Model-Free Measurement of Local Entropy Production and Extractable Work in Active Matter

Uncovering dissipation from coarse observables: A case study of a random walk with unobserved internal states

Inferring underlying microscopic dynamics from low-dimensional experimental signals is a central problem in physics, chemistry, and biology. As a trade-off between molecular complexity and the low-dimensional nature of experimental data, mesoscopic descriptions such as the Markovian master equation are commonly used. The states in such descriptions usually include multiple microscopic states, and the ensuing coarse-grained dynamics are generally non-Markovian. It is frequently assumed that such dynamics can nevertheless be described as a Markov process because of the timescale separation between slow transitions from one observed coarse state to another and the fast interconversion within such states. Here, we use a simple model of a molecular motor with unobserved internal states to highlight that (1) dissipation estimated from the observed coarse dynamics may significantly underestimate microscopic dissipation even in the presence of timescale separation and even when mesoscopic states do not contain dissipative cycles and (2) timescale separation is not necessarily required for the Markov approximation to give the exact entropy production, provided that certain constraints on the microscopic rates are satisfied. When the Markov approximation is inadequate, we discuss whether including memory effects can improve the estimate. Surprisingly, when we do so in a "model-free" way by computing the Kullback-Leibler divergence between the observed probability distributions of forward trajectories and their time reverses, this leads to poorer estimates of entropy production. Finally, we argue that alternative approaches, such as hidden Markov models, may uncover the dissipative nature of the microscopic dynamics even when the observed coarse trajectories are completely time-reversible.

Transport and energetics of bacterial rectification

Randomly moving active particles can be herded into directed motion by asymmetric geometric structures. Although such a rectification process has been extensively studied due to its fundamental, biological, and technological relevance, a comprehensive understanding of active matter rectification based on single particle dynamics remains elusive. Here, by combining experiments, simulations, and theory, we study the directed transport and energetics of swimming bacteria navigating through funnel-shaped obstacles-a paradigmatic model of rectification of living active matter. We develop a microscopic parameter-free model for bacterial rectification, which quantitatively explains experimental and numerical observations and predicts the optimal geometry for the maximum rectification efficiency. Furthermore, we quantify the degree of time irreversibility and measure the extractable work associated with bacterial rectification. Our study provides quantitative solutions to long-standing questions on bacterial rectification and establishes a generic relationship between time irreversibility, particle fluxes, and extractable work, shedding light on the energetics of nonequilibrium rectification processes in living systems.

Noise-Induced Phase Separation and Time Reversal Symmetry Breaking in Active Field Theories Driven by Persistent Noise

Within the Landau-Ginzburg picture of phase transitions, scalar field theories develop phase separation because of a spontaneous symmetry-breaking mechanism. This picture works in thermodynamics but also in the dynamics of phase separation. Here we show that scalar nonequilibrium field theories undergo phase separation just because of nonequilibrium fluctuations driven by a persistent noise. The mechanism is similar to what happens in motility-induced phase separation where persistent motion introduces an effective attractive force. We observe that noise-induced phase separation occurs in a region of the phase diagram where disordered field configurations would otherwise be stable at equilibrium. Measuring the local entropy production rate to quantify the time-reversal symmetry breaking, we find that such breaking is concentrated on the boundary between the two phases.

Information-theoretical limit on the estimates of dissipation by molecular machines using single-molecule fluorescence resonance energy transfer experiments

Single-molecule fluorescence resonance energy transfer (FRET) experiments are commonly used to study the dynamics of molecular machines. While in vivo molecular processes often break time-reversal symmetry, the temporal directionality of cyclically operating molecular machines is often not evident from single-molecule FRET trajectories, especially in the most common two-color FRET studies. Solving a more quantitative problem of estimating the energy dissipation/entropy production by a molecular machine from single-molecule data is even more challenging. Here, we present a critical assessment of several practical methods of doing so, including Markov-model-based methods and a model-free approach based on an information-theoretical measure of entropy production that quantifies how (statistically) dissimilar observed photon sequences are from their time reverses. The Markov model approach is computationally feasible and may outperform model free approaches, but its performance strongly depends on how well the assumed model approximates the true microscopic dynamics. Markov models are also not guaranteed to give a lower bound on dissipation. Meanwhile, model-free, information-theoretical methods systematically underestimate entropy production at low photoemission rates, and long memory effects in the photon sequences make these methods demanding computationally. There is no clear winner among the approaches studied here, and all methods deserve to belong to a comprehensive data analysis toolkit.

