Hidden entropy production and work fluctuations in an ideal active gas

Effects of kinetic energy on heat fluctuations of passive and active overdamped driven particles

To describe the spatial trajectory of an overdamped Brownian particle, inertial effects can be neglected. Yet, at the energetic level of stochastic thermodynamics, changes in kinetic energy must be considered to accurately predict the heat exchanged with the thermal bath. On the other hand, in the presence of external driving forces, one would expect the effects of kinetic energy fluctuations to be reduced, as thermal noise becomes comparatively less relevant. Here, we investigate the competition between the kinetic energy and the external work contributions to the heat statistics of passive and active overdamped Brownian particles subject to external driving forces. We find that kinetic energy effects cause fluctuations in the exchanged heat to become non-Gaussian. To evaluate the relevance of these effects, we compute the excess kurtosis and the Pearson correlation. For fixed parameter values adapted from previous experiments with silica beads in passive and active baths, we identify a crossover transition from a regime in which the stochastic heat of overdamped particles is dominated by external work, where kinetic energy changes can be safely ignored, to a regime dominated by kinetic energy effects. Our results also provide a quantitative analytical way to assess how deep into a particular regime the system is.

Collective motion of energy depot active disks

In the present work we have studied collectives of active disks with an energy depot, moving in the two-dimensional plane and interacting via an excluded volume. The energy depot accounts for the extraction of energy taking place at the level of each particle in order to perform self-propulsion, included in an underdamped Langevin dynamics. We show that this model undergoes a flocking transition, exhibiting some of the key features of the Vicsek model, namely, band formation and giant number fluctuations. These bands, either single or multiple, are dense and very strongly polarised propagating structures. Large density bands disappear as the activity is further increased, eventually reaching a homogeneous polar state. We unravel an effective alignment interaction at the level of two-particle collisions that can be controlled by activity and gives rise to flocking at large scales.

Unified hierarchical relationship between thermodynamic tradeoff relations

Recent years have witnessed a surge of discoveries in the studies of thermodynamic inequalities: the thermodynamic uncertainty relation (TUR) and the entropic bound (EB) provide a lower bound on the entropy production (EP) in terms of nonequilibrium currents; the classical speed limit (CSL) expresses the lower bound on the EP using the geometry of probability distributions; the power-efficiency (PE) tradeoff dictates the maximum power achievable for a heat engine given the level of its thermal efficiency. In this study, we show that there exists a unified hierarchical structure encompassing all of these bounds, with the fundamental inequality given by an extension of the TUR (XTUR) that incorporates the most general range of currentlike and state-dependent observables. By selecting more specific observables, the TUR and the EB follow from the XTUR, and the CSL and the PE tradeoff follow from the EB. Our derivations cover both Langevin and Markov jump systems, with the first proof of the EB for the Markov jump systems and a more generalized form of the CSL. We also present concrete examples of the EB for the Markov jump systems and the generalized CSL.

Power dissipation and entropy production rate of high-dimensional optical matter systems

Entropy production is an essential aspect of creating and maintaining nonequilibrium systems. Despite their ubiquity, calculation of entropy production rates is challenging for high-dimensional systems, so it has only been reported for simple (i.e., l-particle) systems. Moreover, there is a dearth of nontrivial experimental systems where precise measurements of entropy production rate and characterization of the nonequilibrium steady state (NESS) are simultaneously possible. We report an approach to calculate the entropy production rate of overdamped, nonconservative, N-body systems and demonstrate this on a six-particle triangle optical matter (OM) system as a nontrivial example. OM systems consist of (nano-)particles organized into ordered arrays that are bound by electrodynamic interactions associated with the scattering and interference of light, and the associated induced-polarizations in and among the particles in coherent optical beams. The flux of laser light in OM systems in a solution environment necessitates that they dissipate energy, produce entropy, and relax to a NESS. The NESS may have several ordered particle configurations (i.e., isomers) that can interchange by barrier crossing processes. Understanding the power dissipation and entropy production rate of a NESS in an OM system along different (collective) modes of motion can advance understanding of the relative stability of the NESSs as well as inform design and control of OM structures. Therefore, we compute the components of the entropy production rate and power dissipation along the collective coordinates of the 6 Ag nanoparticle triangle OM system from OM NESS trajectory data and verify the Seifert relation [U. Seifert, Rep. Prog. Phys. 75, 126001 (2012)10.1088/0034-4885/75/12/126001] for these complex systems with a nuanced interpretation.

