the loss of electromotility saturation also to shift to more positive potentials. The temporary loss of electromotility associated with channel opening is probably caused by local buckling of the bilayer as the inner half of the channel expands as expected from X-ray structural data.

From valleys to ridges: exploring the dynamic energy landscape of single membrane proteins

Membrane proteins are involved in essential biological processes such as energy conversion, signal transduction, solute transport and secretion. All biological processes, also those involving membrane proteins, are steered by molecular interactions. Molecular interactions guide the folding and stability of membrane proteins, determine their assembly, switch their functional states or mediate signal transduction. The sequential steps of molecular interactions driving these processes can be described by dynamic energy landscapes. The conceptual energy landscape allows to follow the complex reaction pathways of membrane proteins while its modifications describe why and how pathways are changed. Single-molecule force spectroscopy (SMFS) detects, quantifies and locates interactions within and between membrane proteins. SMFS helps to determine how these interactions change with temperature, point mutations, oligomerization and the functional states of membrane proteins. Applied in different modes, SMFS explores the co-existence and population of reaction pathways in the energy landscape of the protein and thus reveals detailed insights into local mechanisms, determining its structural and functional relationships. Here we review how SMFS extracts the defining parameters of an energy landscape such as the barrier position, reaction kinetics and roughness with high precision.

Fluctuation theorems for a molecular refrigerator

We extend fluctuation theorems to a molecular refrigeration system that consists of Brownian particles in a heat bath under feedback control of their velocities. Such control can actively remove heat from the bath due to an entropy-pumping mechanism [Phys. Rev. Lett. 93, 120602 (2004)]. The presence of entropy pumping in an underdamped Brownian system modifies both the Jarzynski equality and the fluctuation theorems. We discover that the entropy pumping has a dual role of work and heat.

Thermo-mechanical responses of a surface-coupled AFM cantilever

Atomic force microscopy (AFM) has been widely used for measuring mechanical properties of biological specimens such as cells, DNA, and proteins. This is usually done by monitoring deformations in response to controlled applied forces, which have to be at ultralow levels due to the extreme softness of the specimens. Consequently, such experiments may be susceptible to thermal excitations, manifested as force and displacement fluctuations that could reduce the measurement accuracy. To take advantage of, rather than to be limited by, such fluctuations, we have characterized the thermomechanical responses of an arbitrarily shaped AFM cantilever with the tip coupled to an elastic spring. Our analysis shows that the cantilever and the specimen behave as springs in parallel. This provides a method for determining the elasticity of the specimen by measuring the change in the tip fluctuations in the presence and absence of coupling. For rectangular and V-shaped cantilevers, we have derived a relationship between the mean-square deflection and the mean-square inclination and an approximate expression for the specimen spring constant in terms of contributions to the mean-square inclination from the first few vibration modes.

Entropy production of Brownian macromolecules with inertia

We investigate the nonequilibrium steady-state thermodynamics of single Brownian macromolecules with inertia under feedback control in an isothermal ambient fluid. With the control being represented by a velocity-dependent external force, we find such an open system can have a negative entropy production rate, and we develop a mesoscopic theory consistent with the second law. We propose an equilibrium condition and define a class of external force, which includes the transverse Lorentz force, leading to equilibrium.

Manipulation of microcantilever oscillations

Experimental observation of self-sustaining oscillations via a delayed feedback system is presented for a rectangular silicon microcantilever. The system is modeled as one and two-dimensional damped oscillator and the resulting delay differential equations are studied in frequency and time domain. The shortcomings of each model are outlined, and an improved formulation of the dynamics of the cantilever is presented.

Measuring and modeling thermal fluctuations at nanometer length scales

The size of mechanical, electrical, and optical devices continues to be reduced. As the length scales of such devices decrease, coupling to the external environment greatly increases. Thermal fluctuations due to momentum exchange between air molecules and micron scale devices under ambient conditions can effect the dynamics of a system. To illustrate this we use an atomic force microscope cantilever and detection system to measure background noise and thermal fluctuations of a micron size beam. The beam is modeled by a Langevin-type equation that is externally forced by a white-noise spectrum having an analytic as opposed to a statistical form. This model is compared with experimental data. It is found that at higher frequencies, a white-noise spectrum is not sufficient to model such a system. We modify the forcing spectrum so that it decays at higher frequencies and subsequently achieve closer agreement between the model and the experimental observations.

High-Q dynamic force microscopy in liquid and its application to living cells

We present a new dynamic force microscopy technique for imaging in liquids in the piconewton regime. The low quality factor (Q) of the cantilever is increased up to three orders of magnitude by the implementation of a positive feedback control. The technique also includes a phase-locked loop unit to track the resonance of the cantilever. Experiments and computer simulations indicate that the tip-sample forces are below 100 pN, about two orders of magnitude lower than in conventional tapping mode atomic force microscopy. Furthermore, the spectroscopic ability is greatly enhanced. Either the phase shift or the resonant frequency shows a high sensitivity to variations in either the energy dissipation or conservative interactions between the tip and the sample, respectively. The potential of this technique is demonstrated by imaging living cells.