Active engines: Thermodynamics moves forward
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Active engines: Thermodynamics moves forward ´Etienne Fodor and Michael E. Cates Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
PACS – Nonequilibrium and irreversible thermodynamics
Abstract – The study of thermal heat engines was pivotal to establishing the principles of equi-librium thermodynamics, with implications far wider than only engine optimization. For nonequi-librium systems, which by definition dissipate energy even at rest, how to best convert such dissi-pation into useful work is still largely an outstanding question, with similar potential to illuminategeneral physical principles. We review recent theoretical progress in studying the performancesof engines operating with active matter, where particles are driven by individual self-propulsion.We distinguish two main classes, either autonomous engines exploiting a particle current, or cyclicengines applying periodic transformation to the system, and present the strategies put forward sofar for optimization. We delineate the limitations of previous studies, and propose some futuresperspectives, with a view to building a consistent thermodynamic framework far from equilibrium.
The foundation of equilibrium thermodynamics waslargely motivated by the study of thermal engines. Duringthe Industrial Revolution of the 19 th century, an impor-tant challenge was to establish a theoretical framework torationalize how to best convert different forms of energy,such as work and heat. The main contributions of Carnotand Stirling was to provide simple protocols, based on aminimal description of engines, for which the performancescan be evaluated exactly in terms of a few parameters [1].These protocols served as a cornerstone to establish theprinciples of equilibrium thermodynamics, thus eclipsingtheir original motivation of optimizing realistic engines.Thermal engines, which operate by manipulating a pas-sive system, can only extract work by changing its tem-perature. Interestingly, many nonequilibrium systems dis-sipate heat despite the absence of any temperature vari-ation. This removes a major constraint, and opens thedoor to designing innovative protocols with no passivecounterpart. Following this route, a main difficulty lies inrationalizing properly the performances of engines whichfall beyond the scope of standard thermodynamics. Thestudy of such innovative engines then requires a search fora systematic recipe to evaluate properly, and to exploit ef-ficiently, the energy fluxes of nonequilibrium systems. Asholds for thermal engines, this target typically amountsto producing maximum work while dissipating minimumheat. The important outstanding challenge is to delineatethe generic principles for the design of nonequilibrium en- gines, encompassing within a unified framework a largeclass of systems beyond equilibrium. Whether or not suchengines are practically useful, one can hope to gain broaderinsights by studying them.In recent decades, active matter has emerged as the classof nonequilibrium systems whose microscopic constituentsextract energy from their environment to produce a sys-tematic, autonomous motion [2–4]. Examples of activematter can be found either in (i) biological systems, suchas swarms of bacteria [5] and cell tissues [6], (ii) social sys-tems, such as groups of animals [7] and human crowds [8],or (iii) synthetic systems, such as self-catalytic colloidsin a fuel bath [9] and vibrated polar particles [10]. Thecombination of individual self-propulsion and interactionsbetween individual particles leads to nonequilibrium col-lective behaviors. Examples include a collective directedmotion with long-ranged polar order [11], and a phase sep-aration for purely repulsive constituents [12].To rationalize these collective behaviors in terms of sim-ple ingredients, minimal models have been proposed. Inmost cases, the details of microscopic interactions allowone to anticipate which type of order emerges at largescale, thus defining different classes of active systems [2].Based on these minimal models, numerous studies havestriven to characterize the properties of active matterwithin a unified framework, by relying mostly on the par-tial applicability of thermodynamic concepts beyond equi-librium. This includes scrutiny of observables directly in-p-1 a r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n Etienne Fodor and Michael E. Cates