Statistical mechanics of passive Brownian particles in a fluctuating harmonic trap

We consider passive Brownian particles trapped in an "imperfect" harmonic trap. The trap is imperfect because it is randomly turned off and on, and as a result particles fail to equilibrate. Another way to think about this is to say that a harmonic trap is time dependent on account of its strength evolving stochastically in time. Particles in such a system are passive and activity arises through external control of a trapping potential, thus, no internal energy is used to power particle motion. A stationary Fokker-Planck equation of this system can be represented as a third-order differential equation, and its solution, a stationary distribution, can be represented as a superposition of Gaussian distributions for different strengths of a harmonic trap. This permits us to interpret a stationary system as a system in equilibrium with quenched disorder.

Efficiency of an autonomous, dynamic information engine operating on a single active particle

The Szilard engine stands as a compelling illustration of the intricate interplay between information and thermodynamics. While at thermodynamic equilibrium, the apparent breach of the second law of thermodynamics was reconciled by Landauer and Bennett's insights into memory writing and erasure, recent extensions of these concepts into regimes featuring active fluctuations have unveiled the prospect of exceeding Landauer's bound, capitalizing on information to divert free energy from dissipation towards useful work. To explore this question further, we investigate an autonomous dynamic information engine, addressing the thermodynamic consistency of work extraction and measurement costs by extending the phase space to incorporate an auxiliary system, which plays the role of an explicit measurement device. The nonreciprocal coupling between active particle and measurement device introduces a feedback control loop, and the cost of measurement is quantified through a suitably defined auxiliary entropy production. The study considers different measurement scenarios, highlighting the role of measurement precision in determining engine efficiency.

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.

Force renormalization for probes immersed in an active bath

Langevin equations or generalized Langevin equations (GLEs) are popular models for describing the motion of a particle in a fluid medium in an effective manner. Here we examine particles immersed in an inherently nonequilibrium fluid, i.e., an active bath, which are subject to an external force. Specifically, we consider two types of forces that are highly relevant for microrheological studies: A harmonic, trapping force and a constant, "drag" force. We study such systems by molecular simulations and use the simulation data to extract an effective GLE description. We find that within this description, in an active bath, the external force in the GLE is not equal to the physical external force, but rather a renormalized external force, which can be significantly smaller. The effect cannot be attributed to the mere temperature renormalization, which is also observed.

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.

Entropy production and collective excitations of crystals out of equilibrium: The concept of entropons

We study the collective vibrational excitations of crystals under out-of-equilibrium steady conditions that give rise to entropy production. Their excitation spectrum comprises equilibriumlike phonons of thermal origin and additional collective excitations called entropons because each of them represents a mode of spectral entropy production. Entropons coexist with phonons and dominate them when the system is far from equilibrium while they are negligible in near-equilibrium regimes. The concept of entropons has been recently introduced and verified in a special case of crystals formed by self-propelled particles. Here we show that entropons exist in a broader class of active crystals that are intrinsically out of equilibrium and characterized by the lack of detailed balance. After a general derivation, several explicit examples are discussed, including crystals consisting of particles with alignment interactions and frictional contact forces.

Minimum entropy production by microswimmers with internal dissipation

The energy dissipation and entropy production by self-propelled microswimmers differ profoundly from passive particles pulled by external forces. The difference extends both to the shape of the flow around the swimmer, as well as to the internal dissipation of the propulsion mechanism. Here we derive a general theorem that provides an exact lower bound on the total, external and internal, dissipation by a microswimmer. The problems that can be solved include an active surface-propelled droplet, swimmers with an extended propulsive layer and swimmers with an effective internal dissipation. We apply the theorem to determine the swimmer shapes that minimize the total dissipation while keeping the volume constant. Our results show that the entropy production by active microswimmers is subject to different fundamental limits than the entropy production by externally driven particles.

Effects of the self-propulsion parity on the efficiency of a fuel-consuming active heat engine

We propose a thermodynamically consistent, analytically tractable model of steady-state active heat engines driven by both temperature difference and a constant chemical driving. While the engine follows the dynamics of the active Ornstein-Uhlenbeck particle, its self-propulsion stems from the mechanochemical coupling with the fuel consumption dynamics, allowing for both even- and odd-parity self-propulsion forces. Using the standard methods of stochastic thermodynamics, we show that the entropy production of the engine satisfies the conventional Clausius relation, based on which we define the efficiency of the model that is bounded from above by the second law of thermodynamics. Using this framework, we obtain exact expressions for the efficiency at maximum power. The results show that the engine performance has a nonmonotonic dependence on the magnitude of the chemical driving and that the even-parity (odd-parity) engines perform better when the size of the engine is smaller (larger) than the persistence length of the active particle. We also discuss the existence of a tighter upper bound on the efficiency of the odd-parity engines stemming from the detailed structure of the entropy production.

