Experimental Realization of a Minimal Microscopic Heat Engine
Aykut Argun, Jalpa Soni, Lennart Dabelow, Stefano Bo, Giuseppe Pesce, Ralf Eichhorn, Giovanni Volpe
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Experimental Realization of a Minimal Microscopic Heat Engine
Aykut Argun, ∗ Jalpa Soni, ∗ Lennart Dabelow, StefanoBo, Giuseppe Pesce, Ralf Eichhorn, and Giovanni Volpe Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden, EU Theoretisch-Physikalisches Institut, Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany, EU Nordita, Royal Institute of Technology and Stockholm University, SE-10691 Stockholm, Sweden, EU Department of Physics E. Pancini, University of Naples Federico II, via Cintia, 80126-I, Naples, Italy, EU (Dated: October 10, 2018)Microscopic heat engines are microscale systems that convert energy flows between heat reservoirsinto work or systematic motion. We have experimentally realized a minimal microscopic heat engine.It consists of a colloidal Brownian particle optically trapped in an elliptical potential well andsimultaneously coupled to two heat baths at different temperatures acting along perpendiculardirections. For a generic arrangement of the principal directions of the baths and the potential,the symmetry of the system is broken, such that the heat flow drives a systematic gyrating motionof the particle around the potential minimum. Using the experimentally measured trajectories, wequantify the gyrating motion of the particle, the resulting torque that it exerts on the potential,and the associated heat flow between the heat baths. We find excellent agreement between theexperimental results and the theoretical predictions.
During the last two decades, the rapid development ofstochastic thermodynamics has provided scientists witha framework to explore the properties of nonequilibriumphenomena in microscopic systems where fluctuationsplay a prominent role [1–8]. The advancement of experi-mental techniques (in particular optical trapping and dig-ital video microscopy [9]) has made it possible to experi-mentally study thermodynamics at the single-trajectorylevel [10–13]. These tools have been applied, e.g., to in-vestigate the performances of molecular machines [6, 14].Furthermore, microscopic heat engines (i.e. artificial mi-croscopic systems that extract heat from the surroundingthermal bath(s) and turn it into useful work or systematicmotion) have been proposed theoretically[15–22] and re-alized experimentally [23–29], providing insights on fun-damental aspects of non-equilibrium thermodynamics.In this Article, we experimentally realize and investi-gate a minimal microscopic engine constituted of a Brow-nian particle held by a generic potential well and simul-taneously coupled to two heat baths at different tem-peratures acting along perpendicular directions so that anon-equilibrium steady state is maintained. For a genericarrangement of the principal directions of the baths andthe potential, the symmetry of the system is broken, suchthat the heat flow between the two heat baths drives asystematic gyrating motion of the particle around thepotential minimum. Originally, this engine was proposedtheoretically by Filliger and Reimann [17]; it is consid-ered to be minimal because of its intrinsic simplicity, yetgenerating a torque via circular motion, and because itworks autonomously in permanent simultaneous contactwith two heat baths (i.e., without the need for an exter-nal driving protocol).In the experiment, we use a single colloidal particlesuspended in aqueous solution at room temperature and ∗ These two authors contributed equally trap it in an elliptical optical potential. The per se isotropic thermal environment is rendered anisotropic byapplying fluctuating electric signals with an almost whitefrequency spectrum along a specific direction; such tech-niques [30–32] and similar ones [33, 34] have recentlybeen demonstrated to generate in excellent approxima-tion high temperature thermal noise with negligible fric-tion effects. In addition to experimentally confirming theprediction of Ref. [17] for the torque (see eq. (7)), we char-acterize the gyrating motion of the colloid in more detailby measuring the cross-correlation between the spatialcoordinates. Moreover, we analyze the energy exchangesbetween the two heat baths mediated by the particle’smotion, using the tools of stochastic energetics [1, 35].Theoretically, we model the motion of the Brownianparticle using overdamped Langevin equations in two di-mensions [17]: γ ˙ x = − ∂∂x U ( x, y ) + p γk B T x ξ x ( t ) ,γ ˙ y = − ∂∂y U ( x, y ) + p γk B T y ξ y ( t ) . (1)The particle’s motion is confined by an elliptical har-monic potential U ( x, y ) with stiffnesses k x ′ and k y ′ alongits principal axes x ′ and y ′ , which are rotated by an angle θ with respect to the coordinate axes x and y (Fig. 1): U ( x, y ) = 12 (cid:2) x y (cid:3) R ( − θ ) k R ( θ ) (cid:20) xy (cid:21) , (2)where R ( θ ) = (cid:20) cos θ sin θ − sin θ cos θ (cid:21) and k = (cid:20) k x ′ k y ′ (cid:21) .The coordinate axes x and y are aligned with the di-rections of the anisotropic temperatures T x and T y , suchthat the angle θ provides a means to control the sym-metry breaking between clockwise and counter-clockwiseorientation. The corresponding thermal fluctuations aremodeled by mutually independent Gaussian white noisesources ξ x ( t ) and ξ y ( t ) with h ξ x ( t ) i = h ξ y ( t ) i = 0 and h ξ x ( t ) ξ x ( t ′ ) i = h ξ y ( t ) ξ y ( t ′ ) i = δ ( t − t ′ ). In the following, T y is equal to the temperature of the aqueous solution,i.e. room temperature T y = 292 K, while T x is eitherroom temperature or hotter due to the effective heatingfrom the electric noise signals [30–32]. The viscous fric-tion in Eq. (1) is given by the isotropic Stokes coefficient γ = 6 πνR , where ν is the viscosity of the watery solutionand R the particle radius.Experimentally, we use polystyrene particles with di-ameter 2 R = 1 . µ m (Microparticles GmbH) held ina potential generated using an optical tweezers [9]: wefocus a laser beam (wavelength λ = 532 nm) using ahigh-numerical aperture objective (60 × , NA 1.40), whilewe introduce the ellipticity in the potential by alter-ing the intensity profile of the laser beam using a spa-tial light modulator (PLUTO-VIS, Holoeye GmbH). Wetrack the position of the particle at 400 fps by digitalvideo microscopy using the radial symmetry algorithm[36]. The values of the optical trapping stiffnesses, k x ′ =1 .