Fig. 1: (Color online) Illustration of autonomous active en-gines. (Top) Snapshots of an experimental active ratchet, madeof a wheel with asymmetric teeth immersed in a bath of bacte-ria. The breakdown of both time-reversal symmetry, inducedby the nonequilibrium dynamics of bacteria, and spatial sym-metry, induced by the wheel shape, leads to a spontaneouscurrent, given here by rotation of the wheel. Taken from [21].(Bottom) The engine extracts work by applying an externalload opposed to the spontaneous current. Here, the load con-sists in connecting an external mass to the wheel, which en-forces a counter-torque opposed to the spontaneous rotation.Adapted from [26]. spired by equilibrium studies, such as pressure [13, 14],surface tension [15, 16], and chemical potential [17, 18].Although incomplete, this thermodynamic frameworksheds light on many anomalous properties of active mat-ter, allowing one to better delineate its deviation fromequilibrium. For instance, introducing an external wall toprobe the internal pressure perturbs the system so stronglythat the measurement can not be disentangled from thedetails of the wall: At variance with equilibrium, the pres-sure does not follow any equation of state in most dryactive systems [14]. Moreover, embedding asymmetric ob-stacles in a bath of active particles leads to spontaneousparticle currents: See fig. 1. These ratchets exploit the si-multaneous breakdown of spatial symmetry, encoded inthe obstacle shape, and of time-reversal symmetry, in-duced by any nonequilibrium dynamics [19]. A large va-riety of ratchets have been proposed and studied, bothexperimentally [20–22] and theoretically [23–26].Many works have focused on how to characterize theirreversibility of the dynamics even in the absence of ob-stacles [27–31]. To this aim, one can rely on the frameworkof stochastic thermodynamics, which allows one to eval-uate irreversibility in terms of some specific observables. This framework was first established in passive systemsto extend standard notions of thermodynamics, such aswork and heat, to small scales [32, 33]. It offers system-atic methods to identify the energy transfers between asystem and its environment using a bottom-up approach,based on analyzing directly the dynamics of the micro-scopic constituents. Stochastic thermodynamics therebycreates a useful framework to build thermodynamic prin-ciples out-of-equilibrium, since it does not require us todefine a priori any macroscopic state function. Various re-cent studies have shown how to extend such a frameworkfrom passive to active systems, thus providing definitionsof work and heat in active matter [34–36].Recent developments in active matter and in stochasticthermodynamics have thus led to many important resultson the thermodynamic properties of active systems. Bynow, our understanding of active matter is sufficiently ad-vanced and well-established to rationalize (i) how to de-tect unambiguously the nonequilibrium properties of ac-tive systems, and (ii) how to disentangle properly the en-ergy transfers induced either by the microscopic activity orby any external protocol. This knowledge fosters unprece-dented hopes for building a systematic, versatile frame-work of design principles for the reliable control of activesystems. Towards such a framework, a first and importantstep consists in studying a specific class of protocols whichextract energy from active matter. In doing so, one mo-tivation is to go beyond simply characterizing nonequilib-rium properties, to exploit the deviation from equilibriumto design active engines. A second motivation is the hopethat, as happened for equilibrium thermodynamics, studyof specific engines can illuminate general principles.In what follows, we present a short review of various ac-tive engines proposed in the literature. The challenge is tosearch for appropriate protocols which convert the randommotion of self-propelled particles into extractable work,and to establish a systematic road-map for the optimiza-tion of engine performances. Some protocols are inspiredby experimental realizations, others rely on innovativesettings which might motivate future experiments. Weconsider minimal models of self-propelled particles, whichcontain the essential ingredients at play in active matter,and we discuss how to exploit properly their nonequilib-rium properties. We distinguish two main classes of activeengines: (i) autonomous engines, which exploit the ratchetcurrents, see fig. 1, and (ii) cyclic engines, which operateby applying periodic transformation to active and/or pas-sive particles, see fig. 2. Within each class, we presentsome simple examples, which could inspire more complexones, and we characterize their performances with univer-sal observables, namely efficiency and power.