Entropons as collective excitations in active solids

The vibrational dynamics of solids is described by phonons constituting basic collective excitations in equilibrium crystals. Here, we consider a non-equilibrium active solid, formed by self-propelled particles, which bring the system into a non-equilibrium steady-state. We identify novel vibrational collective excitations of non-equilibrium (active) origin, which coexist with phonons and dominate over them when the system is far from equilibrium. These vibrational excitations are interpreted in the framework of non-equilibrium physics, in particular, stochastic thermodynamics. We call them "entropons" because they are the modes of spectral entropy production (at a given frequency and wave vector). The existence of entropons could be verified in future experiments on dense self-propelled colloidal Janus particles and granular active matter, as well as in living systems, such as dense cell monolayers.

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.

Single active particle in a harmonic potential: Question about the existence of the Jarzynski relation

The interest in active matter stimulates the need to generalize thermodynamic description and relations to active matter systems, which are intrinsically out of equilibrium. One important example is the Jarzynski relation, which links the exponential average of work done in an arbitrary process connecting two equilibrium states with the difference of the free energies of these states. Using a simple model system, a single thermal active Ornstein-Uhlenbeck particle in a harmonic potential, we show that if the standard stochastic thermodynamics definition of work is used, the Jarzynski relation is not generally valid for processes connecting stationary states of active matter systems.

Most probable paths for active Ornstein-Uhlenbeck particles

Fluctuations play an important role in the dynamics of stochastic systems. In particular, for small systems, the most probable thermodynamic quantities differ from their averages because of the fluctuations. Using the Onsager Machlup variational formalism we analyze the most probable paths for nonequilibrium systems, in particular, active Ornstein-Uhlenbeck particles, and investigate how the entropy production along these paths differs from the average entropy production. We investigate how much information about their nonequilibrium nature can be obtained from their extremum paths and how these paths depend on the persistence time and their swim velocities. We also look at how the entropy production along the most probable paths varies with the active noise and how it differs from the average entropy production. This study would be useful to design artificial active systems with certain target trajectories.

Inertial and geometrical effects of self-propelled elliptical Brownian particles

Active particles that self-propel by transforming energy into mechanical motion represent a growing area of research in mathematics, physics, and chemistry. Here we investigate the dynamics of nonspherical inertial active particles moving in a harmonic potential, introducing geometric parameters which take into account the role of eccentricity for nonspherical particles. A comparison between the overdamped and underdamped models for elliptical particles is performed. The model of overdamped active Brownian motion has been used to describe most of the basic aspects of micrometer-sized particles moving in a liquid ("microswimmers"). We consider active particles by extending the active Brownian motion model to incorporate translation and rotation inertia and account for the role of eccentricity. We show how the overdamped and the underdamped models behave in the same way for small values of activity (Brownian case) if eccentricity is equal to zero, but increasing eccentricity leads the two dynamics to substantially depart from each other-in particular, the action of a torque induced by external forces, induced a marked difference close to the walls of the domain if eccentricity is high. Effects induced by inertia include an inertial delay time of the self-propulsion direction from the particle velocity, and the differences between the overdamped and underdamped systems are particularly evident in the first and second moments of the particle velocities. Comparison with the experimental results of vibrated granular particles shows good agreement and corroborates the notion that self-propelling massive particles moving in gaseous media are dominated by inertial effects.

Dynamics of active particles with translational and rotational inertia

Inertial effects affecting both the translational and rotational dynamics are inherent to a broad range of active systems at the macroscopic scale. Thus, there is a pivotal need for proper models in the framework of active matter to correctly reproduce experimental results, hopefully achieving theoretical insights. For this purpose, we propose an inertial version of the active Ornstein-Uhlenbeck particle (AOUP) model accounting for particle mass (translational inertia) as well as its moment of inertia (rotational inertia) and derive the full expression for its steady-state properties. The inertial AOUP dynamics introduced in this paper is designed to capture the basic features of the well-established inertial active Brownian particle model, i.e. the persistence time of the active motion and the long-time diffusion coefficient. For a small or moderate rotational inertia, these two models predict similar dynamics at all timescales and, in general, our inertial AOUP model consistently yields the same trend upon changing the moment of inertia for various dynamical correlation functions.