Deep learning probability flows and entropy production rates in active matter

Active matter systems, from self-propelled colloids to motile bacteria, are characterized by the conversion of free energy into useful work at the microscopic scale. They involve physics beyond the reach of equilibrium statistical mechanics, and a persistent challenge has been to understand the nature of their nonequilibrium states. The entropy production rate and the probability current provide quantitative ways to do so by measuring the breakdown of time-reversal symmetry. Yet, their efficient computation has remained elusive, as they depend on the system's unknown and high-dimensional probability density. Here, building upon recent advances in generative modeling, we develop a deep learning framework to estimate the score of this density. We show that the score, together with the microscopic equations of motion, gives access to the entropy production rate, the probability current, and their decomposition into local contributions from individual particles. To represent the score, we introduce a spatially local transformer network architecture that learns high-order interactions between particles while respecting their underlying permutation symmetry. We demonstrate the broad utility and scalability of the method by applying it to several high-dimensional systems of active particles undergoing motility-induced phase separation (MIPS). We show that a single network trained on a system of 4,096 particles at one packing fraction can generalize to other regions of the phase diagram, including to systems with as many as 32,768 particles. We use this observation to quantify the spatial structure of the departure from equilibrium in MIPS as a function of the number of particles and the packing fraction.

Machine-Learning Optimized Measurements of Chaotic Dynamical Systems via the Information Bottleneck

Deterministic chaos permits a precise notion of a "perfect measurement" as one that, when obtained repeatedly, captures all of the information created by the system's evolution with minimal redundancy. Finding an optimal measurement is challenging and has generally required intimate knowledge of the dynamics in the few cases where it has been done. We establish an equivalence between a perfect measurement and a variant of the information bottleneck. As a consequence, we can employ machine learning to optimize measurement processes that efficiently extract information from trajectory data. We obtain approximately optimal measurements for multiple chaotic maps and lay the necessary groundwork for efficient information extraction from general time series.

Defect turbulence in a dense suspension of polar, active swimmers

We study the effects of inertia in dense suspensions of polar swimmers. The hydrodynamic velocity field and the polar order parameter field describe the dynamics of the suspension. We show that a dimensionless parameter R (ratio of the swimmer self-advection speed to the active stress invasion speed [Phys. Rev. X 11, 031063 (2021)2160-330810.1103/PhysRevX.11.031063]) controls the stability of an ordered swimmer suspension. For R smaller than a threshold R_{1}, perturbations grow at a rate proportional to their wave number q. Beyond R_{1} we show that the growth rate is O(q^{2}) until a second threshold R=R_{2} is reached. The suspension is stable for R>R_{2}. We perform direct numerical simulations to characterize the steady-state properties and observe defect turbulence for R

Key Words Collapse

MESH Headings Collapse

Grants Collapse

Affiliation(s)

Navdeep Rana Max Planck Institute for Dynamics and Self-Organization (MPIDS), D-37077 Göttingen, Germany
Rayan Chatterjee Stanford Medicine, Stanford University, Stanford, California 94305, USA
Sunghan Ro Department of Physics, Technion-Israel Institute of Technology, Haifa 3200003, Israel
Dov Levine Department of Physics, Technion-Israel Institute of Technology, Haifa 3200003, Israel
Sriram Ramaswamy Department of Physics, Indian Institute of Science, Bengaluru 560 012, India
Prasad Perlekar Tata Institute of Fundamental Research, Hyderabad 500046, India

Collapse

Irreversible Mesoscale Fluctuations Herald the Emergence of Dynamical Phases

We study fluctuating field models with spontaneously emerging dynamical phases. We consider two typical transition scenarios associated with parity-time symmetry breaking: oscillatory instabilities and critical exceptional points. An analytical investigation of the low-noise regime reveals a drastic increase of the mesoscopic entropy production toward the transitions. For an illustrative model of two nonreciprocally coupled Cahn-Hilliard fields, we find physical interpretations in terms of actively propelled interfaces and a coupling of eigenmodes of the linearized dynamics near the critical exceptional point.

Optimal Power Extraction from Active Particles with Hidden States

We identify generic protocols achieving optimal power extraction from a single active particle subject to continuous feedback control under the assumption that its spatial trajectory, but not its instantaneous self-propulsion force, is accessible to direct observation. Our Bayesian approach draws on the Onsager-Machlup path integral formalism and is exemplified in the cases of free run-and-tumble and active Ornstein-Uhlenbeck dynamics in one dimension. Such optimal protocols extract positive work even in models characterized by time-symmetric positional trajectories and thus vanishing informational entropy production rates. We argue that the theoretical bounds derived in this work are those against which the performance of realistic active matter engines should be compared.