Particle-based description of active engines. – Weconsider a generic setting where the engine can be de-scribed in terms of (i) the coordinates of some passiveelements { x a } , for instance the positions of obstacles, and(ii) a set of active coordinates { r i , θ i } , typically the po-sition and orientation of active particles in two spatialp-2ctive engines: Thermodynamics moves forward Fig. 2: (Color online) Illustration of cyclic active engines.(Top) Colloidal engine made of a passive tracer, in blue, im-mersed in a bath of active particles, in red, and confined in aharmonic trap whose stiffness can be varied externally. A typ-ical cycle consists in synchronizing the trap stiffness with someparameters of the bath. Taken from [37]. (Bottom) Macro-scopic engine made of active particles confined in a container.The rightmost boundary is connected to an external wheel,hence periodically compressing and expanding the assemblyof active particles. A typical cycle consists in synchronizingthe volume of the container with either some properties of theboundaries, such as their stiffness, or some properties of theconfined particles, such as their temperature or activity. dimensions. Provided that inertial effects are negligible,their dynamics is given by coupled, overdamped Langevinequations. Each of these encodes a balance between theforces stemming from a thermostat, namely viscous dragand thermal noise, that deriving from external potential,and some non-conservative forces. Introducing the poten-tial U ( { x a , r i , θ i } ), the dynamics of passive elements reads˙ x a = µ p (cid:18) f a − ∂U∂ x a (cid:19) + (cid:112) D p ζ a , (1)and that of active particles is (with e i = (cos θ i , sin θ i ))˙ r i = v e i + µ (cid:18) f i − ∂U∂ r i (cid:19) + √ D ξ i , ˙ θ i = µ r (cid:18) ω i − ∂U∂θ i (cid:19) + (cid:112) D r η i . (2)The terms { f a , f i , ω i } represent non-conservative forcesand torques which can in principle be applied on thevarious elements of the system by an external agent oroperator. The active force v e i has a constant norm v ,and its orientation is given by the unit orientation vec-tor e i . This force represents the conversion of some en-ergy fuel present in the environment, for instance ATPin living systems [6], into an autonomous self-propulsion. The terms { ζ a , ξ i , η i } are uncorrelated Gaussian whitenoises, with zero mean and unit variance. The diffu-sion coefficients { D p , D, D r } and the mobilities { µ p , µ, µ r } are generally constrained so that their respective ratio isset by the temperature T of the surrounding thermostat: T = D p /µ p = D/µ = D r /µ r . Note that these equa-tions do not conserve momentum and therefore representa “dry” system, such as a 2D layer of active particles ona solid support with which momentum is exchanged.Importantly, we take the potential U to be under ex-ternal control, as captured by a set of parameters { α k } which determine the shape of U . By changing { α k } as afunction of time, the operator can vary the shape of anexternal potential applied on each particle (or a subset ofthem), and also in principle control how particles interactbetween themselves. For autonomous engines, the opera-tor typically applies some constant forces and/or torques { f a , f i , ω i } which are opposed to the spontaneous currentof the corresponding passive and/or active coordinates,without changing U , see fig. 1. In contrast, for cyclic en-gines, the parameters { α k } are varied periodically in timewithout applying any external force or torque, see fig. 2.While both features could be combined in a single device,this division into two basic types is conceptually helpful. Extracted work and dissipated heat. – Followingstandard definitions of stochastic thermodynamics [32,33],the work W produced during a protocol time τ is W = (cid:90) τ dt (cid:20) (cid:88) k ˙ α k ∂U∂α k + (cid:88) a f a · ˙ x a + (cid:88) i (cid:0) f i · ˙ r i + ω i ˙ θ i (cid:1)(cid:21) . (3)The sign convention is that energy is extracted from thesystem whenever W <
0. The work W is a stochasticprocess which fluctuates from one realization of the proto-col to another. The challenge in designing an engine is toensure that the work remains negative on average . Impor-tantly, the definition in (3) only involves terms describinghow the operator perturbs the dynamics. It is agnosticto the presence of any other non-conservative force, andthus (3) is the same for a passive system.To evaluate the amount of heat Q dissipated by thedegrees of freedom { x a , r i , θ i } , it suffices to quantify thepower of the forces and torques exerted by the thermostaton each one of them [32, 33]. Indeed, by definition, thethermostat can only exchange energy with the system inthe form of heat. Therefore, given the viscous drag termsand the thermal noises in (1-2), the heat follows as Q = (cid:90) τ dt (cid:26) (cid:88) a ˙ x a µ p · (cid:0) ˙ x a − (cid:112) D p ζ a (cid:1) + (cid:88) i (cid:20) ˙ r i µ · (cid:0) ˙ r i − √ D ξ i (cid:1) + ˙ θ i µ r (cid:0) ˙ θ i − (cid:112) D r η i (cid:1)(cid:21)(cid:27) . (4)The heat Q is a stochastic process, like the work W , andis defined such that energy is transferred from the system to the thermostat whenever Q >
