Active Brownian Heat Engines

Viktor Holubec,1, 2, ∗ Stefano Steffenoni,1, 3 Gianmaria Falasco,1, 4 and Klaus Kroy11Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany

2Charles University, Faculty of Mathematics and Physics, Department ofMacromolecular Physics, V Holešovičkách 2, CZ-180 00 Praha, Czech Republic

3Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany4Complex Systems and Statistical Mechanics, Department of Physics and

Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg(Dated: January 29, 2020)

We investigate active Brownian heat engines consisting of a Brownian particle confined in amodulated harmonic potential and immersed in a non-equilibrium bath of variable activity. Weshow that their average energetics determined by the second moment of the particle position can bemapped onto that of a model with a passive equilibrium bath at a suitably defined time-dependenteffective temperature. Formal limitations for the thermodynamic performance, including maximumefficiency, efficiency at maximum power, and maximum efficiency at fixed power ensue. They helpto clarify the degree to which such active heat engines can outperform passive-bath designs, whichhas been a debated issue for recent experimental realizations. And they can guide the designof new micro-machines. To illustrate the general principles, we analyze a specific realization ofan active heat engine based on the paradigmatic Active Brownian Particle (ABP) model. Explicitanalytical and numerical results are provided and discussed for quasi-static and finite-rate protocols.They reveal some non-intuitive features of the dynamical effective temperature, which complicatethe implementation of classical cycles (like Carnot or Stirling) with active baths, illustrate variousconceptual and practical limitations of the effective-equilibrium mapping, and clarify the operationalrelevance of various coarse-grained measures of dissipation.

PACS numbers: 05.20.-y, 05.70.Ln

I. INTRODUCTION

The study of heat engines is as old as the industrializa-tion of the world. Its practical importance has promptedphysicists and engineers to persistently improve their ex-periments and theories to eventually establish the consis-tent theoretical framework of classical thermodynamics.It allows to quantify, on a phenomenological level, howwork is transformed to heat, and to what extent thisprocess can be reversed. Heat is the most abundant formof energy, namely “disordered” energy dispersed amongmicroscopic degrees of freedom, and turning it into themacroscopically coherent form called work has been acentral aim since the days of Carnot, Stirling, and otherpioneers, after whom some common designs have beennamed.

Recent advances in technology have allowed and alsorequired to extend this success story into two new majordirections. First, towards microscopic designs that areso small that their operation becomes stochastic ratherthan deterministic [1–4]. And secondly to cases wherethe degrees of freedom of the heat bath are themselvesdriven far from equilibrium, which potentially mattersfor small systems operating in a biological context, e.g.,inside living cells or bacterial colonies [5].

The analysis of small systems requires an extension ofthe theory and basic notions of classical thermodynam-

∗ viktor.holubec@mff.cuni.cz

ics to stochastic dynamics, which goes under the name ofstochastic thermodynamics [6–9]. It seeks to define heat,work, and entropy on the level of individual stochastictrajectories. While the resulting average thermodynam-ics recovers the 2nd law, the theory allows to addition-ally quantify the probability of rare large fluctuations [9].Many recent studies have been devoted to realizationsof microscopic thermodynamic cycles. See, for example,Refs. [10–15] for experimental studies and Refs. [16–25]for theoretical ones. In this field, Brownian heat en-gines play a paradigmatic role [13–15, 24, 25]. Theyare based on a colloidal particle representing the workingsubstance. It diffuses in an equilibrium bath but is con-fined by a time-dependent potential, realizable in prac-tice by optical tweezers, in place of a volume-regulatingpiston [13, 14, 26].

The following discussion concerns an idealized minimalmodel of such Brownian heat engines, but combines itwith the mentioned second recent extension of the classi-cal field of heat engines. Namely, over the last few years,increasing effort has been devoted to study the thermody-namics of quantum [27, 28] and classical (colloidal) [5, 29–31] heat engines that operate in contact with a non-equilibrium “active” bath. Paradigmatic realizations ofsuch active baths are, for example, suspensions of self-propelling bacteria or synthetic microswimmers [5, 32].They are driven far from equilibrium on the level of theindividual particles and not merely by externally imposedoverall boundary or body forces. The corresponding “ac-tive heat engines” can outperform classical designs. Themain trick is to evade the zeroth law of thermodynamics

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by operating very far from equilibrium, so that variousdegrees of freedom do not mutually thermalize. Exploit-ing this unconventional property, these engines can oper-ate between hugely different (effective) temperatures andthereby at unconventionally high efficiencies — withoutrisking the evaporation or freezing of the laboratory. Ac-tive heat engines have moreover been claimed to tran-scend the universal performance bounds set by the sec-ond law of thermodynamics [5], a notion that is criticallyexamined below. To elucidate our general results andconclusions by a specific model, we provide a detailedanalysis of a special active Brownian heat engine. Its de-sign is based on the standard minimal model for activeparticle suspensions, namely the so-called ABP (“activeBrownian particle”) model [33], which is why we wantto refer to it by the reminiscent acronym ABE (“activeBrownian engine”), in the following. The possibility toperform explicit analytical and numerical computationsallow the potential merits and limitations of active heatengines to be analyzed and illustrated quantitatively andin considerable detail.

II. SETUP AND MAIN RESULTS

In all of the following, we consider a heat engine con-sisting of a particle confined to a time-dependent har-monic potential

V (x, y, t) =1

2k(t)r2 =

1

2k(t)(x2 + y2), (1)

with an externally controlled stiffness k(t), and immersedin a (possibly) non-equilibrium bath, described by a zero-mean additive noise η(t). We assume that the dynamicsof the particle position r = (x, y)> obeys the overdampedlinear Langevin equation

r = −µk(t)r(t) + η(t). (2)

Depending on the noise correlations, which remain to beprescribed and need not be Markovian, and dependingon the physical interpretation, this model can describevarious experimentally relevant situations. In Fig. 1 wedepict two of them that we discuss further below: namely,an active particle or “microswimmer” immersed in a pas-sive equilibrium bath (left) [34, 35], and a (passive) col-loid immersed in an active non-equilibrium bath that isitself composed of active particles swimming in a thermalbackground solvent (right) [36–40]. Further examples areprovided by devices that share the same formal descrip-tion on a suitably coarse-grained level, such as noisy elec-tric circuits and similar Langevin systems [41].

In line with such realizations, the trapping potential(1) has the harmonic standard form experimentally cre-ated with the help of optical tweezers [5, 13, 14]. Wehave also taken advantage of the fact that such experi-ments are typically designed in a quasi two-dimensionalgeometry, in narrow gaps between two glass coverslips.

For simplicity, the particle mobility is represented by aconstant scalar µ and the two-time correlation matrixCij(t, t

′) ≡ 〈ηi(t)ηj(t′)〉 ∝ δij of the noise η = (ηx, ηy) bya diagonal form. Our analysis can of course straightfor-wardly be generalized to arbitrary dimensions and mo-bility matrices. Also, with the formalism of Ref. [42, 43],one could extend it to linear memory kernels to representunderdamped or otherwise correlated dynamics.

If η in Eq. (2) stands for the white noise, the modelprovides a good description for existing experimental re-alisations of Brownian heat engines [14, 44]. Their ther-modynamics has been thoroughly analyzed in the litera-ture [24, 45, 46]. An example for an experimental real-isation of the non-equilibrium-noise version is the activeBrownian engine with a bacterial bath, studied by Kr-ishnamurthy et al. [5]. The performance of a quasi-staticStirling heat engine based on the latter design was al-ready nicely analyzed by Zakine et al. [29]. Its finite-timeperformance was numerically investigated in Ref. [31].With respect to these studies, which employ specific pro-tocols, our approach is valid for arbitrary driving proto-cols at arbitrary speeds.

As a main result, we show in the following that thethermodynamics of the system described by Eq. (2) witha non-equilibrium noise η, to which we refer as the activemodel, can be mapped onto the well-investigated modelwith a passive equilibrium bath [24, 45, 46], to which werefer as the passive/equilibrium model:

r(t) = −µk(t)r(t) +√

2Deff(t)ξ(t). (3)

Its bath is characterized by the Gaussian white noise ξ(t)with zero mean, 〈ξ(t)〉 = 0, the unit correlation matrixmatrix, 〈ξi(t)ξj(t′)〉 = δijδ(t− t′), and a time-dependent(effective) temperature [47]

Teff(t) =Deff(t)

kBµ=

1

2µ〈r(t) · η(t)〉 . (4)

Below, the latter is shown to follow solely from the two-time correlation matrix C(t, t′) of the noise η = (ηx, ηy).Since the passive model (3) and the corresponding tem-perature (4) describe the active model only effectively,in terms of its average thermodynamic properties, (3)and (4) are referred to as an effective passive/equilibriummodel and an effective temperature, respectively.

The existence of this mapping immediately impliesthat the performance of the active heat engine in termsof its output power and efficiency is precisely that of thecorresponding effective equilibrium model. Therefore,the known bounds on (finite-time) performance of cyclicBrownian heat engines described by Eq. (3), such as theultimate Carnot efficiency bound [48], the efficiency atmaximum power [24], the maximum efficiency at arbi-trary power [45, 49], and the possibility to almost attainthe reversible efficiency at nonzero power [46], directlycarry over to the active heat engine. Furthermore, theeffective equilibrium model also sets bounds on averagethermodynamic variables for non-cyclic and even tran-sient processes. Yet, the non-equilibrium character of the

3

FIG. 1. Schematic designs of microscopic heat engines based on colloids in modulated harmonic traps, playing the roles of theworking substance and the movable piston, respectively. Left: active particle in a “passive” equilibrium bath. Right: passiveparticle in an “active” non-equilibrium bath composed of energy consuming micro-swimmers immersed into a passive backgroundfluid. To operate the heat engine, the bath temperature and/or activity and the confinement strength are modulated cyclically.Thereby “disordered” energy dispersed in the bath and randomly propelling the colloid against its confinement is concentratedin a degree of freedom that can be externally harnessed to perform (mechanical) work.

underlying dynamics reveals itself upon closer inspection,as detailed in the remainder of the paper.

III. OUTLINE OF THE PAPER

The paper is organized as follows. In the next Sec. IV,we define the thermodynamic variables describing theperformance of the heat engine. In Section V, the ef-fective temperature (4) is derived and the equivalence ofthe active and effective passive models, (2) and (3), isproven, which are our main results.

The following Sec. VI describes consequences of themapping to equilibrium. Specifically, Sec. VIA containsa detailed description of the passive model, and Sec. VIBreviews its thermodynamic behavior both for quasi-staticand finite-time cyclic driving. In Sec. VIC, we discuss arecent experimental realization of an active heat enginewith a bacterial bath in the light of our results, and clarifythe status of the reported extraordinary efficiency values.

