Harnessing non-equilibrium forces to optimize work extraction

Domain: physics
Published: 2025-12-09
Editorial outcome: accept

Abstract
While optimal control theory offers effective strategies for minimizing energetic costs in noisy microscopic systems over finite durations, a significant opportunity lies in exploiting the temporal structure of non-equilibrium forces. We demonstrate this by presenting exact analytical forms for the optimal protocol and the corresponding work for any driving force and protocol duration. We also derive a general quasistatic bound on the work, relying only on the coarse-grained, time-integrated characteristics of the applied forces. Notably, we show that the optimal protocols automatically harness information about non-equilibrium forces and an initial state measurement to extract work. These findings chart new directions for designing adaptive, energy-efficient strategies in noisy, time-dependent environments, as illustrated through our examples of periodic driving forces and active matter systems. By exploiting the temporal structure of non-equilibrium forces, this largely unexplored approach holds promise for substantial performance gains in microscopic devices operating at the nano- and microscale.

Introduction
Overtwocenturiesago,thedevelopmentofthermodynamicslaidthe foundation for the Industrial Revolution. In recent decades, major advances, particularly through the development of stochastic thermodynamics, have extended thermodynamic principles to microscopic systems, where thermal fluctuations play a dominant role1–4. Thisemergingframeworkenablesustorigorouslyaddresstwocentral challenges: how to optimally control small-scale processes under constraintsofaccuracy,speed,andminimalenergyexpenditure;and howtoefficientlyharvestenergyfromstronglyfluctuating,far-from- equilibriumenvironments.

Harvestingenergyfromnon-equilibriumforcesandfluctuations hasalreadyprovensuccessfulinavarietyofmacroscopictechnolo- gies,includingwave-energyconvertersthatutilizeoscillatoryforces5, piezoelectricdevicespoweredbybiomechanicaldeformation6–8,and wearablesthatgenerateenergyfromhumanmotion9.Atmicroscopic scales,thisprinciplemaybeevenmoreconsequential,asbothbiolo- gical and synthetic systems routinely operate in dynamic, out-of- equilibrium conditions. There is growing evidence that non- equilibrium fluctuations are not just unavoidable noise, but can be harnessed as a resource. For example, biological systems such as molecularmotorsperformmicro-andnanoscaletaskswithremark- ableefficiencydespiteoperatingundernoisyanddrivenconditions10,11. Likewise,non-equilibriumstochasticengineshaveinsomecasesbeen shown to outperform their equilibrium counterparts by exploiting fluctuations12–17.Adeeperunderstandingoftheseeffectsmayprove crucialforthefuturedesignofefficient,robustsmall-scaleengines18.

Astechnologicalinnovationcontinuestopushtheboundariesof miniaturization,identifyingthefundamentallimitsoftheseprocesses anddesigningcontrolstrategiesthatminimizeenergeticandtemporal costs,hasbecomeessentialfortheoptimaloperationanddesignof next-generation microscopic machines. Of particular interest is the developmentofoptimalstrategiesforvaryingcontrolparametersover timetodriveasystembetweentwostateswhileminimizingcostssuch asenergy,dissipation,orduration19–22.Recentadvanceshaveextended theseconcepts to morecomplex,non-homogeneous environments, includingdisorderedmedia23,24,stochasticresettingprocesses25,and viscoelasticbackgrounds26.Optimalprotocolshavealsobeenstudied in systems involving multiple or constrained control parameters27. Additionally, approaches based on information geometry and ther- modynamicmetricshaveyieldedbroad,unifyinginsightsintooptimal controlinfar-from-equilibriumsystems28.

Despitetheseadvances,acriticalquestionremainslargelyunex- plored:howtoexploitthetemporalstructureofforcesandfluctua- tions in non-equilibrium environments. Unlike equilibrium systems, whichlackintrinsictimedependence,non-equilibriumsettingsoften exhibitrichtemporalfeatures,suchascharacteristictimescalesand frequencyspectra,thatmaybeharnessedforimprovedcontroland energyextraction.Indeed,generalizedLandauerboundssuggestthat excess work can be reduced by leveraging the information content separating a system’s non-equilibrium state from a referenceequili- briumstate29,30.

In this work, we address this critical question by identifying optimalprotocolsthatextractenergyusinginformationabouttime- dependentforcesactingonaparticle.Sinceforcesinducetranslational displacements,itisnaturaltoconsiderprotocolsthatmanipulatethe trapviaatranslationaldegreeoffreedom,allowingthecontrolstrat- egytoadapttoforce-inducedeffects.Theseforcesmayoriginatefrom externalfieldssuchasfluidflowsorelectromagneticfields,orarise internally,asinthecaseofself-propulsionforcesdrivingactiveparti- cles (see Fig. 1). Taking a unified perspective, we present a general solutiontotheoptimalcontrolproblemthatappliesbroadlytosuch temporallydrivensystems.Weillustratethisframeworkthroughthe paradigmaticexampleofaparticleconfinedinaharmonicpotential, wheretheprotocolgovernsthetrap’sposition.

Results
Optimalcontrolofacolloidalparticleinaharmonictrapfar fromthermalequilibrium
Toillustrateourapproach,weexamineawell-establishedmodelsys- tem:aparticleconfinedwithinaharmonicpotential,wherethecontrol protocolgovernsthepositionofthetrapcenter.Anobject,underthe combinedeffectofasmoothexternaldrivingforcef(t),aharmonic potentialwithstiffnesskandtrapcenterλ(t),andathermalbathat temperatureT,obeystheLangevinequation

\gamma\dot{x}(t) = -k[x(t) - \lambda(t)] + f(t) + \sqrt{2k_BT\gamma}\xi(t), (1)

where \xi(t) is Gaussian white noise with \langle\xi(t)\rangle=0 and \langle\xi(t)\xi(t')\rangle=\delta(t-t'). The bath is unaffected by the forces f(t)31. This model could represent a colloidal particle manipulated by optical tweezers,orserveasatoymodelforamotionprotocol\lambda(t)foramicro- botwithspring-likecouplingtoacargoatpositionx(t).Theforcesf(t), whichfornowarearbitraryandmaybedeterministicorstochastic, genericallydrivethesystemoutofequilibrium.Weemphasizethatitis assumedthatthesedrivingforcesarespatially-homogeneous,asother scenarios would render the problem non-linear and analytically untractable. Yet, a wide range of non-equilibrium systems can be capturedbytheabovemodel,aswedescribeinfuturesections.We denotebyq(t)\equiv\langle x(t)\rangle themeanparticletrajectory,wheretheaverage is taken with respect to the Gaussian noise as well as any possible stochastic effectsinthe forces.We alsolet F(t)\equiv\langle f(t)\rangle denote the averaged force. The mean particle trajectory can be obtained by averagingtheequationofmotion,yielding

\gamma\dot{q}(t) = -k[q(t) - \lambda(t)] + F(t). (2)

We emphasize that if the forces are deterministic and known with precision, F(t)=f(t). Furthermore, anystochasticeffectmaybeeither inherent to the forces, such as in the case of active self-propulsion forces,orrepresenteffective forces resultingfromerrors inexperi- mentalmeasurementsorinference.Recenteffortsinforceinference haveemployedavarietyofapproaches,includingmachinelearning, Bayesian methods, and information-theoretic techniques32–39. This underscores the importance of information-limited optimization, where protocols are derived based on the perceived driving forces- asappliedfieldsorflows,orinternallygeneratedforcessuchasinactivematter. Theoptimaltransportproblemcanalsobeinterpretedintermsofmicro-robotics andcargotransport,where\lambda^*(t)representsthetrajectoryofamicro-robottrans- portingacargothatisexposedtonon-equilibriumconditions.

Controlling the position of the harmonic trap
V[x,\lambda(t)] = \frac{1}{2}k[x - \lambda(t)]^2 with a protocol \lambda(t) comes at a thermo- dynamiccost,whichisgivenbythemeanwork

W[\{ \lambda(t) \}] = \int_0^{t_f} dt \dot{\lambda}(t) \frac{\partial V[x, \lambda(t)]}{\partial \lambda(t)} : (3)

Weusetheconventionthatapositiveworkimpliesenergyhastobe paid,whilenegativeworkcorrespondstoenergyextracted.InEq.(3), averagesaretakenovernoiseandstochasticforcerealizationsaswell asovermeasurementerrors.Undertheboundaryconditions\lambda(t_i)=\lambda_i andto\lambda(t_f)=\lambda_f,weseektheoptimalprotocol\lambda^*(t)thatminimizesthe workperformedoverafixedtimeinterval.Intheabsenceofexternal forces(F=0),theoptimalprotocolisknowntobelinear,withsym- metric discontinuous jumps at the very beginning and end of the protocol19.UsingEq.(2)asadynamicalconstrainttogetherwiththe work in Eq. (3), the optimization problem can be solved exactly, resulting in an Euler–Lagrange equation \gamma\ddot{q}(t) = \dot{F}(t)/2. This can be solved with the aforementioned boundary conditions, from which one can calculate the optimal protocol, mean particle position, and work.