Most probable path of active Ornstein-Uhlenbeck particles

Using the path integral representation of the nonequilibrium dynamics, we compute the most probable path between arbitrary starting and final points that is followed by an active particle driven by persistent noise. We focus our attention on the case of active particles immersed in harmonic potentials, where the trajectory can be computed analytically. Once we consider the extended Markovian dynamics where the self-propulsive drive evolves according to an Ornstein-Uhlenbeck process, we can compute the trajectory analytically with arbitrary conditions on position and self-propulsion velocity. We test the analytical predictions against numerical simulations and we compare the analytical results with those obtained within approximated equilibriumlike dynamics.

Tuning nonequilibrium phase transitions with inertia

In striking contrast to equilibrium systems, inertia can profoundly alter the structure of active systems. Here, we demonstrate that driven systems can exhibit effective equilibrium-like states with increasing particle inertia, despite rigorously violating the fluctuation-dissipation theorem. Increasing inertia progressively eliminates motility-induced phase separation and restores equilibrium crystallization for active Brownian spheres. This effect appears to be general for a wide class of active systems, including those driven by deterministic time-dependent external fields, whose nonequilibrium patterns ultimately disappear with increasing inertia. The path to this effective equilibrium limit can be complex, with finite inertia sometimes acting to accentuate nonequilibrium transitions. The restoration of near equilibrium statistics can be understood through the conversion of active momentum sources to passive-like stresses. Unlike truly equilibrium systems, the effective temperature is now density dependent, the only remnant of the nonequilibrium dynamics. This density-dependent temperature can in principle introduce departures from equilibrium expectations, particularly in response to strong gradients. Our results provide additional insight into the effective temperature ansatz while revealing a mechanism to tune nonequilibrium phase transitions.

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.

Entropy production of active particles formulated for underdamped dynamics

The present work investigates the effect of inertia on the entropy production rate Π for all canonical models of active particles for different dimensions and the type of confinement. To calculate Π, the link between the entropy production and dissipation of heat rate is explored, resulting in a simple and intuitive expression. By analyzing the Kramers equation, alternative formulations of Π are obtained and the virial theorem for active particles is derived. Exact results are obtained for particles in an unconfined environment and in a harmonic trap. In both cases, Π is independent of temperature. For the case of a harmonic trap, Π attains a maximal value for τ=ω^{-1}, where τ is the persistence time and ω is the natural frequency of an oscillator. For active particles in one-dimensional box, or other nonharmonic potentials, thermal fluctuations are found to reduce Π.

Fluctuations of entropy production of a run-and-tumble particle

Out-of-equilibrium systems continuously generate entropy, with its rate of production being a fingerprint of nonequilibrium conditions. In small-scale dissipative systems subject to thermal noise, fluctuations of entropy production are significant. Hitherto, mean and variance have been abundantly studied, even if higher moments might be important to fully characterize the system of interest. Here, we introduce a graphical method to compute any moment of entropy production for a generic discrete-state system. Then, we focus on a paradigmatic model of active particles, i.e., run-and-tumble dynamics, which resembles the motion observed in several micro-organisms. Employing our framework, we compute the first three cumulants of the entropy production for a discrete version of this model. We also compare our analytical results with numerical simulations. We find that as the number of states increases, the distribution of entropy production deviates from a Gaussian. Finally, we extend our framework to a continuous state-space run-and-tumble model, using an appropriate scaling of the transition rates. The approach presented here might help uncover the features of nonequilibrium fluctuations of any current in biological systems operating out-of-equilibrium.

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

Time-reversal symmetry breaking and entropy production are universal features of nonequilibrium phenomena. Despite its importance in the physics of active and living systems, the entropy production of systems with many degrees of freedom has remained of little practical significance because the high dimensionality of their state space makes it difficult to measure. Here we introduce a local measure of entropy production and a numerical protocol to estimate it. We establish a connection between the entropy production and extractability of work in a given region of the system and show how this quantity depends crucially on the degrees of freedom being tracked. We validate our approach in theory, simulation, and experiments by considering systems of active Brownian particles undergoing motility-induced phase separation, as well as active Brownian particles and E.coli in a rectifying device in which the time-reversal asymmetry of the particle dynamics couples to spatial asymmetry to reveal its effects on a macroscopic scale.