Readers familiar with the concept of effective tempera-ture and the standard stochastic thermodynamics of col-loidal heat engines may want to skip (parts of) this sec-tion and continue directly with Sec. VII, where we ex-emplify our results by a worked example based on theparadigmatic active Brownian particle (ABP) model, towhich we refer to as to ABE. The following Sec. VIII in-troduces alternative interpretations of the ABP modeland the corresponding dissimilar contributions to theentropy production that denounces the non-equilibriumcharacter of the engine that persists during nominallyreversibly operation. Readers familiar with this modelmay wish to jump to Sec. IXA, where we analyze thequasi-static performance of the ABE and some peculiar-ities of the effective bath temperature. The finite-time

performance of the ABE is discussed in Sec. IXB. Forbetter readability, various details concerning the ABPdynamics, the definitions of the various entropies, andthe positional distribution function have been deferredto an Appendix. We conclude in Sec. X.

For the remainder of the paper and in the figures, weset the Boltzmann constant to unity, kB = 1, whichamounts to measuring energies in Kelvin. The ubiqui-tous time argument t of the thermodynamic variables isoccasionally tacitly omitted.

IV. THERMODYNAMIC VARIABLES

In standard macroscopic thermodynamics, quantitiessuch as work and heat are functionals on system trajec-tories in the thermodynamic state-space. For small sys-tems, described by stochastic differential equations andstochastic trajectories {r(t)}, also work and heat becomestochastic functionals. See for example Refs. [8, 9] forsystems in contact with equilibrium baths and Ref. [50]for a system in contact with an active bath. Here, we areinterested in noise-averaged values of these functionals,which can be constructed from the internal energy of theprocess (2), namely

U(t) = 〈V 〉 =1

2k(t) [σx(t) + σy(t)] =

1

2k(t)σ(t), (5)

where σx =⟨x2⟩, σy =

⟨y2⟩, and σ = 〈r · r〉. For periodic

driving, the average state of the system eventually, aftera transient period, attains a time-periodic steady state.And due to the symmetry of the potential, the averageparticle displacements 〈x〉 and 〈y〉 during the cycle willbe zero, so that the mean square displacements σx(t)and σy(t) also determine the long-time variances for thex- and y-coordinates, respectively.

4

Combining Eq. (5) with the first law of thermodynam-ics, U(t) = W (t) + Q(t), we identify the work done onthe particle during the time interval (ti, tf),

W (ti, tf) =

∫ tf

ti

dt W (t)

=1

2

∫ tf

ti

dt k(t)σ(t) =1

2

∫ k(tf )

k(ti)

dk σ, (6)

as the energy flowing into the system from an externalsource controlling the potential, and the heat,

Q(ti, tf) =

∫ tf

ti

dt Q(t)

=1

2

∫ tf

ti

dt k(t)σ(t) =1

2

∫ σ(tf )

σ(ti)

dσ k, (7)

as the energy flowing into the system from the reservoir.The performance of an engine is most commonly quan-

tified in terms of output power P and efficiency η. In thepresent case, these variables read

P ≡ Wout

tp, η ≡ Wout

Qin. (8)

Here, Wout = −W (0, tp) is the work done by the engineper cycle of duration tp. We employ the unit step func-tion Θ(x) to express the heat flowing from the bath intothe system as

Qin =

∫ tp

0

dt Q(t)Θ[Q(t)] (9)

While the definition of the output power bares no con-troversy, it must be stressed that our definition of theefficiency neglects the energy required to maintain thenon-equilibrium state of the bath, which is sometimesreferred to as the housekeeping heat.

However, we argue that this is the correct way to assesthe efficiency of engines in contact with non-equilibriumreservoirs. In analogy with standard heat engines, wetreat energy exchanged with degrees of freedom underprecise experimental control as work and energy ex-changed with not explicitly resolved degrees of freedomas heat. For the situation depicted in Fig. 1, the degreeof freedom under control is the particle position and theremaining degrees of freedom (particle orientation, stateof the surrounding fluid, etc.) represent the bath. Impor-tantly, this means that the boundary between work andheat depends on the skills of the experimentalist. Takeas an example the active particle depicted in Fig. 1a).For an experimentalist controlling just the stiffness ofthe potential, the distinction between heat and work isas described above. In contrast, if the active particlecould signal its future path, this additional informationcould be utilized to exploit its self-propulsion to performuseful work (without need for the potential). In such a

case, however, the definition of efficiency would have tobe modified to acknowledge the house-keeping power re-quired for the self-propulsion. It would thereby turn intoa measure of efficiency for the work-to-work conversionin the noisy environment.

V. EFFECTIVE TEMPERATURE

A. General formulation

It is noteworthy that all of the above thermodynamicquantities are determined solely by the variance σ, whichobeys the ordinary differential equation

σ(t) = −2µk(t)σ(t) + 2 〈r(t) · η(t)〉 . (10)

The latter follows from Eq. (2) by taking the scalar prod-ucts of r and both sides and averaging over the noise. Forarbitrary additive noise η, Eq. (2) has the formal solution

r(t) = r0e−K(t,t0) +

∫ t

t0

dt′ η(t′)e−K(t,t′), (11)

with K(t, t′) ≡ µ∫ tt′dt′′ k(t′′) and r0 ≡ r(0) denoting an

arbitrary initial position of the particle. Together withthe two-time noise correlation matrix, C(t, t′), the aver-age in Eq. (10) evaluates to

〈r(t) · η(t)〉 = 2Deff(t) ≡ 〈r0 · η(t)〉 e−K(t,t0)+

2

∫ t

t0

dt′ Tr[C(t, t′)]e−K(t,t′) , (12)

where Tr denotes the trace operation. A crucial ob-servation is that it therefore assumes a form thatwould also result from the Gaussian white noise η =√

2Deff(t)ξ(t) with the correlation matrix Cij(t, t′) =

2√Deff(t)Deff(t′)δijδ(t − t′). This implies that the av-

erage thermodynamic behavior of the active model (2)with arbitrary additive noise is the same as that of thepassive model (3) with an effective equilibrium bath tem-perature

Teff(t) =Deff(t)

µ=

1

2µ〈r(t) · η(t)〉 =

kσ

2+

σ

4µ. (13)

The last expression follows from Eq. (10). It shows thatalso the effective temperature is uniquely given by thevariance σ. Notably, the result (13) is valid arbitrarilyfar from equilibrium and it does not follow from any close-to-equilibrium linear-response approximation like in theGreen-Kubo formula [51].

Also note that for positive effective temperatureTeff(t) ≥ 0, Eq. (13) establishes the announced map-ping between the active and passive heat engine and thusproves our main result. Negative effective temperaturescan however be obtained, for example, during transientsdeparting from initial conditions with 〈r0 · η(t)〉 < 0. At

5

late times, the sign of the effective temperature is deter-mined by the integral in Eq. (12), which is positive forstandard correlation matrices C(t, t′) with non-negativediagonal elements. For a quasi-static process, where thesystem parameters vary slowly compared to the intrin-sic relaxation times, one can neglect σ(t) relative to theother terms in Eq. (13). The effective temperature thenreduces to the well-known form [5]

Teff(t) = k(t)σ(t)/2. (14)

For slowly driven systems, the effective temperature isthus always positive, thanks to the positivity of the trapstiffness k and variance σ.

B. Cyclic heat engines

The general definition (13) of the effective temperatureapplies both under transient and stationary conditions.Cyclic heat engines operate time-periodically by virtue oftheir periodic driving. Accordingly, we assume that thepotential stiffness k(t) is a periodic function with periodtp and that the noise correlation matrix is of the form

Cij(t, t′) = 2δijI(t)I(t′)fi(t− t′), (15)

where I(t) stands for a tp-periodic intensity of the noise,and fi(t) are arbitrary functions obeying fi(0) = 1 anddecaying towards zero as t → ∞. The system dynamicsthen settles onto a time-periodic attractor, independentof the initial condition r0, at late times. From now on,we assume that the engine operates in this “steady state”regime, to which we refer as the limit cycle. During thecycle, the effective temperature Teff(t) takes the form [seeEqs. (12) and (13)]

1

µI(t)

∫ t

−∞dt′ I(t′)[fx(t− t′) + fy(t− t′)]e−K(t,t′). (16)

Importantly, for positive diagonal elements of the correla-tion matrix, the effective temperature is then manifestlypositive, as required to map the active onto the passivemodel.

C. (Im)possible generalizations

The simplifying power of the present approach cru-cially relies on two main features. Firstly, on the linearityof Eq. (2), and secondly on the fact that thermodynamicsis predominantly concerned with average energetics.

For the active heat engines discussed in the presentcontribution, the pertinent microscopic degree of free-dom is the position of the colloid. Its thermodynamicsis contained in the variance σ = 〈r · r〉, which controlsthe complete average energetics (work and heat) of theengine through Eqs. (6) and (7). However, the describedmapping to a passive-bath model cannot be extended be-yond such average energetics, since the active (2) and

passive (19) models differ in variables which depend onhigher moments of the position r or its complete dis-tribution. This is for example the case for the totalentropy or the fluctuations of work, heat and entropy.Without further ado, one thus cannot take for grantedthe results obtained under the assumption of a perfectcontact with an equilibrium bath, such as the Jarzynskiequality [52], the Crooks fluctuation theorem [53], theHatano-Sasa equality [54], and various inequalities con-taining higher moments of work, heat, and entropy, suchas thermodynamic uncertainty relations [55–58].

Also note that, for a true equilibrium noise η, the (ef-fective) temperature Teff in Eq. (4) would agree with allother possible definitions of temperature, thereby tyingtogether many a priori unrelated dynamical quantities(e.g. by their structurally identical Boltzmann distribu-tions or fluctuation-dissipation theorems, etc.). However,for a non-equilibrium noise, differently defined tempera-tures can (and generally will) have different values. Werefer to Refs. [59–62] for various (complementary) ap-proaches to effective temperatures and Refs. [63–65] forsome reviews. Moreover, as illustrated by the ABP re-sults (35)–(36) in App. D, typical non-equilibrium dis-tributions deviate strongly from Boltzmann’s Gaussianequilibrium distribution, such as the one characterizingthe long-time limit of the equilibrium process, Eq. (3), atconstant Teff — namely ρ(r) ∝ exp[−kr2/2Teff ]. There-fore, in order to build an effective thermodynamic de-scription from a non-equilibrium statistical-mechanicsmodel, one generally has to calculate precisely the effec-tive temperatures corresponding to the relevant degreesof freedom, under the prescribed conditions.

This leads to the mentioned second limitation of thepresented effective-temperature mapping, namely that ithinges on the linearity of the model. To make the point,let us consider a one-dimensional setting with the poten-tial U(x, t) = k(t)xn/n when the Langevin equation forposition x reads

x(t) = −k(t)[x(t)]n−1 + η(t) (17)

and the internal energy, work, and heat (per unit time)are given by U(t) = k(t) 〈[x(t)]n〉, W (t) = k(t) 〈[x(t)]n〉,and, Q(t) = k(t)d 〈[x(t)]n〉 /dt, respectively. In order todescribe the average thermodynamics, we thus have toconsider the dynamics of the nth moment 〈[x(t)]n〉. Mul-tiplying Eq. (17) by xn−1 and averaging the result overthe noise, we find that

d

dt〈[x(t)]n〉 = −nk(t)

⟨[x(t)]2n−2

⟩+ n

⟨[x(t)]n−1η(t)

⟩.