UndergeneralF(t),wederiveanoptimalprotocolwhichcanbe decomposedintotwocontributions

\lambda^*(t) = \lambda_{eq}(t) + \lambda_{neq}(t): (4)

Thefirsttermisanequilibriumcontribution(F=0),andtheseconda non-equilibriumcontribution(F\neq 0)thatisdeterminedsolelybythe perceived drive and protocol duration. Respectively, these take the form

\lambda_{eq}(t) = q_i + \frac{1+\omega t}{2+\omega t_f}(\lambda_f - q_i), (5)

\lambda_{neq}(t) = \frac{1}{2\gamma} \int_0^t dt' F(t') - \frac{1+\omega t}{2+\omega t_f} \frac{1}{2\gamma} \int_0^{t_f} dt' F(t') - \frac{F(t)}{2\gamma\omega}, (6)

where \omega=k/\gammaistheinverserelaxationtimescaleoftheharmonictrap. Inthefreediffusivelimit,F=0werecovertheprotocolfirstobtained inRef.19inthecaseofaninitialequilibriumstatewithq_i=0.Inthis partoftheprotocol,\lambda_{eq}(t),consistsofastraightlineasafunctionof timebutwithdiscontinuousjumpsatthebeginningandendofthe protocol. The non-equilibrium contribution to the protocol \lambda_{neq}(t), containingtheforce,dependsonlyonthedurationoftheprotocol,but notontheinitialandfinallocationoftheprotocol.Hence,Eq.(4)can be interpreted as the equilibrium protocol with superimposed correctionsthatcompensateforthedrivingforcesF(t).Duringthe protocol,themeanparticlepathisgivenby

\langle q(t) \rangle = q_i + \frac{2\lambda_f - \omega q_i t_f}{2+\omega t_f} + \frac{1}{2\gamma} \int_0^t dt' F(t') - \frac{1}{2+\omega t_f} \frac{1}{2\gamma} \int_0^{t_f} dt' F(t') \omega t. (7)

We note that therearetwo contributions to the path; first,a linear contribution thatdepends on both the detailsof the dynamics and the protocol, and a second term, coming from a largely non-linear time-dependence of Eq. (7).

Weemphasizethat,justlikeinref.19,theprotocolonlyensures thatthepotentialisatthefinallocation\lambda_f attimet_f,withoutanycon- straintsonparticlelocation.Onecouldinprinciple,alsoconstrainthe particlelocationattheendoftheprotocol,leadingtoincreasedcontrol, butatthecostoflessenergyextraction.Inthisrelationbetweencontrol andcost,weconsiderprotocolsthatareabletoextractmaximalwork fromthenon-equilibriumforces.Theparticlepositionattheendofthe protocolq(t_f)will,inthequasistaticregime,begivenas

\lim_{t_f \to \infty} q(t_f) = \lambda_f + \lim_{t_f \to \infty} \frac{F(t_f)}{k}. (8)

Surprisingly,thisdeviationisnotdeterminedbythevalueoftheforce atthelaterstagesoftheprotocol,butbythefulltime-averagedforce since the initial time t=0. The memory of the full dynamics is a consequence of the way in which the optimization intertwines the forces,theprotocol,andtheparticleposition.

Asshowninthe“Methods”section,theworkassociatedwiththe optimalprotocolcanbeshowntotaketheform

\begin{aligned} W = &\frac{1}{2}k(\lambda_f - q(t_f))^2 - \frac{1}{2}k(\lambda_i - q_i)^2 \\ &+ \frac{\omega}{2+\omega t_f} \lambda_f - q_i - \frac{1}{2\gamma} \int_0^{t_f} dt F(t) \end{aligned} (9)

whichisanexactresultvalidforarbitraryprotocoldurationst_f andfor arbitraryforcesF(t).

Quasistaticboundonworkextraction
Inmanycases,knowingthequasistaticlimitisinformative,asitpro- videsboundsontheworkexchanged.Whetherthisboundispositive (costingwork)ornegative(extractingwork)andboundedorinfiniteis ofhighpracticalrelevance.

In the quasistatic limit t_f \to \infty, Eq. (9) is useful in contributing to the work and provides physical insights into the mechanisms by whichworkisextractedfromknowledgeoftheforces.Secondly,the quasistatic limit allows us to optimally adapt to the particle, even for finite-time experiments. When the potential is shapeduringtheprotocol,theequilibriumfreeenergydifferenceis zero. Consequently, in the quasistatic limit, only non-equilibrium effects contribute, arising either from non-equilibrium initial condi- tions captured by q_i or from non-equilibrium driving forces F(t). Takingtheslowlimit(t_f \to \infty)ofEq.(9),wefind

W = W_i + W_{ta} + W_d, (10)

wheretheworkhasbeendecomposedintothreeparts;aninformation theoreticcontribution,acontributionfromtime-averagedforces,and a contribution from forces deviating from its time-average, respec- tivelytakingtheform

W_i = -\frac{1}{2}k(\lambda_i - q_i)^2, (11)

W_{ta} = -(q_f - q_i) \lim_{t_f \to \infty} F(t_f), (12)

W_d = -\frac{1}{4\gamma} \lim_{t_f \to \infty} t_f \text{Var}(F; t_f). (13)

In these expressions, we also used the time-averaged mean and variance

\bar{F}(t_f) = \frac{1}{t_f} \int_0^{t_f} dt' F(t'), (14)

\text{Var}(F; t_f) = \frac{1}{t_f} \int_0^{t_f} dt' F^2(t') - \left( \frac{1}{t_f} \int_0^{t_f} dt' F(t') \right)^2. (15)

Beforeweinterpreteachterminthequasistaticwork,itisworth emphasizingthesimplicityofthisresult.AlltermsinEq.(10)maybe calculateddirectlywithouttheneedtogoexplicitlythroughtheoptimizationprocedurefortheforcesactingontheparticle \{\lambda_i, \lambda_f, F(t)\}. Hence, this formula can be applied to a wide range of systems withouthavingtogothroughtheoptimizationprocedureexplicitly. Furthermore,inthequasistaticlimit,theworkisdeterminedsolely by the firsttwo time-integrated cumulants, rendering higher-order fluctuationsirrelevant.

InEq.(10),thefirsttermisdeterminedbytheinitialconditionof theparticleandnaturallyadmitsaninformation-theoreticinterpreta- tion.Becausewespecifyonlythemeanoftheinitialdistributionp_i(x), the system may start in an arbitrarily complex state. However, the harmonictrapcanonlyaccesstheportionofthisinformationthatis compatiblewithitsfixedshapeorposition40,41.

Tomakethismoreprecise,weintroducetheM-projection\pi[p_i], definedbyref.42as

\pi[p_i](x) = \text{argmin}_{\rho \in B} D_{KL}(p_i || \rho), (16)

where D_{KL} istheKullback–Leiblerdivergence.Thisoperationprojects theinitialdistributionp_i(x)ontothespace B=\{Z^{-1}e^{-\beta V[x, \mu]}, \mu \in R\} of Boltzmannstatescompatiblewiththetrapmanipulation-here,fixed- variance(i.e.,constant\beta andstiffness)Gaussiandensitieswithvariable location\mu(oftencalledshiftmeasures).ByminimizingD_{KL}, \pi[p_i]isthe least-information-lossGaussianapproximationofp_i.

Inourexample,\pi[p_i](x)becomesaGaussianwithcenterq_i and variancedeterminedbythetrapshape.Onecanthenshowthat

\frac{1}{2} k_B T D_{KL}(\pi[p_i] || p_{eq}) = \frac{1}{2} k(\lambda_i - q_i)^2 (17)

where p_{eq} isthetrueBoltzmannstateoftheinitialtrap.Thus,thefirst terminEq.(10)quantifiestheaccessibleinformationwithinthenon- equilibrium initial state that can be transformed into negative, i.e., extracted, work. A detailed derivation of this can be found in theSupplementaryInformation.

ThetwolasttermsofEq.(10)arecontributionsoriginatinginthe non-equilibriumdrivingforces.Moreprecisely,thecontributionW_{ta} originatesinthetime-averagedforceandissimplytheworkneededto moveaparticleadistanceq_f - q_i inthepresenceofF.Thelasttermis determinedbythedeviationsoftheforcearoundthetime-averaged mean,andcanbewritten

W_d = -\frac{1}{4\gamma} \lim_{t_f \to \infty} \int_0^{t_f} d\tau \delta F(\tau)^2 (18)

where \delta F(\tau) = F(\tau) - \bar{F}(t_f) are the deviations of the force from its time averaged value. This term encodes how much work can be extractedbyutilizingtheinformationaboutthetemporaldetailsoftheforce.Weemphasizeherethatonecanextractmoreworkfrom time-varying forces than from stationary ones, especially in situations where the deviations from the time-averaged mean arestronglypersistentintime.

Precisionvs.riskinerroneousforceinference
Hereweconsidertheerrorsoccurringifoneratherthanusingthetrue force F uses a coarse-grained version \tilde{F} coming for example,from errors in inference. Since exact inference is hard or impossible, knowledge about what determines the error in the estimated work extraction could be crucial in realistic implementations. Here we derive a simple bound on this error, expressed fully in terms of experimentallyaccessiblequantities.

Inthefollowing,wewillby\lambda(t)alwaysmeantheprotocolderived usingtheinferredorestimatedforces\tilde{F}.Letq(t)and\tilde{q}(t)respectively denote the mean position under the effect of the true force (actual meantrajectory)andtheexpectedtrajectoryunderthebeliefthatthe forcesare\tilde{F}(t).Theworkislinearintheposition,sowecanwrite

\Delta W = W[q] - W[\tilde{q}] = \int_0^{t_f} dt \dot{\lambda}(t) [\tilde{q}(t) - q(t)]. (19)

Theparticlepositiontakesthegeneralform

\langle q(t) \rangle = q_i e^{-\omega t} + \int_0^t ds e^{-\omega(t-s)} [\gamma^{-1}F(s) + \omega\lambda(s)]. (20)

Usingthisexpressionintheworkerrorabove,wefind

\Delta W = \omega \int_0^{t_f} dt \dot{\lambda}(t) \int_0^t ds e^{-\omega(t-s)} [\tilde{F}(s) - F(s)]. (21)

IfoneknewthetrueforcesF,therewouldbenoissuewithinference. Hence,oftenonecanatbestknow|\tilde{F}(s) - F(s)| \leq \epsilon, with \epsilon a uniform boundontheinferenceerrors.Hence

|\Delta W| \leq \omega \epsilon \int_0^{t_f} dt |\dot{\lambda}(t)| \int_0^t ds e^{-\omega(t-s)} (22)

= \epsilon \int_0^{t_f} dt |\dot{\lambda}(t)| (1 - e^{-\omega t}) (23)

\leq \epsilon \int_0^{t_f} dt |\dot{\lambda}(t)| (24)

whereweusedmonotonicityofintegrals.Hence,wehavetheerror bound

|\Delta W| \leq \epsilon \int_0^{t_f} dt |\dot{\lambda}(t)|. (25)

The abRove integral is sometimes called the total variation TV[\lambda] = \int_0^{t_f} dt |\dot{\lambda}(t)|, andhence the work error is bounded by the protocol’s total variation times the force inference errors |\Delta W| \leq \epsilon TV[\lambda]. Thelinearityintheforceerrorsisadirectconsequenceofthelinearity of the problem in the presence of harmonic traps, and we would expectnon-lineardependenceonforceerrorsinotherscenarios.