Active Refrigerators Powered by Inertia

We present the operational principle for a refrigerator that uses inertial effects in active Brownian particles to locally reduce their (kinetic) temperature by 2 orders of magnitude below the environmental temperature. This principle exploits the peculiar but so-far unknown shape of the phase diagram of inertial active Brownian particles to initiate motility-induced phase separation in the targeted cooling regime only. Remarkably, active refrigerators operate without requiring isolating walls opening the route toward using them to systematically absorb and trap, e.g., toxic substances from the environment.

Passive probe particle in an active bath: can we tell it is out of equilibrium?

We study a passive probe immersed in a fluid of active particles. Despite the system's non-equilibrium nature, the trajectory of the probe does not exhibit non-equilibrium signatures: its velocity distribution remains Gaussian, the second fluctuation dissipation theorem is not fundamentally violated, and the motion does not indicate breaking of time reversal symmetry. To tell that the probe is out of equilibrium requires examination of its behavior in tandem with that of the active fluid: the kinetic temperature of the probe does not equilibrate to that of the surrounding active particles. As a strategy to diagnose non-equilibrium from probe trajectories alone, we propose to examine their response to a small perturbation which reveals a non-equilibrium signature through a violation of the first fluctuation dissipation theorem.

Signatures of irreversibility in microscopic models of flocking

Flocking in d=2 is a genuine nonequilibrium phenomenon for which irreversibility is an essential ingredient. We study a class of minimal flocking models whose only source of irreversibility is self-propulsion and use the entropy production rate (EPR) to quantify the departure from equilibrium across their phase diagrams. The EPR is maximal in the vicinity of the order-disorder transition, where reshuffling of the interaction network is fast. We show that signatures of irreversibility come in the form of asymmetries in the steady-state distribution of the flock's microstates. These asymmetries occur as consequences of the time-reversal symmetry breaking in the considered self-propelled systems, independently of the interaction details. In the case of metric pairwise forces, they reduce to local asymmetries in the distribution of pairs of particles. This study suggests a possible use of pair asymmetries both to quantify the departure from equilibrium and to learn relevant information about aligning interaction potentials from data.

Inertial particle under active fluctuations: Diffusion and work distributions

We study the underdamped motion of a passive particle in an active environment. Using the phase space path integral method we find the probability distribution function of position and velocity for a free and a harmonically bound particle. The environment is characterized by an active noise which is described as the Ornstein-Uhlenbeck process (OUP). Taking two similar, yet slightly different OUP models, it is shown how inertia along with other relevant parameters affect the dynamics of the particle. Further we investigate the work fluctuations of a harmonically trapped particle by considering the trap center being pulled at a constant speed. Finally, the fluctuation theorem of work is validated with an effective temperature in the steady-state limit.

Experimental investigation of active noise on a rotor in an active granular bath

In an active bath, besides thermal noise, immersed passive objects also persistently experience collisions from active particles, which are often coarse-grained into a colored active noise with an assumed exponential time correlation. The exponentially correlated active noise extremely simplifies the theoretical description of immersed passive objects but so far lacks direct experimental verification. Here, we experimentally investigate the active noise subjected by a passive rotor confined in an active granular bath. On the basis of Langevin dynamics, we extract the characteristic of the active noise by analyzing the power spectrum of the rotor trajectory. Our experimental results find that the active noise experienced by the granular rotor does show an exponential time correlation to a good extent, even though due to the small experimental system and low collision frequency, the profile of the active noise in our system is non-Gaussian. Our findings give direct experimental evidence, which supports the widely-used active Ornstein-Uhlenbeck particle model in our dry active system.

Intuitive view of entropy production of ideal run-and-tumble particles

This work investigates the entropy production rate, Π, of the run-and-tumble model with a focus on scaling of Π as a function of the persistence time τ. It is determined that (i) Π vanishes in the limit τ→∞, marking it as an equilibrium. Stationary distributions in this limit are represented by a superposition of Boltzmann functions in analogy to a system with quenched disorder. (ii) Optimal Π is attained in the limit τ→0, marking it as a system maximally removed from equilibrium. Paradoxically, the stationary distributions in this limit have the Boltzmann form. The value of Π in this limit is that of an unconfined run-and-tumble particle and is related to the dissipation energy of a sedimenting particle. In addition to these general conclusions, this work derives an exact expression of Π for the run-and-tumble particles in a harmonic trap.