(18)Thus, in order to get an exact closed dynamical equa-tion for 〈[x(t)]n〉, we also need a dynamical equationfor⟨[x(t)]2n−2

⟩which, in turn, depends on the moment⟨

[x(t)]3n−4⟩, and so on. However, out of equilibrium each

degree of freedom (and, also each moment 〈[x(t)]n〉) has,in general, its own effective temperature, if such a set ofeffective temperatures can consistently be defined at all.

6

Recall that the development of a useful (finite-time)thermodynamic description based on a time-dependenteffective temperature requires a system with equilibriumnoise that yields the same (time-resolved) dynamics ofthe relevant moments, for which our above discussionof the variance of the linear model (2) provides theparadigm. This means that we would have to developa passive model with an equilibrium noise that gives riseto precisely the same dynamics of all the moments inEq. (18) as the original nonlinear active model. This canget very difficult if not impossible if the moments rep-resent independent effective degrees of freedom so thattheir effective temperatures differ [66].

Despite these limitations, there are also many impor-tant properties that can successfully be captured by theeffective-temperature mapping. In the next section, wereview its general consequences for the performance of ac-tive heat engines. Readers familiar with stochastic ther-modynamics may wish to continue directly with Sec. VII,where we derive and discuss more specific analytical re-sults based on the so-called active Brownian particle(ABP) model with an exponential correlation matrix.

VI. EFFICIENCY

A. Effective entropy production

As described above, the dynamics of the variance in theactive model (2) can be mimicked exactly by the effectivepassive model

r(t) = −µk(t)r(t) +√

2Deff(t)ξ(t) (19)

with an equilibrium bath at the time-dependent temper-ature Teff(t), as long as the latter does not transientlyturn negative. The noise intensity Deff(t) and the mobil-ity 1/µ in Eq. (19) are thus related by the fluctuation-dissipation relation Deff = µTeff . As discussed in Sec. IV,the variance determines the average thermodynamics ofthe active engine in terms of work, heat, and efficiency. Inparticular, due to our interpretation of thermodynamicvariables above, the (average) performance of the activeheat engine is the same as that of a passive heat enginebased on Eq. (19) and can thus be taken over from theknown thermodynamics of the passive model [8, 24, 25].In fact, such a consistent thermodynamic framework iscrucial for a meaningful concept of efficiency, far fromequilibrium.

For pedagogical reasons and for completeness, webriefly review the thermodynamics of the effective passivemodel, here. The main result, to be derived below, is theexplicit formulation of the second law of thermodynam-ics. It states that the active engine has a non-negativetotal effective entropy-production rate

Sefftot(t) = µTeff(t)σ(t)

[2

σ(t)− k(t)

Teff(t)

]2

≥ 0. (20)

Thermodynamically, the entropy production can al-ways be decomposed into the contributions

Sefftot(t) = Seff(t) + Seff

R (t) (21)

due to the working substance itself and due to the entropychange in the (effective) heat bath, respectively. Since,by definition, the heat flow from/into an equilibrium heatbath is reversible, the entropy change of the bath obeysthe Clausius equality,

SeffR (t) = − Q(t)

Teff(t)= −k(t)σ(t)

2Teff(t), (22)

where the last step exploits the definition (7) of the mu-tual heat-exchange rate Q(t). For the system entropy,one merely has the weaker Clausius inequality

Seff(t) ≥ −dSeffR (t) = Q(t)/Teff(t) . (23)

It can be turned into an equality if a quasi-static driv-ing protocol is employed, which then also optimizes thethermodynamic efficiency of the active heat engine.

We now show how these results follow from thestatistical-mechanics description. First and foremost,note that the linearity of Eq. (19) ensures that thestochastic process r(t) is a linear functional of the Gaus-sian white noise ξ(t). The probability density for theparticle position r = (x, y) at time t is therefore alsoGaussian, namely

peff(x, y, t) =1

πσ(t)exp

[− (x2 + y2)

σ(t)

], (24)

and can easily be seen to solve the Fokker-Planck equa-tion

∂peff

∂t= ∇r · [µ∇rV (r) +Deff∇r] peff (25)

with ∇r = (∂x, ∂y). The corresponding Gibbs-Shannonentropy

Seff(t) = −∫ ∞−∞

dx

∫ ∞−∞

dy peff log peff

= log σ(t) + log π + 1 (26)

is thus solely determined by the variance σ(t) of thePDF (24), and therefore changes with the rate

Seff(t) =σ(t)

σ(t). (27)

The second law in the form given in Eq. (20) now followsby inserting Eqs. (22) and (27) into Eq. (21) and usingEq. (10) for the time derivative of the variance in theform σ = 4µTeffσ(1/σ − k/2Teff), after rearranging theresulting terms.

To make the entropy production vanish, which corre-sponds to the equal sign in Eqs. (20) and (23), one has to

7

drive the engine quasi-statically. This amounts to settingσ = 0 in Eq. (13), which yields

σ(t) = σ∞(t) ≡ 2Teff(t)/k(t) . (28)

For a quasi-static driving, the rates of change (22) and(27) of the reservoir and system entropies also both van-ish, since they are proportional to the vanishing timederivative σ = 0. However, this feature alone might notbe enough for concluding that the entire entropy

∆Sefftot(tp) =

∫ tp

0

dt′Sefftot(t

′) (29)

throughout the whole cycle vanishes as tp → ∞, sinceit is sensitive to how large tp has to be chosen to en-sure quasi-static conditions, which depends on the in-trinsic relaxation behavior of the working substance (inour case the trapped colloid) [67]. It is a consequence ofthe fluctuation dissipation relation fulfilled by the effec-tive equilibrium model that the rates of change (22) and(27) of the reservoir and system entropies converge toeach other fast enough that the whole quasi-static cycleis reversible, i.e. that (29) vanishes. We come back tothis issue in Sec. VII, where we analyze an explicit modelrealization.

B. Efficiency bounds

For an arbitrary cycle, the Clausius inequality (23) can,via standard manipulations [68], be rewritten in terms ofthe quasi-static (qs) bounds for the output work Wout

and efficiency η, respectively,

Wout ≤W qsout, (30)

η ≤ ηqs ≤ ηC = 1− min(Teff)

max(Teff). (31)

According to the discussion in the previous section, theseconditions identically constrain the active heat engine.Given any driving protocol for the variation of the con-trol parameters k(t/tp) and Teff(t/tp), etc., along the cy-cle, the largest output work per cycle and the largestefficiency are thus attained for quasi-static driving withtp → ∞. The ultimate (Carnot) efficiency limit ηC forthe active engine is thus reached in a quasi-static Carnotcycle composed of two “isothermal” branches, with con-stant Teff , interconnected by two “adiabatic branches”,with constant entropy (26) and variance σ∞.

Similarly, the mapping to the passive model (19) im-plies that the finite-time performance of the active heatengine is the same as that of its effective passive replace-ment. For convenience, we summarize some consequencesof this observation, here. The quasi-static conditions,needed to reach the upper bound ηC on efficiency exactly,imply infinitely slow driving and thus vanishing outputpower. Naturally, such powerless heat engines are un-interesting for practical purposes [45], where only finite-time processes are relevant, and, thus, other measures of

engine performance have been proposed. A prominentrole among them plays the maximum power condition.Schmiedl and Seifert [24] showed that overdamped Brow-nian heat engines deliver maximum power if they operatein the so called low-dissipation regime [69]. Their analy-sis implies that the efficiency at maximum power of theactive heat engine is given by

ηMP = 1−

√min(Teff)

max(Teff). (32)

This result applies if the engine is driven along a finite-time Carnot cycle composed of two isotherms of constantTeff and two infinitely fast adiabatic state changes at con-stant σ, with a suitable protocol for the trap stiffness k(t)that minimizes the work dissipated during the isothermalbranches. We also note that the maximum-power condi-tion was investigated for a specific class of active colloidalheat engines in Ref. [30].

Actual technical realizations of heat engines are usu-ally designed for a certain desired power output. Thus,even more useful than the knowledge of the efficiencyat maximum power is the knowledge of maximum effi-ciency at a given power. Like the former, the latter is,for a Brownian heat engine of fixed design, attained whenoperating in the low-dissipation regime along a finite-time Carnot cycle [49, 70, 71]. The exact numerical andapproximate analytical value of the maximum efficiencyat arbitrary power for our setting can be obtained us-ing the approach of Ref. [49]. Another universal result,applicable to the active Brownian heat engine, is that,for powers P close to the maximum power P ?, the ef-ficiency increases infinitely fast with decreasing P (i.e.|dη/dP |P→P? | → ∞) [49, 72]. Therefore, it is usuallyadvantageous to operate heat engines close to maximumpower conditions [small δP = (P ?−P )/P ?], rather thenexactly at these conditions (δP = 0) [70]. Moreover,the results of Refs. [49, 70, 71] show that ηC can be at-tained only in the limit δP → 1, where either the powerP completely vanishes, or it is negligible with respect tothe maximum power P ?. Recently, this insight led toa proposition of protocols yielding very large maximumpower, thus allowing Brownian heat engines to operateclose to (and practically with) Carnot’s efficiency at largeoutput power [45, 46]. As discussed in the following para-graph, active Brownian heat engines offer an alternativeroute for achieving this.

C. Experiments

A relevant experimental realization of a system de-scribed by our model (2) is the bacterial heat engine ofRef. [5]. In this impressive experimental study, a col-loidal particle trapped in a harmonic potential with aperiodically modulated stiffness k(t) was immersed in abath of self-propelled bacteria with periodically modu-lated activity. The authors controlled the system param-eters quasi-statically, along a Stirling cycle composed of

8

two isochoric state changes at constant trap stiffness kand two “isothermal” state changes at constant solventtemperature T and constant activity of the bacteria (inour notation corresponding to a constant strength Teff ofthe noise η). Driven by the periodic parameter modula-tion, the dynamics converged to a limit cycle, cyclicallytransforming energy absorbed from the disordered bac-terial bath into a colloidal work. The authors measuredthe work done per cycle as well as the energy (heat) ob-tained per cycle from the bath and determined the effi-ciency of the machine as their ratio. They also measuredthe (quasi-static) effective temperature (14), which couldbe as high as 2000 K, suggesting that the engine couldpractically be operated at maximum efficiency.

We note, however, that the authors described theirmost surprising result as follows. “At high activities, theefficiency of our engines surpasses the equilibrium satu-ration limit of Stirling efficiency, the maximum efficiencyof a Stirling engine where the ratio of cold to hot reser-voir temperatures is vanishingly small.” This is indeedextraordinary, because it contradicts the limitations im-posed by the effective passive model and the second law ofthermodynamics, as described above. A possible expla-nation could be that the actual cycle does not correspondto the definition of a Stirling cycle [29]. For instance, afailure to control the pertinent effective temperature Teffduring the “isothermal” state changes could readily causeimportant deviations from the Stirling efficiency. How-ever, Fig. 2a) of Ref. [5] strongly suggests that the effec-tive temperature was actually carefully controlled duringthe state changes performed at constant bath tempera-ture and bacterial activity, so that the actual effectivecycle should indeed qualify as a proper Stirling cycle inthe sense of our mapping to an effective passive bath.