Thisresulttellsusthatmoreerraticcontrolprotocolswithalarge totalvariationhaveahigherpotentialforproducingerrors.Theerroris alsoadditiveintheprotocolduration;increasingt_f givesmorechances tomakeunfavorablemovesbasedonfaultyknowledge.Thisinsight could also aid in choosing the type of inference one should use. Indeed,ifmultiplemethodsareavailable,\lambda andthereafter TV[\lambda] should becalculatedineachscenario,andtheonewithsmallesttotalvariation should be applied. This analysis demonstrates an inherent tradeoff between precision and gain: more erratic control protocols can, in principle,exploitfine-scaledetailsoftheforcestoextractadditional work,buttheyalsocarryahigherriskoferror.Conversely,smoother protocolswithlowertotalvariationreducethelikelihoodofmistakes, atthepotentialcostofleavingsomeworkunextracted.Understanding thisbalanceallowsonetotailorcontrolstrategiesaccordingtowhe- therthepriorityismaximizingworkextractionorminimizingerrors, effectivelyframingtheproblemasarisk-rewardoptimization.

Casestudies
Our above framework leverages information about non-equilibrium forcestoenhanceenergeticefficiency.Thisisnotunlikeinformation engines,whichtraditionallyutilizeinformationthroughmeasurement andfeedbacktorectifyfluctuationsandextractwork30,43,44.Leveraging priorknowledgeoftheforcesthatdrivethesystemawayfromequi- librium,Euler–Lagrangeminimizationgivesrisetoprotocolsthatboth anticipate and adapt to prescribed particle dynamics, enabling the spontaneous extraction of maximal energy from non-equilibrium processes. In the Supplementary Information, a baseline scenario withoutforcesisdiscussed.Below,weexploretwocasestudiesthat illustratetheversatilityofourapproachwithrespecttodifferentforce types: externally applied periodic forces and internally generated activeself-propulsionforces.

CaseI:periodicforces
Many energy harvesting solutions are based on periodic forces or motion,suchaswave-energyconverters,wearablefabricsthatextract energyfrommovement,andpiezoelectricgeneratorsthatcancharge pacemakers through heartbeats5,6,9,45,46. These classical approaches typicallyutilizemechanicaloscillatorscoupledtotransducers,aiming toconvertdisplacement,velocity,orstrainintoelectricalpower.Here, weconsiderasimplemicroscopicanalogy,whichincontrastrelieson optimal control to identify limits and optimal strategies for energy harvesting, independent of any specific transducer design. We con- sider a Brownian particle effectively confined to one-dimensional movement,exposedtoperiodicdrivingforces

F(t) = f_0 \sin(t/\tau_p). (26)

Heref_0 istheamplitudeoftheforcewhile\tau_p determinesitsperiodicity. Figure3showstheworkasafunctionofprotocoldurationandforce periodicity. The associated optimal protocol is shown in regions of bothpositiveandnegativework.Asiscommonforoptimalprotocols, discontinuousjumpsareseeninthebeginningandfinalpartsofthe protocol.

Weseethatforsufficientlyslowprotocols,comparedtothefor- cingperiod,theoptimalcontrolisabletoutilizetheoscillationssuch that work can be extracted. The optimal protocol harnesses the dynamic information available and automatically extracts as much workaspossible.Thisisaconsequenceofpreciseknowledgeofthe forceatalltimes.Theprotocolconsistsofrepeatedlylettingtheforce movetheparticleintoahigh-energystatebeforeshiftingthepotential accordinglytoextractthestoredenergyaswork.Letusrecallthatin Brownian information engines, feedback is used to rectify thermal noise and convert measurement information into work. In simple enginedesigns,thepotentialhasthefreedomtomoveleftandright based on the measurement outcome, extracting work is a manner similar to the above protocols47. Figure 4 summarizes the main mechanismbehindtheworkextractioninthecaseofperiodicforces.

While the above repeating cycles are specific to the current example,thewayinwhichworkisextractedoffersinsightsintomore generic situations. Indeed, combining the Euler–Lagrange equation with the equation of motion, we have \dot{\lambda} = \dot{q} - \dot{F}/(2k). When forces increaseinagivendirection \dot{F} > 0 theoptimalprotocolletstheparticle movefasterthanthetrap, \dot{\lambda} < \dot{q}, liftingtheparticletoahigherenergy state.Onceforcesstarttodecrease \dot{F} < 0 theprotocolcatchesupto theparticle \dot{\lambda} > \dot{q}.

Theabovecyclecan,inprinciple,extractarbitrarilylargeamounts ofworkoveranindefinitelylongprotocol.Inourexample,thisfollows fromEq.(10),where \text{Var}(F) = f_0^2/2 remainsconstant,causingW_d \approx t_f \text{Var}(F) togrowunbounded(inthenegativedirection)ast_f \to \infty.For finiteyetlargeprotocoldurations, t_f \gg \tau_p, thetermW_d dominates,and theworkbehavesas

W = -\frac{f_0^2 t_f}{8\gamma} (27)

whichisindependentoftheforcingperiod \tau_p. Thisresultsfromadirect competition between two effects caused by changing \tau_p; firstly, it affects the amplitude of the particle displacements and hence how muchenergycouldbeharvestedpercycle.Secondly,italsochanges how many cycles of harvesting can be completed in a given total duration t_f. For slow protocols, these effects balance, resulting in Eq.(27).

This large potentialfor workextraction relies on precise initial information about the periodic forces. For example, we can easily extend the model to consider random variables representing force phase information. Taking these errors to be normal distributed with zero mean and variance \sigma^2_\epsilon, we obtain the mean force \bar{F}(t) = \exp(-\sigma^2_\epsilon/2) f_0 \sin(t/\tau_p). Hence, incorporating these initial errorseffectivelyrescalestheforceamplitude,andtheworkextracted atlargeprotocoldurationsinsteadtakestheform W = -\frac{f_0^2 \exp(-\sigma^2_\epsilon) t_f}{8\gamma}. Hence,initialmeasurementerrorscanbedetrimentaltothedesignof engines,leadingtoexponentiallyreducedworkextraction.Notably, while phase errors cause an exponential suppression of work, their degreeofsuppressionremainsunchangedovertime.Oncethephase is incorrectly estimated, the mismatch persists, effectively lowering theforceamplitudethroughouttheentireprocess.

In the presence of a periodicity error, modeled as f(t) = f_0 \sin(t/(\tau_p + \epsilon)) theeffectiveforcecanbeapproximatedforsmall \epsilon.When \epsilon isnormallydistributedwithasmallvariance \sigma^2_\epsilon weuse:

F(t) \approx f_0 \sin \left( \frac{t}{\tau_p} - \epsilon \frac{t}{\tau_p^2} \right) = f_0 \exp \left( -\frac{t^2 \sigma^2_\epsilon}{2 \tau_p^4} \right) \sin \left( \frac{t}{\tau_p} \right) (28)

Here, the periodic force experiences a pronounced exponential suppression. In contrasttothecasewheretheerrorisinthephase,theerror inperiodicitygivesrisetoalargerphasemismatchovertime,leadingto strongdestructiveinterference,whichsuppressestheeffectiveforces. Theworkinthequasistaticcasecanbecalculatedasbefore,

W_{qs} \approx -\frac{\pi \tau_p \sigma^2_\epsilon f_0^2}{6\gamma} (1 - e^{-\tau_p^2/\sigma^2_\epsilon}) (29)

Insharpcontrasttothecasewithouterrors,workextractionisnow bounded.Weemphasizethatalthoughthisapproximationmaynotbe quantitativelyexactinthequasistaticapproximation,itclearlyshows howsmallerrorsqualitativelyaffecttheextractedwork.Itisalsoworth emphasizingthatwhenthereareerrorsintheperiodicity,thequasi- static work extraction becomes smaller when the periodicity \tau_p is reduced.Thisreflectsthefactthatforveryrapidforces,evensmall errorswouldmakeworkextractionhard.

CaseII:Workextractionfromactiveforces
Active matter represents a compelling category of non-equilibrium systems, where mechanisms for work extraction have been investi- gated recently48–58. Unlike the externally imposed oscillatory forces discussed previously, active particles generate propulsive forces internallythroughautonomousmechanisms.Wenowapplyourgen- eralframeworktoseveralrepresentativeactivematterscenariosand comparetheresultingworkthatcanbeextractedineachcase.