0. The definition in (4)p-3Etienne Fodor and Michael E. Catesagain holds even for a passive system ( v = 0). Substitutingthe dynamics (1-2) into eq. (4), we deduce Q = (cid:90) τ dt (cid:26) (cid:88) a (cid:18) f a − ∂U∂ x a (cid:19) · ˙ x a + (cid:88) i (cid:20)(cid:18) v e i µ + f i − ∂U∂ r i (cid:19) · ˙ r i + (cid:18) ω i − ∂U∂θ i (cid:19) ˙ θ i (cid:21)(cid:27) . (5)For constant { α k } and in the absence of { f a , f i , ω i } , theexternal operator does not perturb the dynamics. The av-erage heat is then extensive in time: (cid:104)Q(cid:105) = ( vτ /µ ) (cid:80) i (cid:104) e i · ˙ r i (cid:105) , which illustrates that there is always a constant rateof energy dissipated by the active forces, independently ofany external protocol. Finally, combining the expressionof work and heat in (3) and (5), we get W − Q = U ( t = 0) − U ( t = τ ) − (cid:90) τ dt (cid:88) i v e i µ · ˙ r i . (6)For passive systems ( v = 0), this is the first law of ther-modynamics (FLT), as expected. For an active system,the energy budget written in (6) no longer involves solelythe work W , the heat Q , and the potential U . Crucially,it now also accounts for the energy spent sustaining theself-propulsion of active particles.While the definition of work, given in (3), is standardin most of the literature, how heat should be properly de-fined remains widely debated. One reason for the contro-versy is that the thermal noises in the dynamics (1-2), aresometimes omitted in active matter studies. In this con-text, some works have defined heat by replacing the ther-mal noises appearing in (4) with the corresponding activeforces for each degree of freedom [38–40]. This amounts toenforcing that the standard FLT, i.e. eq. (6) with v = 0,should hold in the same form for active and passive sys-tems. Even when the dynamics features thermal noises,other works have defined heat by substituting the sum ofthermal noises and active forces, instead of thermal noisesonly, in (4) [37, 41]. Again, this yields the standard FLT.Importantly, the main drawback of these definitions ofheat, leading to the standard FLT, is that they do not cap-ture the energy dissipated by active forces. Indeed, whenthe operator does not perturb the dynamics, these defi-nitions lead to a vanishing average heat, which suggestsmisleadingly that the self-propulsion can be sustained atzero cost. To avoid such an inconsistency, one should for-mally distinguish the thermal noises from the active forces,as is done in (4). This applies even when (as is often truein active systems) thermal effects are negligible. Indeed,the self-propulsion stems from the underlying consump-tion of fuel energy which, although affected by thermalfluctuations, is mainly not of thermal origin.Yet another way of defining heat imagines the activevelocity v e i to stem from a local advective fluid flow, withthe displacement of a given active particle evaluated withrespect to this flow [31, 42]. In this approach, the velocity˙ r i appearing in (4) is replaced systematically with ˙ r i − v e i , which leads to substituting ∂U/∂ r i instead of ˙ r i /µ in (6).With this definition, the average heat in the absence ofexternal perturbation reads vτ (cid:80) i (cid:104) e i · ( ∂U/∂ r i ) (cid:105) , whichvanishes when there is no potential U . Overall, among thevarious definitions of heat presented above, the only onewhich retains a finite cost of energy even in the absence ofpotential U is the one we have adopted in (4). Efficiency and power. – The efficiency of the engine E compares the output work with the energetic cost ofoperating the engine: E = (cid:104)W(cid:105)(cid:104)W(cid:105) − (cid:104)Q(cid:105) . (7)Since the average heat is always positive (a signature ofthe irreversibility of the dynamics), the efficiency is al-ways smaller than unity, as it should be. The defini-tion in (7) is standard for protocols at constant temper-ature [19, 26, 43–45]. For cycles which vary the tempera-ture, other definitions inspired by that of thermal enginesare sometimes preferred, with a view to comparing theperformances of thermal cycles with and without activ-ity [37, 39, 41, 46, 47]. Under steady cycling of a periodicprotocol with period τ , the U terms in (6) cancel on aver-age, so that by substituting (6) into (7), we get E = |(cid:104)W(cid:105)| (cid:18) (cid:90) τ dt (cid:88) i (cid:68) v e i µ · ˙ r i (cid:69)(cid:19) − . (8)This makes it clear that, with our definition, the effectivecost of operating an engines only comes from the activeforces. Note that this approach deliberately neglects theenergetic cost of converting fuel (or another ambient en-ergy supply) into self-propulsion. Indeed, we only resolvethe energy budget