This leaves us with an apparent paradox. The key toits resolution seems to lie in the definition of the effi-ciency employed in Ref. [5]. The authors measured thestochastic work output and heat input for individual cy-cles. The resulting mean values of work and heat werethen obtained by averaging over several cycles, and theefficiency was formally expressed as their ratio. How-ever, as can be inferred from Fig. 2 of Ref. [5], it wasnot actually evaluated according to this definition, butinstead as the average of the work-over-heat ratio of in-dividual cycles. The latter is not a well-behaved thermo-dynamic quantity, though [17, 73], since the distributionof the work-over-heat ratio per cycle is heavy-tailed andits moments do not exist. It therefore does not convergewith increasing averaging time and can take more or lessarbitrary values.

In summary, active heat engines can experimentally beoperated very close to the optimum efficiency, which isuniversally delineated by the second law. The reason isthat they allow extreme changes of the pertinent effec-tive temperature to be realized with ordinary equipment,thanks to the breakdown of the zeroth law of thermody-namics for the non-equilibrium heat bath. Along the pro-cedures outlined above, the appropriate consistent notion

of effective temperature for the active bath also followsfrom the second law.

In the following sections and in App. D, we explicitlyanalyze a specific realization of an active heat engine toillustrate the merits and limitations of the mapping tothe passive model (19).

VII. WORKED EXAMPLE: THE ABE MODEL

A. Model definition

To exemplify the above findings for a specific model, wenow consider the so-called ABP model. It is the standardminimal model for a particle embedded into an equilib-rium bath at temperature T but actively propelling withvelocity v = vn(θ) in the direction determined by thediffusing unit vector n(θ) at angle θ(t). Encouraged byexperimental evidence [5, 36, 37] and in accord with the-oretical studies based on a rigorous elimination of (fast)active degrees of freedom [74, 75], the ABP model withharmonic confinement (Fig. 1a) has recently also beenused to model passive Brownian colloids embedded in anactive bath (Fig. 1b) [29, 31, 38–40]. Indeed, within thegeneral formalism outlined above, the ABP model pro-vides us with a simple realization of Eq. (2) in terms ofa trapped colloid driven by the non-equilibrium noise

η =√

2D(t)ξ + v(t) . (33)

Here the components of ξ = (ξx, ξy) are mutually inde-pendent zero-mean unit-variance Gaussian white noises,but the velocity term v = v(t)n(θ) prohibits a straight-forward equilibrium interpretation. It contributes an ex-ponential term to the total noise correlation matrix

Cij(t, t′) = 〈ηi(t)ηj(t′)〉 = δij

[2√D(t)D(t′)δ(t− t′)

+1

2v(t)v(t′) exp

{−∫ max(t,t′)

min(t,t′)

dt′′Dr (t′′)

}]. (34)

Such exponential memory has indeed also been found ina weak-coupling model for a passive tracer in an activebath [74, 76]. Besides, it is often employed as a tractablemodel for the complex correlations arising in strongly in-teracting systems. For the following, we assume that thetranslational diffusion coefficient D(t) obeys the Einsteinrelation D(t) = µT (t), but do not constrain the rota-tional diffusion coefficient Dr(t) in the same way. Thelatter describes the free diffusion of the particle orienta-tion n = v/v on a unit circle and is incorporated intothe ABP equations of motion [34, 77, 78] through yetanother independent zero-mean unit-variance Gaussianwhite noise ξθ:

r(t) = −µkr(t) + v(t) +√

2D(t)ξ(t) (35)

θ(t) =√

2Dr(t)ξθ(t) . (36)

9

That the ABP model provides a proper non-equilibriumactive noise, as desired for Eq. (2), is not only apparentfrom the two-time correlation matrix (34), which fixesthe average thermodynamics of the model in a way thatis not consistent with a fluctuation-dissipation relation.It is further manifest in higher order correlation functions[79] that are sensitive to the non-Gaussian character ofthe noise (33). As illustrated in App. D, this for exampleallows for a bimodal distribution of the coordinates x andy, so that the ABP model captures some of the generi-cally non-Gaussian character of non-equilibrium fluctu-ations, lost in another widely employed active-particlemodel that represents the active velocity as an Ornstein-Uhlenbeck process [35]. We note that these propertiesare essentially caused by the variable rotational noise ξθand persist in a constant-speed (v = const. 6= 0) versionof the model.

To emphasize the paradigmatic character of the heatengine described by the ABP-Eqs. (35)–(36) with per-diodically driven parameters k(t), T (t), v(t), Dr(t), werefer to it as the ABE model. It involves three ingredi-ents that can potentially drive it far from equilibrium: (i)If the stiffness k(t) changes on time-scales shorter thanthe intrinsic relaxation time, the particle dynamics is notfast enough to follow the protocol adiabatically. (ii) Ifthe rotational diffusion coefficient Dr is not constrainedby the Einstein relation, the rotational degree of free-dom can be considered connected to a second bath ata temperature distinct from T . In general, connectinga system to several reservoirs at different temperaturesdrives it out of equilibrium. (iii) Finally, the velocityterm in the Langevin system is formally identical to anon-conservative force giving rise to persistent currentsthat prevent equilibration.

B. Cyclic driving protocol

Our driving protocol involves a periodically modulatedstiffness k, reservoir temperature T , rotational diffusioncoefficient Dr, and active velocity v. We let the sys-tem evolve towards the limit cycle, where we analyze itsperformance. While the following theoretical discussionapplies to arbitrary periodic driving, we exemplify ourresults with a specific Stirling-type protocol that mimicsthe experimental setup of Ref. [5] (see Fig. 2). It consistsof four steps of equal duration (tp/4):

(i) “Isothermal” compression A→ B: the stiffness k in-creases linearly from k< to k> at constant noise strengthcorresponding to the temperature T = T< and activity{Dr, v} = {D<

r , v<}.(ii) “Isochoric” heating B → C: the noise strength

{T,Dr, v} increases linearly from {T<, D<r , v<} to

{T>, D>r , v>} at constant stiffness k = k>.

(iii) “Isothermal” expansion C → D: the stiffness de-creases linearly from k> to k< at constant noise strength{T>, D>

r , v>}.(iv) “Isochoric” cooling D → A: the noise strength

0 0.2 0.4 0.6 0.8 1

5

5.2

5.4

5.6

0 0.5 1

1

1.5

2

0 0.5 1

0.05

0.055

0 0.5 1

0

2

4

0 0.2 0.4 0.6 0.8 1

1

2

3

4

FIG. 2. The driving protocol of the ABE (a-d) and the effec-tive temperature (e) that maps it to a passive model: a) trapstiffness, b) bath temperature, c) rotational diffusion coeffi-cient, and d) active velocity, all as functions of time during thelimit cycle. The full blue line in panel e) depicts the effectivetemperature Teff(t) of Eq. (44), the dashed line its limit (45)for a quasi-static (infinitely slow) driving. Parameters used:tp = 1, k< = 5, k> = 5.5, T> = 2, T< = 1, D>

r = 0.055,D<

r = 0.05, v> = 4, v< = 0 and µ = 1.

decreases back to its initial value at constant stiffnessk = k<.

Note that the “isothermal” state changes are character-ized by constant bath temperature and activity, which ingeneral corresponds to a varying effective temperature[see Fig. 2e)]. As explained in Sec. IV, the engine con-sumes (performs) work from A → B (C → D) while heatis absorbed (emitted) from (to) the reservoir from B →C (D → A).

C. Variance dynamics on the limit cycle

During the limit cycle, which is attained at late times,the dynamics of the variance σ(t) = 2σx(t) = 2σy(t) [dueto the symmetry of Eq. (35)] is for arbitrary time-periodicdriving governed by the two coupled ordinary differential

10

0 0.5 1 1.5 2 2.5 3

0

0.4

0.8

1.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

0.5

1

FIG. 3. Positional variance σ(t) = 〈r · r〉 over time for the pro-tocol shown in Fig. 2. Increasing activity v> = 0, 2, 4 yieldsan increasing variance σ. a) BD simulations of the relaxationto the limit cycle. b) c) d) The dynamics on the limit cycle inBD simulations (solid green), numerical solutions [80] (dot-dashed black), and from the analytical formula (40) (dottedred), shows perfect agreement, despite considerable distancefrom the quasi-static limit (43) (broken blue lines).

equations

H(t) = −[µk(t) +Dr(t)]H(t) + v(t), (37)σ(t) = −2µk(t)σ(t) + 4D(t) + 2v(t)H(t) . (38)

Here, the term 2D(t) + v(t)H(t) determines the long-time time-periodic behavior of the average 〈r(t) · η(t)〉.See App. A for details and for the derivation of the time-periodic solution

H = H0e−K(t,0)−F (t,0) +

∫ t

0

v (s) e−K(t,s)−F (t,s),(39)

σ = σ0e−2K(t,0) + 4

∫ t

0

dt′Deff(t′)e−2K(t,t′) (40)

with functions K(t, t0) = µ∫ tt0dt′k(t′), F (t, t0) =∫ t

t0dt′Dr(t

′), and Deff(t) = D(t) + v(t)H(t)/2. The con-

stants

H0 =

∫ tp0dt′ v (t′) e−K(tp,t′)−F(tp,t′)

1− e−K(tp,0)−F (tp,0), (41)

σ0 = 4

∫ tp0dt′Deff(t′)e−2K(tp,t

′)

1− e−2K(tp,0)(42)

secure the time-periodicity of the solution. Quasi-staticconditions correspond to slow driving relative to the re-laxation times τH = 1/(µk+Dr) and τσ = 1/(2µk) for Hand σ, respectively. It allows the dynamics of the func-tions H and σ to be regarded as relaxed, H = σ = 0,from which one gets the quasi-static variance

σ(t) = σ∞(t) ≡ 2

k

(T +

v2

2µ

1

kµ+Dr

). (43)

The leading correction in the driving speed is derived inApp. B. Conversely, if the driving is fast relative to therelaxation times τH and τσ, the colloid cannot respond tothe changing parameters k, T , v and Dr, and its varianceis given by Eq. (43) with time-averaged parameter values.

At intermediate rates, the complete expression (40) hasto be used. To make sure that we calculate the nested in-tegral correctly, we cross-check the obtained results withtwo independent methods, BD simulations and numericalsolutions [80]. The finite-time variances follow the quasi-static ones like carrot-chasing donkeys, i.e., the variancedecreases (increases) if it is larger (smaller) than the sta-tionary value σ∞ corresponding to the given value of thecontrol parameters, cf. Figs. 3 (b)-(d). The discrepancybetween the quasi-static and the finite-time predictionsincreases for faster driving and moreover grows with theactivity ratio v>/v<. As intuitively expected, and sug-gested by the role of v in Eq. (38), larger active velocitieslead to larger variances.