Weconsideranactiveparticleintwodimensions,modeledasan active Brownian particle (ABP), obeying the stochastic equation of motion

\dot{x}(t) = \frac{f_0}{\gamma} \hat{n}(t) - \omega[x(t) - \lambda(t)] + \sqrt{2D}\xi(t) (30)

where f_0 istheconstantmagnitudeoftheactiveself-propulsionforce, and \hat{n}(t) = [\cos\phi(t), \sin\phi(t)] determines the direction of propulsion. Persistenceisexpectedtoplayakeyroleintheenergyextractioninany activesystem,motivatingustoconsidernotonlyreorientationsdriven bywhitenoise,asisusualforABPs,butaslightlymorerealisticsituation wherefinite-timecorrelationsarepresent.Inthesimplestcase,onecan considerorientationsdrivenbyanOrnstein-Uhlenbecknoise

\dot{\phi}(t) = \Omega(t) (31)

\dot{\Omega}(t) = -\frac{1}{\tau_R} \Omega + \sqrt{\frac{2D_r}{\tau_R}} \eta(t) (32)

where D_r isarotationaldiffusioncoefficientand \tau_R arelaxationtime- scaleassociatedwiththerotationaldynamics.Suchmodelshavebeen consideredinthepasttoincludememoryeffectsintheorientational dynamics,resultingforexamplefrommisalignmentswiththeinstan- taneous propulsion direction or inertial effects59–62. When \Omega is sta- tionary,thedistributionof \phi(t) isknowntobeGaussianwithmean \phi_0 andvariance \sigma^2_\phi(t) = 2D_r (t - \tau_R(1 - e^{-t/\tau_R})). Thisresultsinameanforce

F(t|\phi_0) = \langle f_0 \hat{n}(t) | \phi_0 \rangle = f_0 \begin{bmatrix} \cos(\phi_0) e^{-\frac{1}{2}\sigma^2_\phi(t)} \\ \sin(\phi_0) e^{-\frac{1}{2}\sigma^2_\phi(t)} \end{bmatrix} (33)

conditionedonknowingtheinitialforce,i.e.,propulsiondirection,as canbeobtainedfromameasurement.Fromthis,theoptimalprotocol and the associated work can be calculated. Recently, repeating this scheme was proposed as a way of constructing active information engines55,56.Inthecaseofone-dimensionalrun-and-tumbleoractive Ornstein-Uhlenbeckparticles,werecovertheresultsofref.55,while ourgeneralresultscanalsobeusedtostudyawiderangeofother activemattermodels,alsoinhigherdimensions.

Determiningtheoptimalprotocolanditsworkcostrequiresthe time-integratedmeanand varianceoftheforce,whichwecompute exactly (see Supplementary Information). With the time-integrated mean and variance in hand, we compute the optimal protocol \lambda^*(t) directlyfromEqs.(5)and(6),withtheresultsshowninFig.5.Wesee thatshortprotocoldurations(e.g.,panela)resultsinprotocolswith largeinitialjumpandalmostlineardragging.Longerprotocoldura- tions,however,(e.g.,paneld)showssmallerjumpsandacurvedpro- tocol trajectory that utilizes the particle persistence to lower the energeticcost,orevenallowworkextraction.Generally,theprotocol makesajumptoapositionbehindtheparticle,andfollowstheparti- cle’sinitialdirectionofmotionforawhile.Thisslowstheparticledown andaftertheparticle’spersistenceislostduetorotationalnoise,the protocoldragstheparticlebacktothetargetlocation.

Inanalyzingworkextraction,thequasistaticboundW_{qs} canbe directly obtained from Eq. (10). Without loss of generality, we set \phi_0 = 0,which,bysymmetry,impliesF_y = \lambda^*_y(t) = 0.Asaresult,onlythe x-componentoftheprotocolcontributesandneedstobeconsidered. Thisresultsin

W_{qs}(D_0) = W_{qs}(0) \frac{e^{2D_0} \Gamma(2D_0) - \Gamma(2D_0, 2D_0)}{(2D_0)^{2D_0-1}} (34)

which is expressed solely in terms of the dimensionless number D_0 = D_r \tau_R,introducedasadelaynumberinRef.61.Here,wenormal- izedtheworkbyits \tau_R \to 0 value

W_{qs}(0) = W_{ABP} = -f_0^2 / (8D_r \gamma) (35)

which is the bound for the normal ABP model with Gaussian white noisedrivingtheorientations.Figure6,leftpanel,showstheoptimal protocol and the associated mean particle position for a persistent activeparticleinitiallyorientedalongthepositivex-axis.Theoptimal protocolexhibitsaninitialjumpbehindtheparticleposition.Although thisincursahighinitialworkcost(asalsoreportedinref.55),itstra- tegically enables work extraction from the particle’s persistence throughouttheremainderoftheprotocol.TherightpanelofFig.6 showsthattheworkinthequasistaticlimitincreasesmonotonically withthedelaynumberD_0,confirmingthebeneficialeffectoffinite- time angular correlation \tau_R. We emphasize that while the ratio W_{qs}(D_0)/W_{qs}(0) is positive, the work W_{qs}(D_0) remains strictly negative across all delay number values, indicating effective work extractionfromthesystem.

ThequasistaticworkboundgivenbyEq.(10)caneasilybeeval- uatedforawiderangeofotheractiveparticlemodelsaswell.InTable1 we list four other active particle models and the quasistatic work associatedwiththeiroptimalprotocols.Moreprecisely,weconsider activeparticlesdrivenbyfractionalBrowniannoise63,arandomrota- tionaldiffusionmodel,chiralparticles64,65,andparticlesthataccelerate or decelerate in time66. For the random rotationaldiffusivity model with mean fixed to the rotational diffusion of a pure ABP, we find doublethepotentialforworkextraction,whilechiralitydoesnotaffect the bound. For fractional angular noise and accelerating ABPs, the workextractionmaybehigherorlowerthanthepureABPcase, dependingontheparameters.

Methods
Theoptimalprotocoldiscussedinthe“Results”sectionwasderived usingEuler–Lagrangemethods.Here,wederivetheoptimalprotocol for arbitrary driving forces, as well as the associated mean particle trajectoriesandthethermodynamicwork.Wedefinethemeanparticle positionasq(t)=\langle x(t)\rangle.Themeanparticlepositionevolvesas

\gamma\dot{q}(t) = -k[q(t) - \lambda(t)] + F(t) (36)

Welet\omega=k/\gamma,sothattheequationofmotiontakestheform

\dot{q}(t) = -\omega[q(t) - \lambda(t)] + F(t)/\gamma (37)

Foraharmoniccontroltrap,wehavethemeanwork

W = k \int_0^{t_f} dt \dot{\lambda}(t) [\lambda(t) - q(t)] (38)

Using the equation of motion, we can rewrite the work as

W = \int_0^{t_f} dt \dot{\lambda}(t) [\gamma\dot{q}(t) - F(t)] (39)

whereweusedtheequationofmotion.Takingafurtherderivativeof themeanequationofmotion,wecanalsowrite

\ddot{q}(t) = -\omega\dot{q}(t) + \omega\dot{\lambda}(t) + \dot{F}(t)/\gamma (40)

whichwecanusetoeliminate\omega\dot{\lambda}(t)intheexpressionforthework.This gives

W = \int_0^{t_f} dt [\ddot{q}(t) + \omega\dot{q}(t) - \dot{F}(t)/\gamma] [\dot{q}(t) - F(t)/\gamma] (41)

In an example, the terms can be integrated by parts. Similarly, we have

W = \frac{1}{2k} [(\gamma\dot{q})^2(t)]_0^{t_f} + \frac{1}{2k} [F^2(t)]_0^{t_f} - \frac{1}{\omega} [\dot{q}(t)F(t)]_0^{t_f} + \int_0^{t_f} dt L(t, q, \dot{q}) (42)

whereweintroducedtheLagrangian L(t, q, \dot{q}) = \gamma\dot{q}(t) [\dot{q}(t) - F(t)/\gamma]. ThecorrespondingEuler–Lagrangeequationreads

\ddot{q}(t) = \dot{F}(t) / (2\gamma) (43)

whichcaneasilybesolvedby

q(t) = q_i + \phi t + \frac{1}{2\gamma} \int_0^t dt' F(t') (44)

where we used the initial condition q(0)=q_i. To proceed, we must identify the unknown constant \phi, which is typically done by minimizingtheworkwithrespectto\phi19.Forthis,wemustfirstwritethe explicitexpressionforthework,whichwedobyusingtheequationsof motiontodeterminetheboundaryconditionsfortheparticlevelocity:

\dot{q}(0) = \omega(\lambda_i - q_i) + F(0)/\gamma (45)

\dot{q}(t_f) = \omega(\lambda_f - q(t_f)) + F(t_f)/\gamma (46)

Theworkthenexplicitlyreads

W = \frac{1}{2k} [k(\lambda_f - q(t_f)) + F(t_f)]^2 - \frac{1}{2k} [k(\lambda_i - q_i) + F(0)]^2 + \frac{1}{2k} [F^2(t_f) - F^2(0)] - \frac{1}{k} [k(\lambda_f - q(t_f)) + F(t_f)] F(t_f) + \frac{1}{k} [k(\lambda_i - q_i) + F(0)] F(0) + \gamma \phi^2 t_f - \gamma \int_0^{t_f} dt [F(t)/\gamma]^2 (47)

We note that there is here also a dependence on \phi through terms containing q(t_f), from Eq. (44). From this expression, the unknown parameter \phi canbedeterminedbyminimization \partial_{\phi} W = 0, resultingin

\phi^* = \omega \frac{\lambda_f - q_i - \frac{1}{2\gamma} \int_0^{t_f} dt F(t)}{2 + \omega t_f} (48)

This gives the mean particle trajectory, which in turn through the equationofmotion,canbeusedtofindtheoptimalprotocol.Ittakes theform

\lambda^*(t) = \lambda_{eq}(t) + \lambda_{neq}(t) (49)

where

\lambda_{eq}(t) = q_i + \frac{1 + \omega t}{2 + \omega t_f} (\lambda_f - q_i) (50)

and

\lambda_{neq}(t) = \frac{1}{2\gamma} \int_0^t dt' F(t') - \frac{1 + \omega t}{2 + \omega t_f} \frac{1}{2\gamma} \int_0^{t_f} dt' F(t') - \frac{F(t)}{2\gamma\omega} (51)