from the propulsion scale upwards, butnot at smaller scales. Accordingly the dynamics (1-2)makes no attempt to describe how fuel-consuming pro-cesses bring about the active contribution v e i to ˙ r i .Whenever the dynamics of active particles is mostlydominated by free motion, we have (cid:104) e i · ˙ r i (cid:105) ≈ v in (8). Thisholds typically when (i) interactions with passive and/oractive elements are negligible, namely for dilute systems,and (ii) any external force and/or external potential onlyaffects a sub-extensive number of active particles. This en-compasses, for instance, autonomous engines whose load isapplied only on passive elements, and cyclic engines whosevarying potential is located at boundaries of the system.In such cases, it follows from (8) that the efficiency is di-rectly proportional to the extracted power: E ≈ µN v |(cid:104)W(cid:105)| τ ≡ µ (cid:104)P(cid:105) N v , (9)where N denotes here the number of active particles. Thisimportant result implies that these cases avoid any trade-off between efficiency and power. In other words, one canaddress simultaneously the optimization of (i) how to con-vert efficiently the random self-propulsion into extractablep-4ctive engines: Thermodynamics moves forwardwork, and (ii) how to extract maximum power from thegiven active system. This starkly contrasts with the caseof thermal cyclic engines, where the efficiency is alwaysmaximized by long, quasi-reversible cycle times, for whichthe power generally vanishes [48, 49].Moreover, for active engines which satisfy the efficiency-power relation (9), the fluctuations of the power obey thefollowing bound [43]:1 (cid:104)P (cid:105) − (cid:104)P(cid:105) (cid:20)(cid:18) τ ddτ (cid:19) (cid:104)P(cid:105) (cid:21) ≤ µτ D (cid:18) N v µ −(cid:104)P(cid:105) (cid:19) . (10)The derivation of this bound builds on recent studiesof thermodynamic uncertainty relations [50, 51]. Equa-tion (10) suggests that, when the average power ap-proaches N v /µ , or equivalently when the efficiency getsclose to unity, the fluctuations of the power become verylarge. Indeed, the bound requires the variance (cid:104)P (cid:105)−(cid:104)P(cid:105) to be at least of order (cid:104)P(cid:105) in this limit. However, thebound (10) is established only in regimes where (9) shouldhold. As described above, this requires the active par-ticles to spend most of their time dissipating energy bymoving idly, without contributing much to the extractedwork. This corresponds typically to low efficiency, withthe power being far smaller than N v /µ , weakening thebound in (10). It is an open question to what extent (9-10)remain valid beyond this “weak work extraction” regime. Autonomous engines: Ratchets under load. – Animportant class of active engines are those made of ratch-ets [19], where asymmetric potentials lead to a currentof active and/or passive particles. These autonomous en-gines extract work by applying an external load, typicallyeither a constant force or a constant torque, opposed to thespontaneous current. Examples of ratchets in experimen-tal active systems, which operate in general by rectifyingthe random active fluctuations (mainly in living organ-isms so far), include various shapes of obstacles [3]. Ex-amples include gear-wheels with asymmetric teeth, whichspontaneously rotate [21, 22], see fig. 1; fixed chevrons,which establish density differences between spatial com-partments [20]; and micro-patterns on a substrate whichcreate asymmetric potential barriers guiding the migra-tion of cells [52]. These experiments have inspired manytheoretical works on active ratchets [25]. Some of themconsider either dilute regimes or one-body problems with-out any particle interaction [23, 53, 54], yielding explicitanalytical predictions [55]. Other studies include someform of particle interactions, allowing a more quantitativecomparison with experiments [56, 57].Although such ratchets have been studied extensively,only a few works have considered the effect of applying anexternal load [58], with a view to extracting work [26, 40].The performances of these engines are characterized by theloading curves of power and efficiency as functions of theexternal force f . Above a critical loading, the spontaneouscurrent is reversed by the external force, in which case theoperator provides energy to the system instead of extract-ing it. Hence, the engine operates properly only within a Fig. 3: (Color online) Optimization of an autonomous activeengine. Asymmetric obstacles are in contact with a bath ofactive particles, not shown here, which leads to a spontaneousmotion of the obstacles to the right under periodic boundaryconditions. The trajectories of particles at high persistenceare shown as dashed straight lines for several impact angles θ . (Left) For chevron-shaped obstacles, all trajectories belowa certain angle get trapped, see θ = 50 ◦ , and thus contributeto positive current. Others hinder systematically the