D. Effective temperature

Comparing Eqs. (10), (13), and (38) we find for theeffective temperature of the ABE on the limit cycle

Teff(t) =Deff(t)

µ= T (t) +

v(t)H(t)

2µ. (44)

Its value is always larger than the bath temperature T .Apart from the latter, it also depends on the activityv, mobility µ, trap stiffness k, and rotational diffusioncoefficient Dr. All the parameters, except for T , enterTeff indirectly, and in a complex way, through the differ-ential equation (37) for H. The effective temperaturethereupon acquires the characteristic relaxation time,τH = (µk + Dr)

−1. Its quasi-static limiting form (14)explicitly reads

Teff(t)→ T∞eff (t) ≡ kσ∞2

= T +v2

2µ

1

kµ+Dr. (45)

11

The effective temperature possesses several counter-intuitive features. First, in case of periodically modu-lated activity or trap stiffness, it varies in time, even if thebath temperature is held constant. Moreover, due to itsdynamical nature and finite relaxation time, it generallydoes so even when both bath parameters T and v are heldconstant. Hence, to realize a proper (effectively) isother-mal process with constant Teff , one has to carefully tunethe control parameters. This is most easily achieved un-der quasi-static conditions, as demonstrated in Fig. 2e).There we plot the effective temperature (44) (full blueline) and also its quasi-static limit (45), which would beobtained at very slow driving (black dotted line). For thechosen parameters, the quasi-static effective temperature(45) runs approximately along a Stirling cycle, in accordwith the temperature T (t) and activity v(t) [Figs. 2 (b)– (d)]. Conversely, the finite-time effective temperature(44) exhibits substantially different behavior.

Before going into more details, we now outlinethree thermodynamically consistent interpretations ofthe ABE model and derive the corresponding entropyproductions. In the discussion of quasi-static and finite-time performance of the engine in Secs. IXA and IXB,respectively, we utilize these entropy productions as ex-amples of variables that are not captured by the effective-temperature mapping (19). Another example is the fulldistribution of the particle position, which we discuss inAppendix D.

VIII. ENTROPY PRODUCTION

As a genuinely non-equilibrium system, any active heatengine always produces entropy, even if operated in-finitely slowly. However, how much of that entropy wecan (or care to) track depends on our experimental reso-lution (and interpretation of the engine).

A. User perspective

On the coarsest level of description, which might beadopted by a user of the heat engine, only the suppliedheat and the harvested output work matter. Their ratiois the natural measure of efficiency, which is bounded bythe optimum (Carnot) efficiency determined by the effec-tive temperature Teff. As we have discussed, this temper-ature can experimentally be measured for the model of atrapped Brownian particle, namely by a device sensibleto the variance σ of the particle position; see Fig. 4 a).The thermodynamics of the active heat engine is therebymapped to that of an ordinary engine with an equilib-rium bath and obeys the same limitations. Accordingly,the user would conclude that the total dissipated cycleentropy

∆Sefftot =

∫ tp

0

dtSefftot =

∫ tp

0

dtSeffR (46)

is given by the net entropy change per cycle in the bath,which thus solely controls the degree of irreversibility ofthe cycle. To compute the latter, the user would resortto the expression given in Eq. (22) of Sec. VIA, namely

SeffR = −Q/Teff ≡ Qeff

dis/Teff . (47)

Since the particle dynamics is modelled within an over-damped Stokes approximation, the corresponding “effec-tive” dissipation Qeff

dis to the effective equilibrium bath isstraightforwardly given by the force acting on the particletimes its velocity (averaged)

Qeffdis = −〈∇rV · r〉 = −k(t)σ(t)/2. (48)

Importantly, the user is not concerned with other de-tails of the non-equilibrium bath than the variance σand the effective equilibrium temperature Teff it provides.He would thus adopt the above expressions for arbitrarynoise in Eq. (2), regardless of the underlying physics ofthe bath. For the specific ABE realisation of the activeheat engine, these expressions can explicitly be evaluatedusing Eqs. (38), (40) and (44).

B. Trajectory perspective

In contrast to the user, a heat engineer would possi-bly consider the engine at a higher resolution and haveaccess to the individual stochastic trajectories of the par-ticle position generated by Eq. (2). Thereby, she coulduncover the non-equilibrium character of active heat baththat dissipates energy even if the engine operates underquasi-static conditions. To this end, she could evaluatethe dissipation per cycle in the form 〈logPF(Γ)/PR(Γ?)〉,exploiting a relation often referred to as local detailedbalance condition. It relates the symmetry breaking be-tween the path probabilities PF(Γ) and PR(Γ?) for pathsΓ and their time-reversed images Γ? to dissipation. (Formore details, see App. C and Refs. [81, 82].) The methodcan in principle be applied regardless of the physics un-derlying the noise term in Eq. (2), if one can observe orotherwise guess the time-reversed dynamics. See Ref. [82]for an example of a successful application of such a strat-egy to biological systems. In general, this will howevertechnically require assumptions or knowledge of the time-reversed noise dynamics, i.e., microscopic information be-yond that of the stochastic (forward) trajectories of theparticle position. Such information is seldom availableoutside the realm of detailed models of the mesoscopicphysics. For specificity, we therefore now consider theabove ABE based on the ABP model.

C. ABP perspective: sailboats versus surfboards

For ABP’s, the noise η comprises the (time-symmetric)equilibrium white noise

√2Dξ together with the active

12

Teff

a)

F = v/µT

b)

vT

c)

FIG. 4. Different levels of control over the system imply differ-ent changes in the bath entropy. a) Mere “users” of an activeheat engine are only concerned with its thermodynamic in-put/output characteristics. They judge reversibility and en-tropy changes with respect to an effective equilibrium bath[red particles] at a fictitious temperature Teff larger than thetemperature T of the background solvent [yellow particles inb) and c)]. A more detailed knowledge of the engine’s inter-nal working substance (the ABP particle) and its dynamicsallows one to uncover the non-equilibrium character of thesystem, which depends on the time-reversal properties of itsdynamics. If the active velocity v in the ABE results fromdragging or pushing the particle through the liquid by an ex-ternal force F [panel b)], the particle behaves like a sailboatand the change in bath entropy obeys Eq. (52). If the parti-cle is advected by the surrounding liquid with velocity v likea surfboard [panel c)], the entropy change in the bath obeysEq. (53).

propulsion v. The colloid could be a randomly (self-) pro-pelled active particle or a schematically modelled passivetracer in an active bath [5, 29]. In any case, its activevelocity v is due to a dissipative process and admits twoalternative interpretations, depending on its presumedtime-reversal properties. Namely, it can be understoodas a Stokes velocity caused by an external (random) forcev(t)/µ, the so-called swim force. This very common in-terpretation, depicted in Fig. 4) b), treats the particlelike a sailboat blown around by erratic winds, which iswhy we refer to it as the “sailboat” interpretation. Or, ina second interpretation, depicted in Fig. 4) c), the activeterm v(t) can be interpreted as the actual swim velocityof a microswimmer that either “sneaks” through the qui-escent background solvent by an effective phoretic surfaceslip v(t) [83–85] or is passively advected by a local flowfield v(t) [86]. We refer to it as the “surfboard” inter-pretation. It treats v(t) as a proper dynamic velocity asopposed to the disguised force in the sailboat interpre-tation. Upon time-reversal, forces usually do not change

the sign, while velocities do. The detailed balance condi-tion then implies that the rate of entropy change in thebath reads

S±R = Q±dis/T, (49)

for sailboats (+) and surfboats (−), respectively [87–89].The dissipation rates are

Q+dis = 〈(v/µ−∇rV ) · r〉 = Qeff

dis+v2/µ−〈∇rV ·v〉, (50)

Q−dis = 〈−∇rV · (r− v)〉 = Qeffdis + 〈∇rV · v〉. (51)

We refer to App. C for details of the formal derivation,and discuss these results on a physical basis. In the dissi-pation rate Q+

dis for sailboats, the swim term is added asan additional force (intuitively the wind drag) to the po-tential force. In contrast, for surfboards, it is subtractedfrom the particle velocity corresponding to a reformula-tion of the equation of motion in a frame that is freelyco-moving with the flow velocity v(t).

Since Q+dis(t) and Q

−dis(t) have different reference points

(vanishing for sailboats blown against the quay and surf-boards floating freely with the surf, respectively), thetwo dissipation rates can not generally be ordered ac-cording to their magnitude for the ABE, where both sit-uations may (approximately) be encountered along thecycle. Also note that the detailed balance condition im-poses that the heat is dissipated in the background sol-vent at temperature T (t), which is natural from the pointof view of the ABP model. As a consequence, also dif-ferent amounts of entropy production will be assigned tothe self-propulsion, dependent on the chosen ABP inter-pretation.

They can both be understood as composed of the ef-fective dissipation Qeff

dis(t) over the solvent temperatureT (t) ≤ Teff(t), plus some extra (manifestly active) en-tropy production due to the particle’s excursions off thesurf or off the quay, respectively,

T S+R = Qeff

dis + v2/µ− 2µk(Teff − T ) , (52)

T S−R = Qeffdis + 2µk(Teff − T ) . (53)

In the second case (surfboards), the additional propul-sion contribution to the entropy production beyond Seff

Ris manifestly positive, since Teff ≥ T . Intuitively, this isbecause any failure to float with the flow gives rise todissipation. In the first case (sailboats), the minimumcondition for Seff

R can only be guaranteed under quasi-static conditions. Intuitively, the “wind” may otherwisetransiently prevent dissipation by “arresting the sailboatat the quay”.

While the derivation of the expressions (52) and (53)relies on a deeper knowledge of the system dynamics thanthe behavior of the variance, it is worth noting that σ(t)is still sufficient for their evaluation. The dynamics ofthe variance thus suffices to evaluate the “total” entropy

13

∆S±tot(tp) = S±R (tp) =∫ tp

0dt S±R produced per cycle of

the operation of the ABE. In contrast, the change in thesystem entropy

S(t) = −∫ ∞−∞

dx

∫ ∞−∞

dy

∫ 2π

0

dθ p log p , (54)

which vanishes for a complete cycle but is necessaryfor evaluating the total entropy change within the cy-cle, ∆S±tot(t) = S±R (t) + S(t)− S(0), depends on the fullprobability distribution p(x, y, θ, t) for the position of theparticle at time t. The latter obeys the Fokker-Planckequation

∂p

∂t=

(∇r · [µ∇rV (r)− v] +D∇2

r +Dr∂2

∂θ2

)p (55)

corresponding to Eqs. (35) and (36). One can calculatethe PDF p(x, y, θ, t) either numerically, from Eq. (55), orusing BD simulations of Eqs. (35)–(36) (see App. D fora detailed discussion of the results). The system entropyS(t) is thus the only variable of our thermodynamic anal-ysis which generally cannot be calculated using the meansquare displacement σ alone.

The above results are suitable to fully quantify theengine’s thermodynamic performance. In the followingsection we evaluate the general expressions and discusstheir generic properties.

IX. ABE-PERFORMANCE

In this section, we first focus on the quasi-static regimeof operation of the ABE, where we demonstrate in moredetail some peculiarities connected with the unintuitivebehavior of the effective temperature. For vanishing en-tropy productions ∆S±tot, as defined in the previous sec-tion, the non-equilibrium ABE bath is seen to admit arepresentation as an equilibrium bath. Then, we considerfinite-time effects onto the performance of the ABE, andthe additional entropy production due to the non-quasi-static operation.