Combiningresults,theworkcanbewritten

W = \frac{k}{2} [(\lambda_f - q(t_f))^2 - (\lambda_i - q_i)^2] + \frac{k}{2 + \omega t_f} [\lambda_f - q_i - \frac{1}{2\gamma} \int_0^{t_f} dt F(t)]^2 - \frac{1}{2\gamma} \int_0^{t_f} dt F(t)^2 (52)

whichisthegeneralexpressionanalyzedinthe“Results”section.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Optimal control has emerged as a highly active research area within stochastic thermodynamics in recent years, focusing on\nthe development of driving protocols that minimize thermodynamic costs such as work expenditure. This manuscript extends\nthis line of research to systems experiencing time-dependent nonequilibrium forces.\nThe authors derive closed-form expressions for both the optimal protocol and the associated work, demonstrating that the\ntotal work naturally decomposes into three distinct contributions: (i) work arising from initial disequilibrium, (ii) work required\nto drag the system under time-averaged forces, and (iii) work induced by temporal fluctuations in the forces. The analysis is\nillustrated through two compelling examples -- a system driven by oscillating forces and an active Brownian particle. The\nauthors also discuss how measurement errors impact work extraction efficiency.\nThe manuscript is exceptionally well-written and clear, the research is high quality, and the authors provide insightful\nphysical interpretations. Generally, I recommend publication, with only a few minor comments.\nFirst, I think the authors should be a bit clearer regarding the restrictive assumptions underlying the theoretical results.\nConsistent with most optimal control studies in this field, the authors assume that control is implemented exclusively through\nmanipulation of a harmonic trap with fixed width. In addition, they assume that nonequilibrium forces depend only on time\nand not on position. While relaxing this latter assumption would likely preclude analytical solutions (due to coupling between\nthe forces and protocol), it is a substantial limitation that warrants more prominent discussion. I would recommend that the\nauthors emphasize these limitations and discuss the (im)possibility of generalizing the results to space-dependent forces.\nSecond, I am not really convinced by the interpretation of the optimal protocol as an \"information engine\". While the optimal\nprotocol indeed exploits knowledge of the temporal structure of nonequilibrium forces, this information is provided a priori as\ninput to the optimization algorithm. This differs from (my understanding of) an information engine, which should detect and\nrespond to stochastic fluctuations in real-time during the process. That is, in genuine information engines, the driving\nprotocol can vary depending on the stochastic trajectory realized by the system. By contrast, the authors' optimal protocol is\npredetermined through the optimization procedure before any trajectories are observed. I recommend that the authors de-\nemphasize their interpretation of the optimal protocol as an information engine, or explain it in more depth, to avoid\nconfusion.\nI did not find any typos but there is a typesetting issue near line 404.", "reviewer_2": "This work presents optimal protocols for harvesting energy from a force whose mean values at each time are known, using a\nparticle in a harmonic potential or equivalent first-order dynamical system. Despite a few limitations, explored below, I think\nthis makes a nice contribution and yields a number of insights that can guide future work.\nLet’s start with the strengths. The three-part decomposition into an information-based part and parts using time-integrated\nmean and variance of the force is pretty. The last term involving the integrated variance was surprising to me until I thought\nabout it more, and I that’s enough of an advance to favor publication. The other two terms are more familiar. The first has\nbeen explored in roughly equivalent studies of information engines and based on principles found, e.g., in Parrondo et al.,\n[Ref. 30] and the second is similar to studies on optimal navigation (e.g. by the last author). Still, the deviation-based third\nterm is a nice insight and sets one up to understand the application examples.\nOn the other hand, there are many limitations to the study:\n1) Although Eq. 11 is a general result, almost all the discussion is couched in terms of quasistatic protocols, where the\nphysical interpretation is perhaps clearest. But probably faster protocols are better (if not, why?).\n2) The harvesting is the physicists’ version, where the energy isn’t really stored in any useful way but rather just amounts to\n“cooling the bath” This limits practical applications. I contrast the scheme here with previous (experimental) work from the\nRoichman group (Admon et al., PRL 2018) where the energy was captured (by having a particle move upstream in a flow\nfield).\n3) The force is rarely known, and one might wonder how useful results based heavily on this knowledge can be. The authors\ndo address this in several ways: For example, in the section on periodic forcing with phase errors (not such a big deal) and\nfrequency errors (a more serious problem). A systematic way to address such issues (measurements and then feedback) is\nhinted at in the second application, to active matter, which breaks with the rest of the paper by introducing a single\nmeasurement and feedback. One can think of each measurement as making possible a “burst” of harvested energy that\ndecays exponentially (one has to piece this together by substituting the expression for sigma_\\phI^2 into Eq. 25). The work\nacquired is finite (not proportional to tau_R), which is stated but not really emphasized. Of course, with repeated\nmeasurement and feedback, one could address this issue. But that is beyond the scope of this paper.\nGiven these limitations, I don’t think it is fair for the authors to claim to have “uncover[ed] the fundamental limitations of\nenergy harvesting” — that seems too strong. Nonetheless, they have identified some limitations and have provided a great\ndeal of insight in work that is quite creative, and I would be in favor of its eventual publication in Nat. Commun.\nMinor Points:\n1. [I think this is a minor point, but it could be more important.] Figure 1 talks about “arbitrary time-dependent forces\n\\mathcal{F}(t). (The Discussion contains similar language.) Is this really the case? For example, could \\mathcal{F} be a step\nfunction? Could the sine wave example have arbitrarily high frequencies? One imagines that for this case, there wouldn’t be\nmuch to harvest if the external force is too fast relative to tau_R. Whatever the answer, the issue needs more discussion.\n2. References 54 and 55 (preprints) cover very similar ground. There should be more discussion comparing that work and\nthe present paper.\n3. Ref. 26 was published in 2024 as a PRX paper.\n4. Some other papers might warrant inclusion and at least a brief discussion:\ni) A. Kolchinsky et al., “Maximizing Free Energy Gain”, Entropy 2025.\nii) L. Gammaitoni e tal., “Vibration Energy Harvesting…” from the book “Sustainable Energy Harvesting Technologies” Ed.\nYen Kheng Tan, 2011.\nThis second is a stand-in for a large group of papers from the “classical” world of energy harvesting, which is a well-\ndeveloped engineering topic that has also had some contributions by physicists. It would be nice if the paper had more\ncontact with this part of the literature.\n5. Some of the paper exposition could be clearer. I did not see if it is stated that W<0 corresponds to work extraction [if so,\nit is a bit buried].\n6. Similarly, Eq. 12 would be clearer if the underbrace terms (W_i, W_ta, W_d) were immediately defined. For example, W_i\nis defined only in the caption of Fig. 2, which is one page after. In a first reading, I thought from Line 258 that the “i” in W_i\nwas “initial”, which led to some confusion.\n7. Eq. 15: Define better the space \\mathcal{B} of Boltzmann states. Is it the set of Boltzmann distributions for fixed\ntemperature (the bath) but variable lambda? Related: The authors should show (perhaps in the supplement) how to derive\nEq. 16. Given supplements, there seems no need to skip steps.", "reviewer_3": "The authors present a detailed theoretical investigation focusing on optimal work extraction through optimal control protocols\napplied to single-particle systems driven by time-dependent external forces. While the equilibrium component of these\noptimal protocols (Eq.7 in the manuscript) is well-established in the literature, the authors introduce an explicit analytical\nform for the nonequilibrium correction (Eq.8), which represents the main theoretical contribution of this work. Two concrete\nexamples, periodic external driving forces (Case I) and active Brownian particles (Case II), are also provided. These\nexamples clearly illustrate the derived optimal protocols and would be of interest to specialists working on stochastic\nthermodynamics.\nFrom a technical perspective, the manuscript appears sound and clearly presented. However, I find that the overall novelty\nand broad interdisciplinary appeal is somewhat lacking. Specifically, the optimal control approach to work extraction in\nstochastic thermodynamics is already a well-explored topic at least theoretically. Seminal theoretical studies, such as those\nby Schmiedl and Seifert (PRL, 2007)[19], Sivak and Crooks (PRL, 2012), and more recent works including Blaber and\nSivak (J. Phys. Comm., 2023), have extensively addressed this topic. Thus, while Eq.8 is theoretically novel and interesting,\nits overall conceptual contribution appears rather incremental. Furthermore, there is already an experimental demonstration of\noptimal control, Loos et al. (PRX, 2024)[26], and theorefore the purely theoretical nature of the present manuscript would\nlimit its impact potential for a journal targeting a broad interdisciplinary audience.