Hidden Markov models for stochastic thermodynamics

The formalism of state estimation and hidden Markov models (HMMs) can simplify and clarify the discussion of stochastic thermodynamics in the presence of feedback and measurement errors. After reviewing the basic formalism, we use it to shed light on a recent discussion of phase transitions in the optimized response of an information engine, for which measurement noise serves as a control parameter. The HMM formalism also shows that the value of additional information shows a maximum at intermediate signal-to-noise ratios. Finally, we discuss how systems open to information flow can apparently violate causality; the HMM formalism can quantify the performance gains due to such violations.Comment: 27 pages, 13 figures; submitted to New J. Phy

What is superresolution microscopy?

I explain what is, what is not, and what is only sort of superresolution microscopy. I discuss optical resolution, first in terms of diffraction theory, then in terms of linear systems theory, and finally in terms of techniques that use prior information, nonlinearity, and other tricks to improve performance. The discussion reveals two classes of superresolution: Pseudo superresolution techniques improve images up to the diffraction limit but not much beyond. True superresolution techniques allow substantial, useful improvements beyond the diffraction limit. The two classes are distinguished by their scaling of resolution with photon counts. Understanding the limits to imaging resolution involves concepts that pertain to almost any measurement problem, implying that the framework given here has broad application beyond optics.Comment: 9 pages, 5 figure

Nonequilibrium Phenomena in Liquid Crystals

This paper summarizes a talk presented at the April NATO ASI on Spatiotemporal Chaos in Complex Fluids, in Santa Fe, NM. The paper gives reasons that make complex fluids good material systems for conducting experiments on pattern formation and other nonequilibrium phenomena. Much of the discussion focuses on the different phenomena observed in solidification and how the increasing complexity of fluid systems decreases the velocity scale for achieving "rapid" solidification. Five systems are compared to illustrate this point: simple fluids, simple alloys, thermotropic liquid crystals, lyotropic liquid crystals, and diblock copolymers. Finally, an example is given of the kinds of transitions that may be observed in rapid solidification.Comment: 18 pages, Revtex 3.0, no figure

When memory pays: Discord in hidden Markov models

When is keeping a memory of observations worthwhile? We use hidden Markov models to look at phase transitions that emerge when comparing state estimates in systems with discrete states and noisy observations. We infer the underlying state of the hidden Markov models from the observations in two ways: through naive observations, which take into account only the current observation, and through Bayesian filtering, which takes the history of observations into account. Defining a discord order parameter to distinguish between the different state estimates, we explore hidden Markov models with various numbers of states and symbols and varying transition-matrix symmetry. All behave similarly. We calculate analytically the critical point where keeping a memory of observations starts to pay off. A mapping between hidden Markov models and Ising models gives added insight into the associated phase transitions.Comment: 11 pages, 8 figure

Erasure without work in an asymmetric, double-well potential

According to Landauer's principle, erasing a memory requires an average work of at least $kT\ln2$ 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 $kT\ln2$. 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.Comment: 6 pages, 5 figures, and supplemental materia

A Simple Model for Faraday Waves

We show that the linear-stability analysis of the birth of Faraday waves on the surface of a fluid is simplified considerably when the fluid container is driven by a triangle waveform rather than by a sine wave. The calculation is simple enough to use in an undergraduate course on fluid dynamics or nonlinear dynamics. It is also an attractive starting point for a nonlinear analysis.Comment: 8 pages, revtex, with included, embedded eps figs; to appear in Am. J. Phys. (but don't hold your breath

Nanoscale virtual potentials using optical tweezers

We combine optical tweezers with feedback to impose arbitrary potentials on a colloidal particle. The feedback trap detects a particle's position, calculates a force based on an imposed "virtual potential," and shifts the trap center to generate the desired force. We create virtual harmonic and double-well potentials to manipulate particles. The harmonic potentials can be chosen to be either weaker or stiffer than the underlying optical trap. Using this flexibility, we create an isotropic trap in three dimensions. Finally, we show that we can create a virtual double-well potential with fixed well separation and adjustable barrier height. These are accomplished at length scales down to 11 nm, a feat that is difficult or impossible to create with standard optical-tweezer techniques such as time sharing, dual beams, or spatial light modulators

Split PID control: two sensors can be better than one

The traditional proportional-integral-derivative (PID) algorithm for regulation suffers from a tradeoff: placing the sensor near the sample being regulated ensures that its steady-state temperature matches the desired setpoint. However, the propagation delay (lag) between heater and sample can limit the control bandwidth. Moving the sensor closer to the heater reduces the lag and increases the bandwidth but introduces offsets and drifts into the temperature of the sample. Here, we explore the consequences of using two probes---one near the heater, one near the sample---and assigning the integral term to the sample probe and the other terms to the heater probe. The \textit{split-PID} algorithm can outperform PID control loops based on one sensor.Comment: Rev. Sci. Instrum., to appear. 4 pages, 2 figure

High-precision test of Landauer's principle in a feedback trap

We confirm Landauer's 1961 hypothesis that reducing the number of possible macroscopic states in a system by a factor of two requires work of at least kT ln 2. Our experiment uses a colloidal particle in a time-dependent, virtual potential created by a feedback trap to implement Landauer's erasure operation. In a control experiment, similar manipulations that do not reduce the number of system states can be done reversibly. Erasing information thus requires work. In individual cycles, the work to erase can be below the Landauer limit, consistent with the Jarzynski equality.Comment: 7 pages, 7 figure

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

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, for the first time, we measure experimentally this function. 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.Comment: 9 pages, 3 figures, and supplemental materia