A. Quasi-static regime

In the quasi-static regime, the engine dynamics interms of the variance σ(t) and the effective temperatureTeff(t) are given by Eqs. (43) and (45), respectively. Theythus depend merely parametrically on the driving k(t),T (t), Dr(t), and v(t). The effective entropy production∆Seff

tot (20) then vanishes, and the (effective) efficiencyof the ABE is given by the classical result evaluated interms of the stiffness k(t) and temperature Teff(t). Inparticular, a quasi-static cycle consisting of two brancheswith constant Teff and two adiabats will thus operatewith Carnot efficiency ηC, (31). Equivalently, realizing aStirling cycle in terms of k(t) and the effective tempera-ture Teff(t) will result in the (effective) Stirling efficiency

15 20 25 30 35 40 45 50

5

5.1

5.2

5.3

5.4

5.5

0.2 0.3 0.4 0.5 0.6 0.7

5

5.1

5.2

5.3

5.4

5.5

FIG. 5. Two quasi-static (generalized) Stirling cycles in termsof the trap stiffness k(t) and variance σ(t) of the particle posi-tions ABCD and ABBCDD. (The corresponding energy flowsare evaluated in Fig. 6.) a) In the standard Stirling cycleABCD, the heat flows from the bath into the system along theisochor BC and isotherm CD (Q = kσ/2 > 0), and from thesystem into the bath otherwise (Q < 0). b) In the “nonstan-dard Stirling” cycle ABBCDD, the heat flow reverses (outflowalong BB, inflow along BC) along the isochoric branch BC =BBC and similarly for the isochor DA = DDA. The outputworks Wout = 0.5

∫ tp0dkσ of the individual cycles are given

by the areas they enclose. Similarly, heat input and outputcan be visualized as areas below the curves.

ηC log a/(ηC + log a) with a = min(k)/max(k) [29]. Andone could deal similarly with other thermodynamic cyclicprotocols. However, using the simplifying analogy withthe effective equilibrium bath, one should make sure toactually use k(t) and Teff(t) as control parameters andnot simply rely on an intuition about the behavior of theeffective temperature based on the background solventtemperature T , activity v and rotational diffusivity Dr.Indeed, as mentioned in Sec. VIID, what is a Stirling(or Carnot) cycle in terms of the effective temperaturecan be quite different from the one defined in terms ofT , v, and Dr. To quantify the difference, it is useful tointroduce the parameter

K(t) ≡ kµ/Dr (56)

which compares the characteristic timescales D−1r and

(kµ)−1 for relaxation of the orientation θ and the posi-

14

-1.5

-0.5

0.5

1.6

0 20 40 60 80 100

100

-50

0

50

100

FIG. 6. Energy flows for the quasi-static Stirling cycles de-picted in Fig. 5. Net work and heat W , Q = |Qin| − |Qout|,heat influx and outflux Qin, Qout and internal energy change∆U = U(t)−U(0), as defined in Sec. IV are traced out as func-tions of time during a quasi-static cycle of duration tp = 100significantly larger than the relaxation times τH = 1/(µk+Dr)and τσ = 1/(2µk) for Teff and σ, respectively. Panel a)v> = 4. Panel b) v> = 500, v< = 50, D>

r = 500 and D<r = 5;

other parameters as in Fig. 2.

tion r, respectively. The quasi-static effective tempera-ture (45) can be written as

Teff(t) = T +v2

2µDr

1

1 +K. (57)

Only in the limiting cases K → 0 and K → ∞, a naivequasi-static isothermal process (constant temperature T ,activity v, and Dr) corresponds to an effective equilib-rium isothermal process (constant Teff).

Despite the equilibrium analogy, the bath actually cor-responds to a driven system with its apparent equilib-rium characteristics actively maintained by some dissi-pative processes. So even for quasi-static operation ofthe engine, closer inspection reveals this non-equilibriumnature of the bath. In particular, the sailboat/surfboardinterpretations of the particle motion will reveal some ofthis entropy production, so that ∆S±tot ≥ ∆Seff

tot.For strong confinements, K � 1, the active dynamics

is highly persistent on the confinement scale, so that the

particle moves quasi-ballistically in the potential. Theeffective temperature Teff is therefore given by the tem-perature of the equilibrium solvent T , which is the onlyremaining source of noise. Using the sailboat interpreta-tion of the ABP (where v is interpreted as an externalforce), we find that ∆S+

tot = ∆Sefftot = 0. The sailboat

interpretation is thus consistent with the notion that theABE operates reversibly, in this limit. In contrast, thesurfboard (for which v is interpreted as a velocity) dissi-pates, since its free motion clashes with the confinementby the potential, so that ∆S−tot = ∆Seff

tot+∫ tp

0dt v2(t)/µ =∫ tp

0dt v2(t)/µ > 0.

For weak confinements, K � 1, the particle’s activemotion randomizes on the confinement scale so that it canbe subsumed into the δ-correlated noise (33) via the effec-tive temperature and the corresponding noise correlationmatrix Cij(t, t′) = 2

√Deff(t)Deff(t′)δijδ(t − t′). Its dy-

namics mimics Brownian motion in an effective equilib-rium bath maintained at the (stiffness-independent) tem-perature Teff = T +v2/(2µDr). In this case, confinementand random active motion interfere in such a way thatboth the sailboat and surfboard interpretations can de-tect the positive entropy production, ∆S±tot > ∆Seff

tot = 0,and the actual irreversibility of the operation. Only byimposing the additional limit v2 � 2µDr, when the ro-tational motion completely obliterates the active swim-ming, surfboards cease to be bothered by the confinementand no longer dissipate, so that ∆S−tot = 0. In the sail-boat interpretation, this corrsponds to a complete wasteof the efforts of the external swim force, so that it stillreflects the irreversibility via ∆S+

tot =∫ tp

0dt v2(t)/µ > 0.

For intermediate values of K, the effective temperaturedepends on the stiffness k(t) and the (traditional) defini-tion of heat input along an individual step of the drivingprotocol may not actually yield the correct interpreta-tion. It then also fails to yield a consistent measure ofefficiency. Instead, one should carefully reconsider whatis the actual heat input, based on Eq. (9), which canbe rewritten as Qin = 1

2

∫ tp0dt′kσΘ[σ]. Heat thus flows

into the system whenever the variance σ — and thus theeffective system entropy (26) — increases, and vice versa.

To illustrate this point, recall the definition of the Stir-ling cycle in Sec. VIIB. The standard Stirling cycle con-sists of two isochores (constant trap stiffness k) and twoisotherms (constant solvent temperature T ). Thereforeit forms a rectangle in a k-T diagram, translating to ashape similar to the ABCD cycle in Fig. 5, in a k-T/kdiagram. Actually, Fig. 5 is slightly more general, asit shows two possible interpretations of the quasi-staticABE-Stirling cycle in a k-Teff/k diagram. The “standard”protocol ABCD corresponds to the evolution of the ther-modynamic variables as depicted in Fig. 6a). Note thatthey, in turn, evolve strictly monotonically or remain con-stant during the individual steps of duration tp/4. Hence,during a single step, heat is either only absorbed or onlyreleased by the system, and it is possible to write the in-put heat as Qin = QBC+QCD, where QXY is the amount

15

0

1

2

3

0

1

2

3

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

FIG. 7. Evolution of the various entropies discussed in thetext as functions of time during the limit cycle, depicted inFig. 2, with v> = 4. Panel a) “total” ABE entropy changes∆S+

tot for “sailboats”, (red dashed line), ∆S−tot(t) for “surf-

boards” (yellow dot-dashed line), both from Sec. VIII C, andthe effective entropy change ∆Seff

tot(t) from Eq. (29) (solid blueline). Panel b) shows corresponding changes in the reservoirentropy ∆S+

R (t) from Eq. (52) (red dashed line), ∆S−R (t) from

Eq. (53) (yellow dot-dashed line), and ∆SeffR (t) from integrat-

ing Eq. (22) (solid blue line), and panel c) in the system en-tropy ∆Seff(t) from Eq. (58) (solid blue line) and ∆S(t) fromEq. (59) (red dashed line).

of heat absorbed between the points X and Y . Whichcorresponds to the conventional practice for a Stirlingcycle.

Consider next the cycle ABBCDD corresponding toFig. 6b). In this case, the system releases heat duringthe segment BB (σ decreases from σB to σB), but ab-sorbs heat during the remainder of the state change BC(σ increases from σB to σC). A similar situation occursalso at the end of the cycle. Hence, the conventionalshorthand notion of heat input as heat exchanged be-tween the system and the reservoir during an entire stepof the cycle is not appropriate, in this case. Instead, onehas to use the definition (9), also utilized in Fig. 6. Thedashed red, dashed yellow and full blue lines in Fig. 6b)in the time interval from t = 25 to t = 50, also serve toillustrate the differences in the heat balance. For a fur-ther treatment of efficiency of Stirling engines operatingin contact with active baths in the quasi-static regime,we refer to Ref. [29].

0.05

0.06

0.07

0

5

10

0 10 20 30 40 50 60

0

200

400

600

FIG. 8. Efficiency a), power output b), both from Eq. (8),and total entropy production c), as functions of the maximumactive velocity v> for the protocol from Fig. 2. Panel c) ∆S−

tot

(yellow dashed), ∆S+tot (red dot-dashed), and ∆Seff

tot (solidblue), all from Eq. (60).

B. Finite-time performance

Let us finally investigate the most complex case of non-quasi-static cycles for which the protocol from Sec. VIIBis imposed with cycle durations tp significantly shorterthan the internal relaxation times τH = 1/(µk + Dr)and τσ = 1/(2µk) for Teff and σ, respectively. The ABEmodel provides full control over the finite-time thermody-namics. To check our analytical results for the variancegiven in Sec. VIIC, we compared it to direct numericalsolutions of the equations of motion via the matrix nu-merical method of Ref. [80], and found perfect agreement.We also note that the new features observed in the ana-lytical results for the toy model are generic, and shouldqualitatively also be observed for other heat engines incontact with non-equilibrium reservoirs.

The hallmark of non-quasi-static operation of any ther-modynamic heat engine is the observation of a net en-tropy increase during the cycle. Therefore, Fig. 7 depictsthe individual entropy changes defined in Secs. VIA andVIII as functions of time during the limit cycle. Panel

16

a) shows that both the total effective entropy change∆Seff

tot(t), measured by the ABE user, and the total ABEentropy changes ∆S±tot(t), corresponding to the sailboatand surfboard interpretations, are non-decreasing func-tions of time. They thus meet the expectation for validtotal entropies according to the second law of thermody-namics. It is noteworthy, that the ABE entropy changes∆S±tot(t) are larger than the effective entropy change∆Seff

tot(t), at all times, even during the first part of thecycle, given by t ∈ (0, 0.25), where the active velocity vvanishes.