displace-ment of the obstacles, see θ = 210 ◦ . (Right) For kite-shapedobstacles, some of the trajectories which are not trapped in thewedges still contribute to positive current by gliding over thetail, see θ = 105 ◦ . Therefore, the overall current of obstacles,given as the sum of currents for all angles, is higher than thatof chevron-shaped obstacles. Taken from [26]. finite range of loading below the stalling force: f ∈ [0 , f s ].For dilute systems, such as active particles that encounteran assembly of asymmetric obstacles [26, 58], the stallingforce is typically of the order of the individual active force v/µ , or smaller. In this case, the engine typically operatesclose to the regime of linear response, where the current islinear in the deviation from the stalling force: (cid:104) ˙ x (cid:105) ∼ f s − f .In contrast, at higher density clogging effects can poten-tially lead the system far from this regime [59]. Within lin-ear response, the power is then simply a parabolic functionof the load: (cid:104)P(cid:105) = f (cid:104) ˙ x (cid:105) ∼ f ( f s − f ), and it has a well de-fined maximum at f = f s /
2. At sufficiently small density,the efficiency is therefore proportional to power, see (9),and both are maximal at the same loading point.In this linear regime, the optimization of the engineperformances consists essentially in adjusting the shapeof obstacles and their spatial distribution in the system,see fig. 3, to increase the current as much as possible.Indeed, it is sufficient to optimize the current at zeroload to increase systematically the maxima of power andefficiency. When the engine consists of obstacles withenforced synchronized motion (as implemented compu-tationally via periodic boundary conditions, fig. 3) thisamounts to treating a purely geometrical problem in theshared co-moving frame of all obstacles [26]. For instance,starting from chevrons made of single wedge, it was foundempirically that the current is significantly increased byputting an additional wedge at the tail, see fig. 3, thusforming kite-shaped obstacles. The angles and sizes of thetwo wedges, and also the distance between each kite, canp-5Etienne Fodor and Michael E. Catesthen be adjusted to optimize simultaneously the maximumpower and efficiency. This illustrates how optimizing au-tonomous engines can guide the design of innovative ratch-ets, with a view to inspiring future experiments. In morecomplex settings, machine learning algorithms might helpthe optimization of obstacle shape, by exploring efficientlya high-dimensional space of geometric parameters.
Cyclic engines: Beyond thermal cycles. – Anotherclass of active engines, which do not rely on exploiting arectified current of passive and/or active particles, operateby applying periodic transformations to the system. These cyclic engines are inspired by thermal engines, where onechanges the temperature periodically [1]. Note, however,that active cycles can potentially extract work even atconstant temperature. Several years ago, a pioneering ex-periment [60] realized a cyclic active engine. This wasmotivated by colloidal thermal engines comprising a sin-gle passive colloid confined in a harmonic trap by opticaltweezers [61, 62]. The active counterpart of [60] puts thecolloid in contact with a bath of bacteria, see fig. 2. Here,changing the temperature not only affects the fluctuationscoming from the heat bath, but also that stemming fromthe bath of active particles, whose propulsion is affectedby temperature changes. Interestingly, recent experimen-tal techniques have demonstrated the ability to vary theself-propulsion of living organisms by external illumina-tion [63, 64], which opens the door to cycles (possibly butnot necessarily monothermal) where activity is controlledindependently of temperature.The experiments outlined above have motivated a largenumber of theoretical studies on cyclic active engines.Most consider colloidal engines made of a single activeparticle [37, 38, 41, 44, 46, 47], but a few address the caseof many-body systems [43, 45, 65]. Following the exper-iment in ref. [60], many of these works studied thermalcycles [37–39, 41, 47, 65], essentially to determine whetheradding activity can lead the system to outperform thermalengines. As discussed following (6) above, one difficulty inaddressing this issue is to extend properly the definitionof heat beyond the case of passive systems. Importantly,the results of [60] rely on assuming that the FLT has thesame form in passive and active systems. Although thisapproach allows one to estimate heat without measuringexplicitly the power of active forces, it also disregards theenergy cost of sustaining activity of the system. How toestimate the dissipation induced by active forces in exper-iments, which would typically require measuring directlythese nonequilibrium fluctuating forces, is still open.Interestingly, a number of recent theoretical studieshave proposed cycles without any equilibrium counter-part [43–46, 65], typically operating at constant temper-ature. These protocols exploit specific nonequilibriumproperties of active systems. For instance, given that thepressure exerted by confining walls generally depends onmicroscopic details of the walls [14], in contrast with equi-librium, one can now vary the pressure at fixed volume andtemperature by simply changing some microscopic prop- Potential Position e ( θ ) θy x Volume Stiffness