\nAdditionally, the authors' characterization of their results as demonstrating an \"information engine\" is, in my view, not entirely\nconvincing. Traditional information engines explicitly incorporate measurement and feedback control to extract work. While\nthe authors acknowledge the absence of explicit measurement-feedback loops, their interpretation of \"information\" is\nprimarily relevant to initial nonequilibrium states and precise knowledge of external forces. Thus, the use of the term\n\"information engine\" could be considered somewhat misleading.\nIn summary, although this manuscript presents theoretical results that will be of interest to specialists, its contributions\nappear somewhat incremental within a field already rich in theoretical insights. Therefore, while this study would be well\nsuited for publication in a more specialized journal, I believe that it does not meet the threshold of broad impact and\nsignificant innovation required for publication in Nature Communications."}, "author_response": "Reply to reviewers: Harnessing non-equilibrium forces to optimize work extraction\n(NCOMMS-25-35338)\nDearEditorsandreferees,\nWe thank the reviewers for their careful reading of the manuscript, and for their constructive\nfeedback. Wealsoappreciatethepositivecomments,whichincludesstatingthatthemanuscript\nisexceptionallywell-writtenandclear,thattheresearchisofhighquality,andthatitpresentsa\nnumberofinsightsthatcanguidefuturework.\nFollowing the reviewers suggestions, the manuscript has improved in both clarity and rele-\nvance, and more clearly positions itself within the existing literature. We have also added sub-\nstantial material, including a new section, that distinguishes our framework further from the\nexisting ones and makes a clearer bridge to experiments. We hope that with these changes, the\nmanuscriptissuitedforpublicationinNatureCommunications.\nBelow we address individually the comments and questions raised by the referees. Our re-\nsponsesarehighlightedinblue. Similarly, highlightedmanuscriptchangescanbefoundinthe\nattachedmarkupPDF.\nBestregards,\nKristianStølevikOlsen,RémiGoerlich,YaelRoichmanandHartmutLöwen.\nReviewer1\nOptimal control has emerged as a highly active research area within stochastic thermody-\nnamics in recent years, focusing on the development of driving protocols that minimize ther-\nmodynamic costs such as work expenditure. This manuscript extends this line of research to\nsystemsexperiencingtime-dependentnonequilibriumforces.\nTheauthorsderiveclosed-formexpressionsforboththeoptimalprotocolandtheassociated\nwork,demonstratingthatthetotalworknaturallydecomposesintothreedistinctcontributions:\n(i) work arising from initial disequilibrium, (ii) work required to drag the system under time-\naveraged forces, and (iii) work induced by temporal fluctuations in the forces. The analysis\nis illustrated through two compelling examples – a system driven by oscillating forces and an\nactive Brownian particle. The authors also discuss how measurement errors impact work ex-\ntractionefficiency.\nThe manuscript is exceptionally well-written and clear, the research is high quality, and the\nauthors provide insightful physical interpretations. Generally, I recommend publication, with\nonlyafewminorcomments.\nResponse: Wethankthereviewerforthesummaryofourworkandforthepositiveevalua-\ntionofthemanuscript.\nFirst, I think the authors should be a bit clearer regarding the restrictive assumptions under-\n1\nlying the theoretical results. Consistent with most optimal control studies in this field, the\nauthors assume that control is implemented exclusively through manipulation of a harmonic\ntrap with fixed width. In addition, they assume that nonequilibrium forces depend only on\ntimeandnotonposition. Whilerelaxingthislatterassumptionwouldlikelyprecludeanalytical\nsolutions (due to coupling between the forces and protocol), it is a substantial limitation that\nwarrants more prominent discussion. I would recommend that the authors emphasize these\nlimitationsanddiscussthe(im)possibilityofgeneralizingtheresultstospace-dependentforces.\nResponse: We thank the reviewer for pointing out the restrictive assumptions underlying\nour analysis. As the reviewer notes, relaxing the second assumption would couple the exter-\nnal forces and the control protocol, and potentially make the problem non-linear, precluding\nclosed-form analytical solutions. We have revised the manuscript to highlight this limitation\nmoreprominentlybyaddingtothefollowingintheMethodssection:\n\"We emphasize that it is assumed that these driving forces are spatially-homogeneous, as other sce-\nnarios would render the problem non-linear and analytically untractable. Yet, a wide range of non-\nequilibriumsystemscanbecapturedbytheabovemodel,aswedescribeinfuturesections.\"\nRelatedly, we draw attention to recent work by P. S. Pal et al. [arXiv:2504.09467], who stud-\niedenginesinspace-varyinghydrodynamicflows. Althoughtheirsettingdoesnotyieldoptimal\nprotocols, it illustrates possible directions for extending control strategies beyond the assump-\ntionsweadopthere. Asapartofthereviseddiscussionsection,wemention;\n\"Anintriguingavenueforfutureworkistheapplicationofsimilaroptimizationmethodstosettings\nwith spatially varying forces, such as linear hydrodynamic flows, where analytical progress may still be\nfeasible[69].\"\nSecond, Iamnotreallyconvincedbytheinterpretationoftheoptimalprotocolasan\"infor-\nmationengine\". Whiletheoptimalprotocolindeedexploitsknowledgeofthetemporalstructure\nof nonequilibrium forces, this information is provided a priori as input to the optimization al-\ngorithm. This differs from (my understanding of) an information engine, which should detect\nandrespondtostochasticfluctuationsinreal-timeduringtheprocess. That is, ingenuineinfor-\nmationengines,thedrivingprotocolcanvarydependingonthestochastictrajectoryrealizedby\nthe system. By contrast, the authors’ optimal protocol is predetermined through the optimiza-\ntionprocedurebeforeanytrajectoriesareobserved. Irecommendthattheauthorsde-emphasize\ntheirinterpretationoftheoptimalprotocolasaninformationengine,orexplainitinmoredepth,\ntoavoidconfusion.\nResponse: We thank the reviewer for the comment. In the revised manuscript, we make it\nclear that we are not suggesting that the protocols are genuine information engines, but rather\nthattherearesomeanalogiesworthexploring. Wehaveremovedthesentenceclaimingthatthe\nprotocolsactasautomaticinformationengines,andhaveaddedthefollowingdiscussion:\n\"Leveragingpriorknowledgeoftheforcesthatdrivethesystemawayfromequilibrium,Euler–Lagrange\nminimization gives rise to protocols that both anticipate and adapt to prescribed particle dynamics, en-\nablingthespontaneousextractionofmaximalenergyfromnon-equilibriumprocesses.\"\nWehavealsoremovedtheterm\"informationengine\"fromthesectiontitledealingwithpe-\n2\nriodicforces,aswellasfromtheabstractandconcludingdiscussion. Finally,wehavere-written\ntheparagraphcomparingtheoptimalprotocolwithinformationenginesinthecaseofperiodic\nforces,whichnowreads:\n\"Weseethatforsufficientlyslowprotocols,comparedtotheforcingperiod,theoptimalcontrolisable\ntoutilizetheoscillationssuchthatworkcanbeextracted. Theoptimalprotocolharnessesthedynamicin-\nformationavailableandautomaticallyextractsasmuchworkaspossible. Thisisaconsequenceofprecise\nknowledge of the force at all times. The protocol consists of repeatedly letting the force move the particle\nintoahigh-energystatebeforeshiftingthepotentialaccordinglytoextractthestoredenergyaswork. Let\nusrecallthatinBrownianinformationengines,feedbackisusedtorectifythermalnoiseandconvertmea-\nsurementinformationintowork. Insimpleenginedesigns,thepotentialhasthefreedomtomoveleftand\nrightbasedonthemeasurementoutcome,extractingworkisamannersimilartotheaboveprotocols[47].\nFigure(4)summarizesthemainmechanismbehindtheworkextractioninthecaseofperiodicforces.\"\nIdidnotfindanytyposbutthereisatypesettingissuenearline404.\nResponse: Wethank thereviewer forthe attention todetail. Thetypesettingerror hasbeen\ncorrectedintherevisedmanuscript,bysplittingtheequationintoseparatelines.\nReviewer2\nThisworkpresentsoptimalprotocolsforharvestingenergyfromaforcewhosemeanvalues\nateachtimeareknown,usingaparticleinaharmonicpotentialorequivalentfirst-orderdynam-\nicalsystem. Despiteafewlimitations,exploredbelow,Ithinkthismakesanicecontributionand\nyieldsanumberofinsightsthatcanguidefuturework.\nLet’s start with the strengths. The three-part decomposition into an information-based part\nandpartsusingtime-integratedmeanandvarianceoftheforceispretty. Thelastterminvolving\nthe integrated variance was surprising to me until I thought about it more, and I that’s enough\nof an advance to favor publication. The other two terms are more familiar. The first has\nbeen explored in roughly equivalent studies of information engines and based on principles found,\ne.g.,inParrondoetal.,[Ref. 30]andthesecondissimilartostudiesonoptimalnavigation(e.g.\nby the last author). Still, the deviation-based third term is a nice insight and sets one up to un-\nderstandtheapplicationexamples.\nResponse: Wethankthereviewerforthepositiveevaluationandforhighlightingtheclarity\nand novelty of our decomposition. We are especially pleased that the term involving the inte-\ngratedvariancewasfoundintriguing.\nOntheotherhand,therearemanylimitationstothestudy:\n1) Although Eq. 11 is a general result, almost all the discussion is couched in terms of qua-\nsistaticprotocols,wherethephysicalinterpretationisperhapsclearest. Butprobablyfasterpro-\ntocolsarebetter(ifnot,why?).\nResponse: We thank the reviewer for this comment. Slower (quasistatic) protocols indeed\nallow for the extraction of more work per cycle, but at the cost of longer execution times. In\ncontrast,finite-timeprotocolscouldbeadvantageousinsituationswhereoneaimstorepeatedly\nimplement the process, effectively achieving a higher rate of work extraction, even if the work\n3\nper cycle is somewhat reduced. This can equivalently be seen through thinking of the optimal\nprotocol as an engine, where fixing the duration t fixes the power (work per time) as the ratio\nf\nofworkoutputandduration. Sincetheprotocolsareoptimalforfixedt , wehereconsiderthe\nf\nmostefficientengines. Inthecontextofperiodicforces,asillustratedinFigure3,weprovidethe\nintuition that protocols must be allowed sufficient time to exploit the temporal structure of the\ndrivingforces tobeabletoextractwork.