As gleaned from the panel b), the rates of entropychange in the bath, with Seff

R given by Eq. (22) and S±R (t)given by Eqs. (52) and (53), are in that case all equal.The inequality ∆Seff

tot(t) < ∆S±tot(t) is then solely causedby the different changes of the system entropy

∆Seff(t) = Seff(t)− Seff(0) = logσ(t)

σ(0), (58)

∆S(t) = S(t)− S(0), (59)

shown in the panel c), with Seff(t) and S(t) given byEqs. (26) and (54), respectively. For the remaining time[t ∈ (0.25, 1)] of the cycle, even the changes in the bathentropies ∆S±R (t) of the ABE are larger than ∆Seff

R (t).While S−R (t) ≥ Seff

R (t) and S±tot ≥ Sefftot always hold, we

find that S+R (t) < Seff

R (t) is not ruled out (detailed datanow shown). The figure also corroborates the periodic-ity of the system entropies Seff(t) and S(t), so that thetotal entropy changes ∆Seff

tot(tp) and ∆S±tot(tp) per cycleare solely determined by the (per cycle) entropy changes∆Seff

R (tp) and ∆S±R (tp) in the bath, as it should be.To study the influence of activity on the ABE perfor-

mance, in Fig. 8, we fix tp ≡ 1 for various maximumactive velocities v> according to Fig. 2. For small valuesof v> the efficiency is decreased by the activity, while forlarge values of v> it is increased, and eventually attainsa constant maximum value. This behavior can be under-stood as follows. The efficiency of the heat engine quitegenerally increases with the largest difference in the ef-fective temperature max (Teff)−min (Teff), similarly as inthe Carnot formula. Even beyond the quasi-static regimeone expects that the effective temperature is qualitativelydescribed by Eq. (45). For small values of v>, Eq. (45)implies that the temperature difference can be decreasedby variations of the rotational diffusion coefficient, de-picted in Fig. 2c), while it increases with v> for largev>. More intuitive behavior is observed for the powerFig. 8b) and the entropy productions ∆Seff

tot and ∆S±totFig. 8c) that monotonically increase with v>.

Finally, we assess the effect of the finite-time driv-ing on the ABE operation. Specifically, in Fig. 9, wedepict performance of the ABE as function of the cy-cle duration tp for three values of the maximum activevelocity v>. In panel a), the efficiency monotonouslyincreases with increasing tp and eventually reaches thequasi-static limit (the red line). Notably, whether theefficiency is increased or decreased by the bath activ-ity depends on the cycle duration, as evidenced by the

0

0.05

0.1

-0.02

0.02

0.06

0

0.05

0.1

10-3

10-1

-2

0

210

-4

10-2

100

102

0.002

0.1

10

600

FIG. 9. Efficiency a), output power b), and output work c),defined in Sec. IV, and total entropy production d) for v> = 0(dot-dashed lines), v> = 2 (dashed lines) and v> = 4 (fulllines) as functions of cycle duration tp. The inset in panelc) magnifies the initial part of the plot for tp ∈ [10−3, 10−1].Panel d) ∆S−

tot (yellow), ∆S+tot (red), and ∆Seff

tot (blue); allaccording to Eq. (60). Other parameters as in Fig. 2.

dashed and dot-dashed lines wandering above and be-low the solid line. Namely, apart from enhancing theoutput work and power [panels b) and c)], the activityalso provides an increased heat flow into the system. Asexpected, the output power vanishes for large cycle du-rations and exhibits a maximum for a certain value of tp.On the contrary, the output work is, for large cycle times,an increasing function which converges to the quasi-staticvalue, which monotonously increases with v>. Interest-ingly, for 10−2 . tp . 10−1, the output work exhibits ashallow negative excursion as revealed by the blowup inthe inset. This implies a lower bound tp ≈ 10−1 on thecycle duration, below which the system ceases to operateas a heat engine.

As can be observed in Fig. 9d), for small and largecycle durations, the cycle-time dependence of the to-tal entropy productions ∆Seff

tot(tp) and ∆S±tot(tp) exhibitsasymptotic power-law behavior. Taylor expansions of thetotal entropy productions in tp and 1/tp, respectively,give ∆S±tot ∝ ∆Seff

tot ∝ tp for short tp and ∆S±tot ∝ tp

17

for v 6= 0, and ∆Sefftot ∝ 1/tp regardless of v, for long tp.

To be more specific, all the total entropy productions inquestion assume the form

∆Sxtot = −

∫ tp

0

dt1

T (t)

(Q+ F

)(t), (60)

where T = Teff and F = 0 for x = eff, T = T and F =−2µk(Teff−T ) for x = −, and T = T and F = 2µk(Teff−T ) − v2/µ for x = +. For fast driving of the engine, (tpmuch smaller than the intrinsic relaxation times), thecolloid cannot react to the changing driving and settles ona time-independent state corresponding to a mean valueof the driving. Hence, Eq. (60) can be approximated forall x by ∆Sx

tot ≈ −tp(Q+ F

)/T , where the integrand is

evaluated using the time-independent state attained fortp → 0.

For slow driving (tp much smaller than the intrinsicrelaxation times), the colloid attains its steady state (43)independently of the cycle duration tp. Substituting theintegration time t in Eq. (60) by the dimensionless timeτ = t/tp yields

∆Sxtot = −tp

∫ 1

0

dτ1

T (τ)

(1

tp

dQ(τ)

dτ+ F(τ)

), (61)

where T (τ) = T (τtp), Q(τ) = Q(τtp), and F(τ) =F(τtp). The effective total entropy production ∆Seff

tot

vanishes in the limit tp →∞, and thus the leading contri-bution in Eq. (61) is expected to be of order 1/tp. Indeed,expanding under the integral, we obtain (for F = 0)

1

TdQ

dτ≈ d

dτlog σ∞ +

1

tpC. (62)

Since the first term represents a total derivative, the cor-responding loop integral vanishes and what remains isthe correction C/tp. For x = ±, the leading contributionto the integral (61) is simply determined by the non-zerovalue of limtp→∞ F and thus we find ∆S±tot ∝ tp for largetp. For v = 0, all three definitions of entropy productionare equivalent since then T = T = Teff and F = 0. Thisproves the scalings found in Fig. 9d).

X. CONCLUSION AND OUTLOOK

We have derived a simple strategy to map the averagethermodynamics of a linear Langevin system with arbi-trary additive noise to an effective equilibrium model.The mapping is based on the matching of the dynamicalequations for the second moment of position, which hap-pens to determine the (average) energetics. It is validfor arbitrary protocols imposed by the time-dependentmodel parameters. In the quasi-static limit, the (gen-erally time-dependent) effective temperature Teff(t) (4)that accomplishes the mapping recovers the known ex-pression (14). A benefit of this mapping is that conven-tional bounds on the (both finite-time and quasi-static)

thermodynamic performance of machines, especially heatengines, carry over to those with non-equilibrium (ac-tive) baths [5, 29–31]. As a part of our discussion, wehave therefore been able to provide a new perspective onrecent claims of surprisingly high Stirling efficiencies (es-sentially corresponding to infinite temperature steps) ina bacterial heat engine that was experimentally realizedby Krishnamurthy et al. [5].

We have exemplified these somewhat abstract notionsfor a specific model of a Brownian particle confined ina harmonic potential with periodically varying parame-ters. We call it the ABE, since the particle dynamics isbased on the well-known active Brownian particle (ABP)model. Our qualitative conclusions should carry over toother designs, though. In particular, we find that theexplicitly computed effective temperature Teff has somenon-intuitive features. (i) During the limit cycle, whichis attained by the ABE at long times, it obeys a first-order differential equation and thus acquires some timedependence Teff(t) with a technically relevant character-istic relaxation time. (ii) It is important to realize thatit can therefore vary in time even during those parts ofthe cycle in which the model parameters are held con-stant. (iii) Even in the quasi-static limit, Teff depends onthe strength of the potential. This means that realizingspecific thermodynamic conditions, like an “isothermal”process with respect to the effective temperature, is gen-erally not trivial.

The ABE model is also instructive with respect to somelimitations of the effective-bath mapping. Namely, byconstruction, the latter is blind to the potentially richfeatures of the non-equilibrium bath beyond the secondmoment of the particle position, which we identified asthe working degree of freedom of the engine. The effec-tive description thus misses the non-Gaussian shape ofthe positional probability density and the correspondingShannon entropy, for example, and also all housekeep-ing heat fluxes required to maintain the bath activity.Accordingly, we could demonstrate that the entropy pro-duction in the effective model can be understood as alower bound for all conceivable practical and theoreticalrealizations. Namely, it vanishes upon quasi-static oper-ation, whereas any detailed model of the bath dynamicswould, like the explicitly studied ABE, necessarily revealsome of the housekeeping heat fluxes and their associatedentropy production.

As an outlook, we would like to note that our analy-sis suggests some straightforward generalizations to arbi-trary mobility and correlation matrices, thus also includ-ing under-damped dynamics. Furthermore, it should alsobe possible to generalize it to arbitrary linear memorykernels. Another possible extension could be the appli-cation of the presented method to non-linear systems,e.g., by deriving approximate time-dependent effectivetemperatures via suitable closures of the equations de-scribing the relevant degrees of freedom.

18

ACKNOWLEDGMENTS

We acknowledge funding by Deutsche Forschungsge-meinschaft (DFG) via SPP 1726/1 and KR 3381/6-1, and

by Czech Science Foundation (project No. 20-02955J).VH gratefully acknowledges support by the Humboldtfoundation. S.S. acknowledges funding by InternationalMax Planck Research Schools (IMPRS).

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Appendix A: Analytical solution for variance

Inserting the time correlation matrix (34) for the ABPmodel into Eq. (12), Eq. (10) yields the following dy-namic equation for the variance σ =

⟨x2⟩

+⟨y2⟩:

σ + 2µkσ = 4 〈x0ηx(t) + y0ηy(t)〉 e−K(t,t0) + 4D(t)

+ 2v(t)

∫ t

t0

dt′v(t′)e−K(t,t′)−F (t,t′), (A1)

where

K(t, t0) = µ

∫ t

t0

dt′k(t′), (A2)

F (t, t0) =

∫ t

t0

dt′Dr(t′). (A3)

In order to explicitly evaluate the thermodynamics ofthe particular realization of an active Brownian heat en-gine described in Sec. VII, namely the ABP-based enginethat we refer to as the ABE model, we need the solutionof Eq. (A1). More precisely, we can concentrate onto thetime periodic solution, which is attained by the system atlate times, after transients have relaxed, so that it settlesonto a limit cycle (c.f. Fig. 3). Taking the limit t0 → −∞in the formal solution to Eq. (A1), we obtain

σ(t) = 2 limt0→−∞

∫ t

t0

dt′[2D(t′) + v(t′)H(t′)]e−2K(t,t′)

(A4)

20

with

H(t) = limt0→−∞

∫ t

t0

dt′v(t′)e−K(t,t′)−F (t,t′). (A5)

For the numerical evaluation of Eq. (A4) it is useful toexploit that H(t) is a tp-periodic function and to rewriteK(t, t0) asK(t, t0) = b(t−t0)/tpcK(tp, 0)+K(t, t0+b(t−t0)/tpctp) using the tp-periodicity of k(t) (the symbol bxcdenotes the floor operation) and similarly for F (t, t0). In-terestingly, using a simple trick, the time-periodic late-time limit can be found without considering the (numer-ically inconvenient) limit t0 → −∞, just as in the caseof memoryless dynamics [24, 25]. The key insight is thatnot only the function σ, but also H fulfills a certain dif-ferential equation, which can be obtained by taking thetime derivative of Eq. (A5). The resulting formulas andthe time-periodic solutions for σ and H are given in themain text in Sec. VIIC.