Fig. 4: (Color online) Optimization of a cyclic active engine.(Left) Active elliptical particles, in red, are confined betweentwo parallel soft walls, in blue. The protocol consists in syn-chronous variation of the distance between walls, which de-termines the available volume, and their stiffness. Thus, thecycle extracts work by controlling the system only at bound-aries, without changing any microscopic property of the con-fined particles. (Right) Depending on microscopic details of theparticles, such as their activity and their anisotropy, the cycleshould be operated either clockwise or counter-clockwise. Op-timizing the engine performances then requires considerationof the possible cyclic paths in the space of control parameters,and of how the time-dependent execution of such a path affectsits power and efficiency. Taken from [43]. erty of the boundaries, such as their stiffness. Followingthis line, a protocol has been proposed which, inspiredby the Stirling cycle, replaces the changes of temperaturewith cyclic changes of the stiffness of the walls [43]. Itextracts work by changing volume at fixed stiffness, andchanging stiffness at fixed volume, see fig. 4. Another in-stance of monothermal cycle in nonequilbrium bath is [66].The performances of these engines are quantified bytheir power and efficiency as functions of cycle time. Forcycle times shorter than a characteristic value, the engineoperates too fast to follow properly the instructions of theoperator, and does not yield useful work. As the cycletime increases, the power has a non-monotonic behavior;as for thermal cycles, it vanishes at long cycle times and sois maximal in an intermediate regime. In contrast to ther-mal cycles, the heat expenditure during a cycle containsterms proportional to its duration, due to the dissipationrate induced by active forces. This leads the efficiencyto vanish in the quasistatic regime, see (8). Therefore,since the efficiency is maximal at a finite cycle time, in-creasing this maximum efficiency by improved cycle designnow requires understanding of non-quasistatic protocols.Developing systematic methods to address the optimiza-tion of finite-time cyclic protocols remains an outstandingchallenge, which clearly calls for further investigations.
Perspective: Exploiting collective effects. – Mostof the active engines described above are framed as one-body systems, or (equivalently) systems in which inter-actions among the components are negligible. It remainslargely an open research topic to explore the role of many-body effects in active engines and their consequences forthe emerging thermodynamic framework. This motivatesp-6ctive engines: Thermodynamics moves forwardclose investigation of protocols designed to exploit specificcollective effects in active matter. For instance, previousstudies have investigated how asymmetric obstacles affectthe emergence of flocking in aligning particles [67,68], thuspaving the way to designing autonomous engines with suchparticles as the working substance. As another example,one could study cycles which lead to phase transitionsof the confined active particles, such as a phase separa-tion [12]. It would be interesting to explore how thesetransitions affect the engine efficiency and power output.An alternative way to address how engines might har-ness emergent order, is to consider hydrodynamic descrip-tions of activity instead of particle-based dynamics. Pre-vious investigations of ratchet currents in the frameworkof continuum active fields [24, 69] open the door to study-ing autonomous engines at hydrodynamic level. It wouldbe interesting to explore work extraction using cyclic pro-tocols within, for instance, a scalar field theory capturingactive phase separation [12], or coupled polar and scalarfields describing flocking [11]. A major challenge, in ad-dressing the performances of such engines, is to evaluatethe heat dissipated at hydrodynamic level. To this end,one can build on a recent work which proposes a system-atic derivation of the energetic costs of active field theories,by embedding these within a framework that encodes thechemical energy flows required to create the activity [70]. ∗ ∗ ∗
The authors acknowledge insightful discussions withTimothy Ekeh, Timur Koyuk, Patrick Pietzonka, and UdoSeifert. ´EF acknowledges support from an ATTRACT In-vestigator Grant of the Luxembourg National ResearchFund. MEC is funded by the Royal Society.
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