\n2)Theharvestingisthephysicists’version,wheretheenergyisn’treallystoredinanyuseful\nway but rather just amounts to “cooling the bath” This limits practical applications. I contrast\nthe scheme here with previous (experimental) work from the Roichman group (Admon et al.,\nPRL2018)wheretheenergywascaptured(byhavingaparticlemoveupstreaminaflowfield).\nResponse: We are aware of this important limitation. In our analysis, the extracted work\ncould be viewed as an upper bound, representing the maximal amount of energy that could\nin principle be harvested if one also devised a suitable scheme or device for storing the en-\nergy. Indeed,theresultswepresentcouldbeusedtotransportaparticleagainstabackground\nforce(likegravity),whilesimultaneouslyextractingworkfromthetime-dependentforces. Our\ncalculations therefore do not provide a ready-to-use energy harvesting protocol, but instead\ndemonstrates the fundamental limit of how much work could be made available under these\nconditions.\n3) The force is rarely known, and one might wonder how useful results based heavily on\nthisknowledgecanbe. Theauthorsdoaddressthisinseveralways: Forexample,inthesection\non periodic forcing with phase errors (not such a big deal) and frequency errors (a more seri-\nous problem). A systematic way to address such issues (measurements and then feedback) is\nhintedatinthesecondapplication, toactivematter, whichbreakswiththerestofthepaperby\nintroducingasinglemeasurementandfeedback. Onecanthinkofeachmeasurementasmaking\npossiblea“burst”ofharvestedenergythatdecaysexponentially(onehastopiecethistogether\nbysubstitutingtheexpressionforσ2 intoEq. 25). Theworkacquiredisfinite(notproportional\nϕ\nto τ ), which is stated but not really emphasized. Of course, with repeated measurement and\nR\nfeedback,onecouldaddressthisissue. Butthatisbeyondthescopeofthispaper.\nResponse: Weagreethatinrealisticsettings,theforcesarerarelyfullyknown. Intherevised\nmanuscript,wehaveaddedanewsection(finalsectionin\"Methods\"part)dealingwithresults\nthat bounds the error in predicted work extraction arising from inaccuracies in force inference.\nThiscomplementsourexistinganalysisofamplitudeandfrequencyerrorsandprovidesamore\nsystematicframeworkforunderstandingtheimpactofimperfectforceestimation.\nImportantly,thisanalysisformulizestheintuitivepictureofatradeoffbetweenthevariabil-\nityofthecontrolprotocolandtheriskoferror: erraticprotocolswithalargetotalvariationcan,\nin principle, exploit fine-scale details of the forces to extract more work, but they are also more\nsusceptibletomistakes. Conversely,smootherprotocolsreducethelikelihoodoferrorsbutmay\nleavesomeworkunextracted. Thisinsightprovidesguidancefortailoringcontrolstrategiesde-\npendingonwhetherthepriorityismaximizingworkextractionorminimizingthechanceerrors.\nFinally, we emphasize thatalsofordeterministic forces, aninitialmeasurement maystillbe\nnecessary. Forexample,inthecaseofperiodicforces,ameasurementmayberequiredtodeter-\nminethecurrentphaseandidentifytheoptimalstartingpointfortheprotocol.\nthe fundamental limitations of energy harvesting” — that seems too strong. Nonetheless, they\nhaveidentifiedsomelimitationsandhaveprovidedagreatdealofinsightinworkthatisquite\ncreative,andIwouldbeinfavorofitseventualpublicationinNat. Commun.\nResponse: We thank the reviewer for the comment. We agree that the original wording\nwas too strong, and we have revised the manuscript accordingly. In the revised manuscript,\nwehavereplaced\"Byuncoveringthefundamentallimitsofenergyharvestingfromdynamicenviron-\nments\"with\"Byidentifyingkeyconstraintsandprovidinginsightintotheperformanceandtradeoffsof\nwork-extraction protocols under realistic conditions\". We hope this revised wording better reflects\nthescopeofourwork.\nMinorPoints:\n1. [I think this is a minor point, but it could be more important.] Figure 1 talks about “ar-\nbitrary time-dependent forces F(t). (The Discussion contains similar language.) Is this really\nthe case? For example, could F be a step function? Could the sine wave example have abri-\ntrarily high frequencies? One imagines that for this case, there wouldn’t be much to harvest if\ntheexternalforceistoofastrelativetoτ . Whatevertheanswer,theissueneedsmorediscussion.\nR\nResponse: We thank the Reviewers for the comment. Regarding step-like forces, it is true\nthatweassumesmoothforcesthroughout,nowemphasizedinthemethodssectionbyusingthe\nsentence\"Anobject,underthecombinedeffectofa smoothexternaldrivingforce\"whenintroducing\ntheforces. Forthecaseofperiodicforces(seeforexampleFig. 3),weobservethattheperiodof\ntheforcesmustbesufficientlyshortrelativetotheprotocoldurationtoalloweffectiveworkex-\ntraction. Thisreflectstheneedfortheprotocoltoadapttothetemporalvariationsintheforces.\nInthelimitofverylongprotocols,Eq. (29)showsthattheextractedworkbecomesindependent\nof the periodicity of the forces. This can be understood directly from Eq. (15), where we see\nthat the work in this limit is proportional to the time-integrated variance of the forces. For a\nsimple periodic force, the variance is simply 1/2, without any period dependence. Intuitively,\nrapidforceoscillationscasesmallparticledisplacementandlessworkextraction,butalsoleads\nto more cycles of harvesting per unit time. We emphasize this in the revised manuscript by\nadding the following discussion in the section on periodic forces after noting that the work is\nindependentofτ :\np\n\"This results from a direct competition between two effects caused by changing τ ; firstly, it affects\np\ntheamplitudeoftheparticledisplacementsandhencehowmuchenergycouldbeharvestedpercycle. Sec-\nondly, it also changes how many cycles of harvesting can be completed in a given total duration t .\nFor\nf\nslowprotocols,theseeffectsbalance,resultinginEq. (29).\"\nEq. (32), however, shows that when there are measurement errors in the periodicity, the\nworkextractedgrowswiththeperiodoftheforce. Thus,thereviewer’sintuitioniscorrectand\nexcessivelyrapidforcesleadtoreducedworkextraction. Therevisedmanuscriptaddressesthis\nbyincluding inthesectiononperiodicforces:\n\"Itisalsoworthemphasizingthatwhenthereareerrorsintheperiodicity,thequasistaticworkextrac-\ntion becomes smaller when the periodicity τ is reduced. This reflects the fact that for very rapid forces,\np\nevensmallerrorswouldmakeworkextractionhard. \"\n5\n2. References54and55(preprints)coververysimilarground. Thereshouldbemorediscus-\nsioncomparingthatworkandthepresentpaper.\nResponse: In the revised manuscript, we mention the similarities and differences between\nourworkandthesereferences. Wehavealsoupdatedthereferencestotheirpublishedversions.\nIn the results section when analyzing work extraction from different active matter classes, we\nnowincludethefollowingdiscussion:\n\"Recently, repeating this scheme was proposed as a way of constructing active information engines\n[55,56]. In the case of one-dimensional run-and-tumble or active Ornstein-Uhlenbeck particles, we re-\ncover the results of [55], while our general results can also be used to study a wide range of other active\nmattermodels,alsoinhigherdimensions.\"\n3. Ref. 26waspublishedin2024asaPRXpaper.\nResponse: Thereferencehasbeenupdated.\n4. Some other papers might warrant inclusion and at least a brief discussion: i) A. Kolchin-\nsky et al., “Maximizing Free Energy Gain”, Entropy 2025. ii) L. Gammaitoni e tal., “Vibration\nEnergy Harvesting...” from the book “Sustainable Energy Harvesting Technologies” Ed. Yen\nKhengTan,2011. Thissecondisastand-inforalargegroupofpapersfromthe“classical”world\nof energy harvesting, which is a well-developed engineering topic that has also had some con-\ntributions by physicists. It would be nice if the paper had more contact with this part of the\nliterature.\nResponse: We thank the referee for this out. To clarify, the work by Gammaitoni\net al. (Ref 45) is cited in the section on oscillatory forces. We have re-written the introductory\ndiscussionthatsituatesourresultsrelativetothebodyofworkonclassicalharvestingmethods\nfromoscillatoryforces:\n\"Manyenergyharvestingsolutionsarebasedonperiodicforcesormotion,suchaswave-energycon-\nverters,wearablefabricsthatextractenergyfrommovement,andpiezoelectricgeneratorsthatcancharge\npacemakers through heartbeats [5,6,9,45,46]. These classical approaches typically utilize mechanical os-\ncillatorscoupledtotransducers,aimingtoconvertdisplacement,velocity,orstrainintoelectricalpower.\nHere we consider a simple microscopic analogy, which in contrast relies on optimal control to identify\nlimitsandoptimalstrategiesforenergyharvesting,independentofanyspecifictransducerdesign.\"\nSimilarly, while the work by Kolchinsky et al. was already mentioned in our concluding\ndiscussion,wherewenowmentionthat\n\"More generally, bounding the possible work extraction is critical for developing energy-efficient,\nadaptive control strategies for micro- and nanoscale technologies. Recent studies [70,71] have explored\nthis problem from multiple perspectives, including the optimization of reactions in Markovian systems\nunder topological, kinetic, and thermodynamic constraints, and the role of state preparation in setting\nboundsonextractablework.\"\n5. Some of the paper exposition could be clearer. I did not see if it is stated that W < 0\ncorrespondstoworkextraction[ifso,itisabitburied].\nversion. Whendefiningthework,wenowstate:\n\"Weusetheconventionthatapositiveworkimpliesenergyhastobepaid,whilenegativeworkcorre-\nspondstoenergyextracted.\"\n6. Similarly,Eq. 12wouldbecleareriftheunderbraceterms(W ,W ,W )wereimmediately\ni ta d\ndefined. For example, W is defined only in the caption of Fig. 2, which is one page after. In a\ni\nfirst reading, I thought from Line258 that the “i” inW was “initial”, which led to someconfu-\ni\nsion.