Appendix B: Slow driving limit of variance

For slowly varying driving functions k(t), D(t), Dr(t)and v(t), the variance (A4) can be approximated usinga simple formula which follows from the Laplace typeapproximation of the integral [90, 91]

∫ t

t0

dt f(t′)e∫ tt′ dt

′′ g(t′′) =

∫ t

t0

dt f(t′)etp

∫ t/tpt′/tp

dt′′ g(tpt′′)

=

f(t)

g(t)− 1

g2(t)

[v(t)− v(t)

g(t)

g(t)

]+ o(f , g). (B1)

Applying this approximation first on the function H(t)(A5) and then on the variance σ(t) (A4), we obtain theapproximate result

σ(t) = σ∞ −v2

kµκ2

(v

v− κ

κ

)− D

k2µ2

(D

D− k

k

)

− v2

2k2µ2κ

(2v

v− κ

κ− k

k

)+ o(v, D, k, κ). (B2)

Here, σ∞ is the variance (43) for infinitely slow drivingand κ = κ(t) = kµ + Dr. For discontinuous driving,the limiting solution σ∞ is also discontinuous. The firstorder correction (B2) may also be discontinuous if thefirst derivatives of the driving functions exhibit jumps. Insuch a case, however, the assumption on the smallness ofthe derivatives used in the calculation leading to Eq. (B2)is not valid. In accord with the discussion below Eq. (D1)in Appendix. D, Eq. (B2) reveals that activity-correctionsare at least second order in v.

Appendix C: Entropy production from pathprobabilities

The entropy

∆SR,Γ(t) = log(PF/PR) (C1)

delivered to the bath by a particle moving along a tra-jectory Γ(t) = {r(t′), θ(t′)}tt′=0 of the stochastic process(35), (36) is given by the logarithm of the ratio of con-ditional probabilities PF and PR [81, 92], for the trajec-tory conditioned with respect to its initial point and itstime-reversed image. Up to normalization, the forwardprobability is given by

PF ∝ e−2∫ t0dt′ [ξ·ξ+ξ2θ], (C2)

where the noise terms ξ = [r + µ∇rV − v]/√

2D andξθ = θ/

√2Dr follow from Eqs. (35) and (36) [93]. The

backward probability is given by a similar formula. Onejust has to change the sign before quantities which areodd with respect to time reversal.

Assuming the active velocity v = v(cos θ, sin θ) to betime-reversal even, the odd variables in Eqs. (35) and(36) are time derivatives, giving

(PF/PR)+

= e−∫ t0dt′ (∇rV−v/µ)·r/T , (C3)

whereas, for time-reversal odd v, we find

(PF/PR)−

= e−∫ t0dt′∇rV ·(r−v)/T . (C4)

The entropy delivered to the reservoir during time inter-val (0, t) follows as

∆SR(t) = 〈∆SR,Γ(t)〉Γ = 〈log(PF/PR)〉Γ , (C5)

where the average is taken over the individual realizationsΓ of the stochastic process [92]. With Eq. (C3) for thetime-even active velocity, it yields

∆S+R (t) =

∫ t

0

dt′1

T

⟨(vµ−∇rV

)· r⟩, (C6)

and with Eq. (C4), for the time-odd active velocity,

∆S−R (t) =

∫ t

0

dt′1

T〈(r− v) · (−∇rV )〉 . (C7)

Appendix D: Probability distributions (PDFs)

In the 3-dimensional Langevin system (35)–(36), thex − y coordinates are coupled via the active velocity v.The steady probability distribution (PDF) to find theparticle with orientation θ at position (x, y) thus cannotgenerally be written in the separated form p(x, y, θ) =χ(x, θ)ι(y, θ) = χ(x, θ)χ(y, π/2− θ), where χ(x, θ) solvesthe 2-dimensional Fokker–Planck equation

∂tχ =[D∂2

x +Dr∂2θ + ∂x (µ∂xV − v cos θ)

]χ . (D1)

21

FIG. 10. Probability distribution χ for particle position x andorientation θ at the end of the hot isotherm (t = 3tp/4, seeFig. 2). We take v> = 30 and tp = 104. Other parametersare the same as in Fig. 2.

Inserting the separation ansatz into the 3-dimensionalequation (55) and using the formula (D1) leads to thecondition 2Dr∂θχ(x, θ)∂θι(y, θ) = 0 that cannot be ful-filled in general. Nevertheless, one can still reduce the3-dimensional system to just two degrees of freedom byintroducing the polar coordinates x = r cosφ, y = sinφ.Then, Eq. (35) transforms to

r = −kr + v cos(θ − φ) +√

2Dηr , (D2)

φ =v

rsin(θ − φ) +

√2Dr

r2ηφ , (D3)

while θ still obeys Eq. (36). The symbols ηr and ηφdenote independent, zero-mean, Gaussian white noises.Since Eqs. (D2) and (D3) only depend on the differenceθ − φ, introducing the relative angle ψ = θ − φ, subjectto the zero-mean, Gaussian white noise ηψ renders themin the form

r = −kr + v cosψ +√

2Dηr , (D4)

ψ = −vr

sinψ +

√2

(Dr

r2+D

)ηψ . (D5)

The corresponding Fokker–Planck equation for the PDFρ = ρ(r, ψ, t) reads [94]

∂tρ =

[D∂2

r +

(Dr

r+D

)∂2ψ

]ρ− cosψ∂r(vρ)

−D∂r(ρr

)+ k∂r(rρ) +

v

r∂ψ(sinψρ) . (D6)

In general, the equations (D1) and (D6) [or, equivalently(55)] can not be solved analytically and thus we solvedthem using the numerical method described in Ref. [80].We compared the numerical solution of Eq. (D6) to theseparated ansatz p(x, y, θ) = χ(x, θ)χ(y, π/2 − θ) andfound out that, although not exact, the ansatz describes

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.5

1

-3 -2 -1 0 1 2 3

0

0.5

1

-6 -4 -2 0 2 4 6

0

0.2

0.4

0.6

FIG. 11. Marginal distribution ρ for the particle position xat the end of the individual branches of the cycle for differentvalues of the maximum active velocity a) v> = 0, b) v> = 1, c)v> = 10, and d) v> = 30. We have set tp = 1, correspondingto non-stationary driving, other parameters as in Fig. 2. Notethat the curves at t = 0 and t = 1 are equal, in accord withthe time periodic operation.

the full 3-dimensional PDF p(x, y, θ) sufficiently well.Since the 2-dimensional PDF allows for a more intuitivediscussion and exhibits the main qualitative features ofp(x, y, θ), we restrict the following discussion to χ(x, θ).

Figure 10 shows a snapshot of the PDF χ(x, θ, t), so-lution of (D1), at the end of the third branch of a quasi-static cycle introduced in Sec. VIIB (the hot “isotherm”).The figure reveals the typical shape of the PDF χ, withtwo global maxima located at θ = 0 and π, which sur-vives even for rapid driving protocols. Physically, theshape of the PDF can be understood as follows: 1) forany fixed orientation angle θ, the PDF can be expectedto exhibit a maximum at the position where the activevelocity (which acts in the Langevin Eq. (35) for x as aforce v cos θ/µ) is balanced by the force kx exerted bythe parabolic potential; 2) the projection v cos θ/µ of von the x−coordinate changes slowest around its extrema(0 and π), and thus most trajectories contribute to thesurroundings of these points, making the extrema for 0and π largest.

Figure 11 shows snapshots of the marginal PDFρ(x, t) =

∫dθχ(x, θ, t) for the position x at the beginning

of the individual branches of the cycle, for four values of

22

the maximum active velocity v>. With increasing v>, theresulting PDFs become increasingly non-Gaussian and fi-nally even exhibit two separated peaks. Physically, thisbehavior can be understood by the wall accumulation ef-fect due to the persistence of the active motion [95–97],which creates the double peak during the cycle brancheswith large v>. (For similar PDFs, see Ref. [79, 98].) Qual-itatively similar results are also obtained in the quasi-static limit, as already apparent from Fig. 10.

To get some intuition about these results on analyticalgrounds, we now present several approximate solutions toEq. (D1). Different from the standard diffusion (v = 0) inan external potential, the quasi-static (∂tχ = 0) solutionof the Fokker-Planck equation (D1) is not given by theBoltzmann PDF. This is because one cannot subsumethe activity into a generalized potential V which wouldact as a Lyapunov functional for the dynamics of x andθ. Nevertheless, there are several limiting cases wherethe Boltzmann form χ ∝ exp(−V /T ) is still a usefulapproximation.

The best analytical insight into the described qualita-tive properties of the presented numerical solutions toEq. (D1) with time-dependent parameters is obtainedfor rotational diffusion coefficient Dr much smaller thankµ, corresponding to the limit of large K in Eq. (56).Then, the direction of the active velocity can be treatedas quenched, so that the activity can be subsumed into ageneralized potential V = kx2/2 − vx cos θ/µ. The cor-responding quasi-static solution of Eq. (D1) then reads

χ =1

Zχexp

(vx cos θ

µT− kx2

2T

), (D7)

with a normalization constant Zχ. For each fixed valueof the angle θ, the PDF is then Gaussian with its max-imum value exp[v2 cos2 θ/(2Tµ2k)]/Zχ at the positionv cos θ/(µk). The PDF thus posses two global maximalocated at (x, θ) = [v/(µk), 0] and (x, θ) = [−v/(µk), π],and is qualitatively similar to the PDF shown in Fig. 10.

The marginal PDF for x obtained from (D7) then reads

ρ(x, t) =

∫dθχ =

1

Zρexp

(−kx

2

2T

)I0

(vx

µT

). (D8)

Here, I0(x) denotes the modified Bessel function of thefirst kind and Zρ is another normalization constant. Themarginal PDF is Gaussian for v = 0, and becomes moreand more non-Gaussian with increasing v/(µk). For largevalues of v/(µk), it can even become bimodal. This be-havior can be traced back to the shift of the maxima ofthe PDF χ with increasing v/(µk). For small v/(µk),the two maxima substantially overlap and the integra-tion over the angle θ yields a single peak which is nearlyGaussian. For large values of v/(µk), the two peaks donot overlap any more and the marginal PDF thus alsoexhibits two peaks. The behavior of the marginal PDFobtained in the limit Dr � µk thus shows qualitativelythe same behavior as the solution of Eq. (D1) shown inFig. 11.

ForDr much larger than kµ, corresponding to the limitof small K in Eq. (56), the quasi-static PDF is givenby χ ∝ exp(−V/Teff). This is because the rotationaldiffusion obliterates any persistence of the active mo-tion, and the non-equilibrium bath effectively behaveslike an equilibrium one with the renormalized tempera-ture Teff = T + v2/(2µDr). In this limit, the degrees offreedom x and y also become independent.

Yet another case admitting an analytical solution ofEq. (D1), is that of quasi-static driving at small active ve-locity. Then the quasi-static PDF ρ can be approximatedby the McLennan-type form χ ≈ exp(−U/T )[1 −W (x)][99–103]. Without going into details, the function W (x)is in general proportional to the (average) dissipation inthe driven system [101], which, in our case, is given bythe product of the active “force” µ−1v cos θ and the par-ticle velocity x. Since the average over the angle θ of theactive force is zero, the correction W (x) to the particlePDF is seen to be at least second order in v.