Generalized Zurek's bound on the cost of an individual classical or quantum computation

We consider the minimal thermodynamic cost of an individual computation, where a single input x is mapped to a single output y. In prior work, Zurek proposed that this cost was given by K(x|y), the conditional Kolmogorov complexity of x given y (up to an additive constant that does not depend on x or y). However, this result was derived from an informal argument, applied only to deterministic computations, and had an arbitrary dependence on the choice of protocol (via the additive constant). Here we use stochastic thermodynamics to derive a generalized version of Zurek's bound from a rigorous Hamiltonian formulation. Our bound applies to all quantum and classical processes, whether noisy or deterministic, and it explicitly captures the dependence on the protocol. We show that K(x|y) is a minimal cost of mapping x to y that must be paid using some combination of heat, noise, and protocol complexity, implying a trade-off between these three resources. Our result is a kind of "algorithmic fluctuation theorem" with implications for the relationship between the second law and the Physical Church-Turing thesis.

Inferring scale-dependent non-equilibrium activity using carbon nanotubes

In living systems, irreversible, yet stochastic, molecular interactions form multiscale structures (such as cytoskeletal networks), which mediate processes (such as cytokinesis and cellular motility) in a close relationship between the structure and function. However, owing to a lack of methods to quantify non-equilibrium activity, their dynamics remain poorly characterized. Here, by measuring the time-reversal asymmetry encoded in the conformational dynamics of filamentous single-walled carbon nanotubes embedded in the actomyosin network of Xenopus egg extract, we characterize the multiscale dynamics of non-equilibrium activity encoded in bending-mode amplitudes. Our method is sensitive to distinct perturbations to the actomyosin network and the concentration ratio of adenosine triphosphate to adenosine diphosphate. Thus, our method can dissect the functional coupling of microscopic dynamics to the emergence of larger scale non-equilibrium activity. We relate the spatiotemporal scales of non-equilibrium activity to the key physical parameters of a semiflexible filament embedded in a non-equilibrium viscoelastic environment. Our analysis provides a general tool to characterize steady-state non-equilibrium activity in high-dimensional spaces.

Detecting and characterizing phase transitions in active matter using entropy

A major challenge in the study of active matter lies in quantitative characterization of phases and transitions between them. We show how the entropy of a collection of active objects can be used to classify regimes and spatial patterns in their collective behavior. Specifically, we estimate the contributions to the total entropy from correlations between the degrees of freedom of position and orientation. This analysis pin-points the flocking transition in the Vicsek model while clarifying the physical mechanism behind the transition. When applied to experiments on swarming Bacillus subtilis with different cell aspect ratios and overall bacterial area fractions, the entropy analysis reveals a rich phase diagram with transitions between qualitatively different swarm statistics. We discuss physical and biological implications of these findings.

Improving estimation of entropy production rate for run-and-tumble particle systems by high-order thermodynamic uncertainty relation

Entropy production plays an important role in the regulation and stability of active matter systems, and its rate quantifies the nonequilibrium nature of these systems. However, entropy production is hard to experimentally estimate even in some simple active systems like molecular motors or bacteria, which may be modeled by the run-and-tumble particle (RTP), a representative model in the study of active matters. Here we resolve this problem for an asymmetric RTP in one dimension, first constructing a finite-time thermodynamic uncertainty relation (TUR) for a RTP, which works well in the short observation time regime for entropy production estimation. Nevertheless, when the activity dominates, i.e., the RTP is far from equilibrium, the lower bound for entropy production from TUR turns out to be trivial. We address this issue by introducing a recently proposed high-order thermodynamic uncertainty relation (HTUR), in which the cumulant generating function of current serves as a key ingredient. To exploit the HTUR, we adopt a method to analytically obtain the cumulant generating function of the current we study, with no need to explicitly know the time-dependent probability distribution. The HTUR is demonstrated to be able to estimate the steady state energy dissipation rate accurately because the cumulant generating function covers higher-order statistics of the current, including rare and large fluctuations besides its variance. Compared to the conventional TUR, the HTUR could give significantly improved estimation of energy dissipation, which can work well even in the far from equilibrium regime. We also provide a strategy based on the improved bound to estimate the entropy production from a moderate amount of trajectory data for experimental feasibility.