\nResponse: We thank thereviewer for raising thispoint. Inthe revised manuscript, we now\ndefinethesetermsmoreexplicitly,stating:\n\"Takingtheslowlimit(t → ∞)ofEq. (11)wefind\nf\nW = W +W +W , (R1)\nqs i ta d\nwheretheworkhasbeendecomposedintothreeparts;aninformationtheoreticcontribution,acontribution\nfrom time-averaged forces, and a contribution from forces deviating from its time-average, respectively\ntakingtheform\ni i i\nta f i f\nt→∞\nf\nd 4γ t→∞ f f\nf\n7. Eq. 15: Define better the space B of Boltzmann states. Is it the set of Boltzmann distributions for fixed\ntemperature (the bath) but variable lambda? Related: The authors should show (perhapsinthesupplement)howtoderiveEq. 16. Givensupplements, thereseemsnoneedto\nskipsteps.\nResponse: We thank thereviewer for raising thispoint. Inthe revised manuscript, we now\ndefinethesetermsmoreexplicitly,stating:\n\"Thisoperationprojectstheinitialdistributionp (x)ontothespaceB = {Z−1e−βV[x,µ],µ ∈ R}of\ni\nBoltzmannstatescompatiblewiththetrapmanipulation—here,fixed-variance(i.e.,constantβ)Gaussian\ndensitieswithvariablelocationµ(oftencalledshiftmeasures).\"\nWehavealsoprovidedadetailedderivationoftheresultk TD (π[p ] ∥ p ) = 1k(λ −q )2\nB KL i eq 2 i i\nintheSupplementalInformation,underthename\"DerivationofEq. (19)\"intherevisedversion.\nReviewer3\nTheauthorspresentadetailedtheoreticalinvestigationfocusingonoptimalworkextraction\nthroughoptimalcontrolprotocolsappliedtosingle-particlesystemsdrivenbytime-dependent\nexternalforces. Whiletheequilibriumcomponentoftheseoptimalprotocols(Eq.7inthemanuscript)\nis well-established in the literature, the authors introduce an explicit analytical form for the\n7\nnonequilibrium correction (Eq.8), which represents the main theoretical contribution of this\nwork. Two concrete examples, periodic external driving forces (Case I) and active Brownian\nparticles(CaseII),arealsoprovided. Theseexamplesclearlyillustratethederivedoptimalpro-\ntocolsandwouldbeofinteresttospecialistsworkingonstochasticthermodynamics.\nFrom a technical perspective, the manuscript appears sound and clearly presented. How-\never, I find that the overall novelty and broad interdisciplinary appeal is somewhat lacking.\nSpecifically, the optimal control approach to work extraction in stochastic thermodynamics is\nalready a well-explored topic at least theoretically. Seminal theoretical studies, such as those\nby Schmiedl and Seifert (PRL, 2007)[19], Sivak and Crooks (PRL, 2012), and more recent works\nincludingBlaberandSivak(J.Phys. Comm.,2023),haveextensivelyaddressedthistopic. Thus,\nwhile Eq.8 is theoretically novel and interesting, its overall conceptual contribution appears\nrather incremental. Furthermore, there is already an experimental demonstration of optimal\ncontrol, Loos et al. (PRX, 2024)[26], and theorefore the purely theoretical nature of the present\nmanuscriptwouldlimititsimpactpotentialforajournaltargetingabroadinterdisciplinaryau-\ndience.\nResponse: We thank the reviewer for the positive evaluation of our work as technically\nsound and clearly presented. We also appreciate the constructive comments regarding novelty\nand scope. We would like to clarify that the primary aim of this manuscript is not simply to\npresent another optimal protocol. Rather, we seek to highlight a shift in perspective: optimal\ncontrol can be leveraged not only to minimize dissipation or implement control at lower ener-\ngetic cost, but also to actively extract useful work from non-equilibrium driving. In this way,\nourstudyreframestheroleofoptimalcontrolwithinstochasticthermodynamics.\nIn the revised version of the manuscript, we have made substantial additions to emphasize\nthis perspective. In particular, we now include a new section that quantifies the errors associ-\natedwithforceinferenceandexplorestheresultingtradeoffsbetweenprecisionandreliabilityin\nworkextraction. Theseadditionsfurtherunderscorethebroaderconceptualimplicationsofour\nframeworkbeyondthederivationofasingleprotocol,aswellasmakingitmoreexperimentally\nrelevant.\nAdditionally,theauthors’characterizationoftheirresultsasdemonstratingan\"information\nengine\"is,inmyview,notentirelyconvincing. Traditionalinformationenginesexplicitlyincor-\nporatemeasurementandfeedbackcontroltoextractwork. Whiletheauthorsacknowledgethe\nabsenceofexplicitmeasurement-feedbackloops,theirinterpretationof\"information\"isprimar-\nilyrelevanttoinitialnonequilibriumstatesandpreciseknowledgeofexternalforces. Thus, the\nuseoftheterm\"informationengine\"could beconsideredsomewhatmisleading.\nResponse: Weappreciatethereviewer’sthoughtfulcommentregardingourcharacterization\noftheresultsasan“informationengine.”Asnotedbytheotherreviewersaswell,werecognize\nthat our use of the term may be misleading, since traditional information engines explicitly\ninvolve measurement and feedback. To avoid confusion, we have revised the manuscript\naccordingly(see ourdetailed responseto Reviewer 1, clarifying thedistinction andsoftening our\nuseoftheterm“informationengine”whendiscussingourprotocols.\nIn summary, although this manuscript presents theoretical results that will be of interest to\nspecialists, its contributions appear somewhat incremental within a field already rich in the-\noretical insights. Therefore, while this study would be well suited for publication in a more\n8\nspecializedjournal,Ibelievethatitdoesnotmeetthethresholdofbroadimpactandsignificant\ninnovationrequiredforpublicationinNatureCommunications.\nResponse: Wethankthereviewerfortheircarefulevaluation. Wewouldliketoclarifyafew\npointsinresponsetotheconcernaboutnoveltyandimpact:\n• Whileitistruethattheunderlyingmathematicaltools(e.g.,Euler–Lagrangeminimization)\nare classical, the perspective and context we develop here are new. In particular, only in\nveryfewcaseshaveoptimalcontrolprotocolsbeenusednotmerelytoreducedissipation,\nbut to actively extract energy from nonequilibrium systems. Our work highlights and\nformalizesthisperspective,whichisafieldofgrowinginterest.\n• Followingthereviewers’feedback,wehavesignificantlyextendedthemanuscript. These\nadditionsbroadenthescopeandstrengthentheconceptualcontributionofthestudy.\n• Webelievethetargetaudienceisbroad. Theworkliesattheintersectionofseveralfields,\nwithrelevancetothermodynamicsandstatisticalphysics,controltheoryandappliedmath-\nematics, biological systems such as molecular motors that optimally move according to\nnon-equilibriumfluctuations,andengineeringapplicationsincludingroboticsandmicro-\ndevices. Bybridgingtheseareas,themanuscriptspeakstoaninterdisciplinaryreadership\nbeyondaspecialistaudience,makingNatureCommunicationsanidealarena.\nWehopethattheclarificationsandrevisionswehavemadewillleadthereviewertoreconsider\nthemanuscript’snovelty,interdisciplinaryappeal,andsuitabilityforpublication.\n(NCOMMS-25-35338)\nDearEditorsandreferees,\nWethankthereviewersforcarefullyreadingtherevisedmanuscriptandforthefinalcomments.\nAllminorsuggestionshasbeenincorporatedintherevisedmanuscript.\nBestregards,\nKristianStølevikOlsen,RémiGoerlich,YaelRoichmanandHartmutLöwen.\nReviewer1\nTheauthorshaveaddressedmyconcerns. Irecommendthemanuscriptforpublication.\nResponse: Wethankthereviewerforthepositiveevaluation.\nReviewer2\nI think the authors have done a good job in responding to my concerns and (as far as I can\ntell) those of the other reviewers. I particularly appreciated the new section on “Precision vs.\nrisk”,withitsdiscussionoftradeoffs.\nRegardingthegeneralquestionofsignificance,itisimportanttobefair. Thatis,theoriginal\nversionofthepaperwascriticizedforhavingmadeclaimsthataretoostrong. Theclaimshave\nhere been moderated to an appropriate level. But it would be a mistake to go too far the other\nwayanddismisstheinsightspresented: astheauthorsstate,theliteratureonoptimalcontrolof\nstochasticthermodynamicprocesseshasbeenfocusedinrecentyearsonminimizingdissipation.\nHere, the authors make a good case for using that formalism to discuss energy extraction, and\nthen they establish first results that will lead to a program of future research on nonlinearities,\nfasterprotocols,etc. Thiscontrastswithcurrent“practical”effortsforenergyharvesting,which\nare strongly linked to the idea of resonant structures interacting with known periodic forces.\nAlthough such a strategy overlaps at least a bit with “Case I”, it does not work well in other\nsituationssuchas“CaseII”. Becauseofthisenlargedperspective, Ithinkthepaperdeservesto\nbeinNatureCommunications.\nResponse: We are grateful for the positive evaluation, and for highlighting the novelty of\nour work. We are glad that the reviewers shares our conviction that energy extraction is an in-\nterestingdirectioninoptimalthermodynamiccontrol.\nMinorcorrections:\nLine231: addref. toSMSec. 1\nLine235: quasistatic(typo)\nLine361: protocol’s(typo)\nResponse: ThereferencetotheSupplementaryInformationhasbeenadded,andtyposhave\nbeencorrected.\nTheauthorshaveaddressedtheconcernsraisedinmypreviousreport. Inparticular,themean-\ning of \"information engine\" is now clearer, and the manuscript has been revised satisfactorily.\nNevertheless,Ihaven’tbeenfullyconvincedaboutthelevelofconceptualnovelty. Forinstance,\nin their response the authors state that \"In particular, only in very few cases have optimal con-\ntrol protocols been used not merely to reduce dissipation, but to actively extract energy from\nnonequilibriumsystems.\"However,whennonconservativeforcesareabsent,reducingdissipa-\ntion is exactly equivalent to enhancing work extraction. Thus, the novelty of this study ap-\npearstolieintheinclusionofexternaldriving(\"active\")force,whichishoweverrestrictedtobe\nspatially homogeneous. The remaining question is whether this addition constitutes sufficient\nconceptual novelty for Nature Communications. Considering the other reports, my overall as-\nsessment is marginal: I do not offer a strong recommendation for publication, but I would not\nopposeacceptanceiftheotherrefereesrecommend.\nResponse: Wearegladthatthereviewthinkswehaveaddressedtheconcernsintheprevi-\nous report, and thank the reviewer again for pushing us to clarify the conceptual issue relating\ntoinformationengines.\n2"}]

Decision letter