Efficiency of isothermal active matter engines: Strong driving beats weak driving

We study microscopic engines that use a single active particle as their "working medium." Part of the energy required to drive the directed motion of the particle can be recovered as work, even at a constant temperature. A wide class of synthetic active particles can be captured by schematically accounting for the chemical degrees of freedom that power the directed motion without having to resolve the exact microscopic mechanism. We derive analytical results for the quasistatic thermodynamic efficiency, i.e., the fraction of available chemical energy that can be recovered as mechanical work. While this efficiency is vanishingly small for colloidal particles, it increases as the dissipation is increased beyond the linear-response regime and goes through a maximum at large propulsion speeds. Our results demonstrate that driving beyond the linear-response regime has nontrivial consequences for the efficiency of active engines.

Designing circle swimmers: Principles and strategies

Various microswimmers move along circles rather than straight lines due to their swimming mechanisms, body shapes, or hydrodynamic effects. In this paper, we adopt the concepts of stochastic thermodynamics to analyze circle swimmers confined to a two-dimensional plane and study the trade-off relations between various physical quantities, such as precision, energy cost, and rotational speed. Based on these findings, we predict principles and strategies for designing microswimmers of special optimized functions under limited energy resource conditions, which will bring new experimental inspiration for designing smart motors.

Physical bioenergetics: Energy fluxes, budgets, and constraints in cells

Cells are the basic units of all living matter which harness the flow of energy to drive the processes of life. While the biochemical networks involved in energy transduction are well-characterized, the energetic costs and constraints for specific cellular processes remain largely unknown. In particular, what are the energy budgets of cells? What are the constraints and limits energy flows impose on cellular processes? Do cells operate near these limits, and if so how do energetic constraints impact cellular functions? Physics has provided many tools to study nonequilibrium systems and to define physical limits, but applying these tools to cell biology remains a challenge. Physical bioenergetics, which resides at the interface of nonequilibrium physics, energy metabolism, and cell biology, seeks to understand how much energy cells are using, how they partition this energy between different cellular processes, and the associated energetic constraints. Here we review recent advances and discuss open questions and challenges in physical bioenergetics.

Statistical mechanics of active Ornstein-Uhlenbeck particles

We study the statistical properties of active Ornstein-Uhlenbeck particles (AOUPs). In this simplest of models, the Gaussian white noise of overdamped Brownian colloids is replaced by a Gaussian colored noise. This suffices to grant this system the hallmark properties of active matter, while still allowing for analytical progress. We study in detail the steady-state distribution of AOUPs in the small persistence time limit and for spatially varying activity. At the collective level, we show AOUPs to experience motility-induced phase separation both in the presence of pairwise forces or due to quorum-sensing interactions. We characterize both the instability mechanism leading to phase separation and the resulting phase coexistence. We probe how, in the stationary state, AOUPs depart from their thermal equilibrium limit by investigating the emergence of ratchet currents and entropy production. In the small persistence time limit, we show how fluctuation-dissipation relations are recovered. Finally, we discuss how the emerging properties of AOUPs can be characterized from the dynamics of their collective modes.

Phase Transitions of Oppositely Charged Colloidal Particles Driven by Alternating Current Electric Field

We study systems containing oppositely charged colloidal particles under applied alternating current electric fields (AC fields) using overdamped Langevin dynamics simulations in three dimensions. We obtain jammed bands perpendicular to the field direction under intermediate frequencies and lanes parallel with the field under low frequencies. These structures also depend upon the particle charges. The pathway for generating jammed bands follows a stepwise mechanism, and intermediate bands are observed during lane formation in some systems. We investigate the component of the pressure tensors in the direction parallel to the field and observe that the jammed to lane transition occurs at a critical value for this pressure. We also find that the stable steady states appear to satisfy the principle of maximum entropy production. Our results may help to improve the understand of the underlying mechanisms for these types of dynamic phase transitions and the subsequent cooperative assemblies of colloidal particles under such non-equilibrium conditions.

Entropy production fluctuations encode collective behavior in active matter

We derive a general lower bound on distributions of entropy production in interacting active matter systems. The bound is tight in the limit that interparticle correlations are small and short-ranged, which we explore in four canonical active matter models. In all models studied, the bound is weak where collective fluctuations result in long-ranged correlations, which subsequently links the locations of phase transitions to enhanced entropy production fluctuations. We develop a theory for the onset of enhanced fluctuations and relate it to specific phase transitions in active Brownian particles. We also derive optimal control forces that realize the dynamics necessary to tune dissipation and manipulate the system between phases. In so doing, we uncover a general relationship between entropy production and pattern formation in active matter, as well as ways of controlling it.

Collective motion in large deviations of active particles

We analyze collective motion that occurs during rare (large deviation) events in systems of active particles, both numerically and analytically. We discuss the associated dynamical phase transition to collective motion, which occurs when the active work is biased towards larger values, and is associated with alignment of particles' orientations. A finite biasing field is needed to induce spontaneous symmetry breaking, even in large systems. Particle alignment is computed exactly for a system of two particles. For many-particle systems, we analyze the symmetry breaking by an optimal-control representation of the biased dynamics, and we propose a fluctuating hydrodynamic theory that captures the emergence of polar order in the biased state.

Non-Markovian data-driven modeling of single-cell motility

Trajectories of human breast cancer cells moving on one-dimensional circular tracks are modeled by the non-Markovian version of the Langevin equation that includes an arbitrary memory function. When averaged over cells, the velocity distribution exhibits spurious non-Gaussian behavior, while single cells are characterized by Gaussian velocity distributions. Accordingly, the data are described by a linear memory model which includes different random walk models that were previously used to account for various aspects of cell motility such as migratory persistence, non-Markovian effects, colored noise, and anomalous diffusion. The memory function is extracted from the trajectory data without restrictions or assumptions, thus making our approach truly data driven, and is used for unbiased single-cell comparison. The cell memory displays time-delayed single-exponential negative friction, which clearly distinguishes cell motion from the simple persistent random walk model and suggests a regulatory feedback mechanism that controls cell migration. Based on the extracted memory function we formulate a generalized exactly solvable cell migration model which indicates that negative friction generates cell persistence over long timescales. The nonequilibrium character of cell motion is investigated by mapping the non-Markovian Langevin equation with memory onto a Markovian model that involves a hidden degree of freedom and is equivalent to the underdamped active Ornstein-Uhlenbeck process.

Target Search of Active Agents Crossing High Energy Barriers

Target search by active agents in rugged energy landscapes has remained a challenge because standard enhanced sampling methods do not apply to irreversible dynamics. We overcome this nonequilibrium rare-event problem by developing an algorithm generalizing transition-path sampling to active Brownian dynamics. This method is exemplified and benchmarked for a paradigmatic two-dimensional potential with a high barrier. We find that even in such a simple landscape the structure and kinetics of the ensemble of transition paths changes drastically in the presence of activity. Indeed, active Brownian particles reach the target more frequently than passive Brownian particles, following longer and counterintuitive search patterns.

Optimizing energetic cost of uncertainty in a driven system with and without feedback

Many biological functions require dynamics to be necessarily driven out of equilibrium. In contrast, in various contexts, a nonequilibrium dynamics at fast timescales can be described by an effective equilibrium dynamics at a slower timescale. In this work, we study two different aspects: (i) the energy-efficiency tradeoff for a specific nonequilibrium linear dynamics of two variables with feedback and (ii) the cost of effective parameters in a coarse-grained theory as given by the "hidden" dissipation and entropy production rate in the effective equilibrium limit of the dynamics. To meaningfully discuss the tradeoff between energy consumption and the efficiency of the desired function, a one-to-one mapping between function(s) and energy input is required. The function considered in this work is the variance of one of the variables. We get a one-to-one mapping by considering the minimum variance obtained for a fixed entropy production rate and vice versa. We find that this minimum achievable variance is a monotonically decreasing function of the given entropy production rate. When there is a timescale separation, in the effective equilibrium limit, the cost of the effective potential and temperature is the associated "hidden" entropy production rate.

Active matter: Quantifying the departure from equilibrium

Active matter systems are driven out of equilibrium at the level of individual constituents. One widely studied class are systems of athermal particles that move under the combined influence of interparticle interactions and self-propulsions, with the latter evolving according to the Ornstein-Uhlenbeck stochastic process. Intuitively, these so-called active Ornstein-Uhlenbeck particle (AOUP) systems are farther from equilibrium for longer self-propulsion persistence times. Quantitatively, this is confirmed by the increasing equal-time velocity correlations (which are trivial in equilibrium) and by the increasing violation of the Einstein relation between the self-diffusion and mobility coefficients. In contrast, the entropy production rate, calculated from the ratio of the probabilities of the position space trajectory and its time-reversed counterpart, has a nonmonotonic dependence on the persistence time. Thus, it does not properly quantify the departure of AOUP systems from equilibrium.

Entropy production of an active particle in a box

A run-and-tumble particle in a one-dimensional box (infinite potential well) is studied. The steady state is analytically solved and analyzed, revealing the emergent length scale of the boundary layer where particles accumulate near the walls. The mesoscopic steady state entropy production rate of the system is derived from coupled Fokker-Planck equations with a linear reaction term, resulting in an exact analytic expression. The entropy production density is shown to peak at the walls. Additionally, the derivative of the entropy production rate peaks at a system size proportional to the length scale of the accumulation boundary layer, suggesting that the behavior of the entropy production rate and its derivatives as a function of the control parameter may signify a qualitative behavior change in the physics of active systems, such as phase transitions.

Quantifying the non-equilibrium activity of an active colloid

Active matter systems exhibit rich emergent behavior due to constant injection and dissipation of energy at the level of individual agents. Since these systems are far from equilibrium, their dynamics and energetics cannot be understood using the framework of equilibrium statistical mechanics. Recent developments in stochastic thermodynamics extend classical concepts of work, heat, and energy dissipation to fluctuating non-equilibrium systems. We use recent advances in experiment and theory to study the non-thermal dissipation of individual light-activated self-propelled colloidal particles. We focus on characterizing the transition from thermal to non-thermal fluctuations and show that energy dissipation rates on the order of ∼kBT s-1 are measurable from finite time series data.

Active noise experienced by a passive particle trapped in an active bath

We study the properties of active noise experienced by a passive particle harmonically trapped in an active bath. The active bath is either explicitly simulated by an ensemble of active Brownian particles or abstractly represented by an active colored noise in theory. Assuming the equivalence of the two descriptions of the active bath, the active noise in the simulation system, which is directly extracted by fitting theoretical predictions to simulation measurements, is shown to depend on the constraint suffered by the passive tracer. This scenario is in significant contrast to the case of thermal noise that is independent of external trap potentials. The constraint dependence of active noise arises from the fact that the persistent force on the passive particle from the active bath can be influenced by the particle relaxation dynamics. Moreover, due to the interplay between the active collisions and particle relaxation dynamics, the effective temperature of the passive tracer quantified as the ratio of fluctuation to dissipation increases as the constraint strengthens, while the average potential and kinetic energies of the passive particle both decrease.

Collective forces in scalar active matter

Large-scale collective behavior in suspensions of active particles can be understood from the balance of statistical forces emerging beyond the direct microscopic particle interactions. Here we review some aspects of the collective forces that can arise in suspensions of self-propelled active Brownian particles: wall forces under confinement, interfacial forces, and forces on immersed bodies mediated by the suspension. Even for non-aligning active particles, these forces are intimately related to a non-uniform polarization of particle orientations induced by walls and bodies, or inhomogeneous density profiles. We conclude by pointing out future directions and promising areas for the application of collective forces in synthetic active matter, as well as their role in living active matter.

Approach to equilibrium and nonequilibrium stationary distributions of interacting many-particle systems that are coupled to different heat baths

A Hamiltonian-based model of many harmonically interacting massive particles that are subject to linear friction and coupled to heat baths at different temperatures is used to study the dynamic approach to equilibrium and nonequilibrium stationary states. An equilibrium system is here defined as a system whose stationary distribution equals the Boltzmann distribution, the relation of this definition to the conditions of detailed balance and vanishing probability current is discussed both for underdamped as well as for overdamped systems. Based on the exactly calculated dynamic approach to the stationary distribution, the functional that governs this approach, which is called the free entropy S_{free}(t), is constructed. For the stationary distribution S_{free}(t) becomes maximal and its time derivative, the free entropy production S[over ̇]_{free}(t), is minimal and vanishes. Thus, S_{free}(t) characterizes equilibrium as well as nonequilibrium stationary distributions by their extremal and stability properties. For an equilibrium system, i.e., if all heat baths have the same temperature, the free entropy equals the negative free energy divided by temperature and thus corresponds to the Massieu function which was previously introduced in an alternative formulation of statistical mechanics. Using a systematic perturbative scheme for calculating velocity and position correlations in the overdamped massless limit, explicit results for few particles are presented: For two particles localization in position and momentum space is demonstrated in the nonequilibrium stationary state, indicative of a tendency to phase separate. For three elastically interacting particles heat flows from a particle coupled to a cold reservoir to a particle coupled to a warm reservoir if the third reservoir is sufficiently hot. This does not constitute a violation of the second law of thermodynamics, but rather demonstrates that a particle in such a nonequilibrium system is not characterized by an effective temperature which equals the temperature of the heat bath it is coupled to. Active particle models can be described in the same general framework, which thereby allows us to characterize their entropy production not only in the stationary state but also in the approach to the stationary nonequilibrium state. Finally, the connection to nonequilibrium thermodynamics formulations that include the reservoir entropy production is discussed.