63 pN /µ m and k y ′ = 0 .
86 pN /µ m, are measured fromthe acquired particle trajectories by using the equiparti-tion method and the autocorrelation methods [9] in theabsence of electric noise (i.e. when T x = T y = 292 K).The Figures 1(a)-(c) show the experimental equilibriumprobability density p ss ( x, y ) of the particle in the ellipti-cal trap with θ = 0 (Fig. 1(a)), θ = π/ θ = − π/ T x = T y = 292 K are bothequal to room temperature.We can now establish a non-equilibrium steady stateby introducing different temperatures along the x - and y -directions. Due to the colloidal particle being electri-cally charged in solution, a randomly oscillating field ap-plied along the x -direction produces a fluctuating elec-trophoretic force on the particle, which increases its ran-dom fluctuations along the x -direction, leading to aneffective increase of the temperature [30]. The electricfield is generated by driving with an electric white noisetwo parallel thin wires (gold, diameter 30 µ m) placed oneither side of the optical trap at a distance of 1 mm.The effective temperature along x is then proportionalto the variance of the particle position along the x -direction when the principal axes of the optical trap arealigned with the Cartesian axes x and y . Figures 1(d)-(f)present the resulting stationary probability distributionsfor the cases θ = 0 (Fig. 1(d)), θ = π/ θ = − π/ T x = 1750 K and T y = 292 K:they are elongated along the x -direction (in comparisonwith Figs. 1(a)-(c)), because of the presence of the ex-tra noise. Furthermore, we can measure the stationaryprobability density current according to [6] (cid:20) J x ( x, y ) J y ( x, y ) (cid:21) = "(cid:28) x ( t + ∆ t ) − x ( t ) y ( t + ∆ t ) − y ( t ) (cid:29) x ( t )= x, y ( t )= y + (cid:28) x ( t ) − x ( t − ∆ t ) y ( t ) − y ( t − ∆ t ) (cid:29) x ( t )= x, y ( t )= y p ss ( x, y )2∆ t , (3) where the averages are taken over all particle displace-ments during a sampling time interval ∆ t , which start(first line) or end (second line) at position (cid:20) xy (cid:21) . Thiscurrent is represented by the blue arrows in Figs. 1(e)-(f) and clearly indicates the presence of a gyrating mo-tion; the strength and the direction of this rotational mo-tion depend on the rotary asymmetry induced by θ and,importantly, they vanish for θ = 0 (Fig. 1(d)), becausethe principal directions of the baths and of the potentialare aligned and therefore there is no symmetry breaking.Note also that there is essentially no flux in Figs. 1(a)-(c),as expected at thermal equilibrium.In order to quantify this rotational behavior, we calcu-late the differential cross correlation function between x and y [37, 38]: D ( t ) = h x ( t ∗ ) y ( t ∗ + t ) i − h y ( t ∗ ) x ( t ∗ + t ) i = h r ( t ∗ ) r ( t ∗ + t ) sin( φ ( t ∗ + t ) − φ ( t ∗ )) i , (4)where the angular brackets indicate the average over thesteady-state distribution, for which D ( t ) is independentof the reference time point t ∗ . Its representation in thesecond line using polar coordinates r = p x + y and φ = arctan( y/x ) illustrates that it vanishes if there isno net motion of the colloid and that it is positive (nega-tive) for counter-clockwise (clockwise) net gyrating move-ments.Since the model described by Eq. (1) can be solvedanalytically, we can calculate an exact closed expressionfor D ( t ) (see Appendix A for details), D ( t ) = sign( t ) k B ( T x − T y ) e − | t | kx ′ γ − e − | t | ky ′ γ k x ′ + k y ′ sin(2 θ ) . (5)Experimentally, D ( t ) can be directly evaluated from therecorded trajectories without explicit knowledge of thetrap parameters and temperatures [37, 38], using the ex-pression in Eq. (4). Figure 2 presents the experimen-tal D ( t ) (red symbols) for different values of θ and tem-perature anisotropy, which are in good agreement withthe theoretical predictions from Eq. (5) (black lines): D ( t ) vanishes when θ = 0 and is maximized when θ = ± π/ D ( t ) increases as the tem-perature anisotropy increases (Fig. 2(b)).The rotational motion of the particle around the origincan also be measured by studying its weighted angular ve-locity r d φ/ d t . Its average is proportional to the strength M of the average torque h x ( ∂U/∂y ) − y ( ∂U/∂x ) i exertedby the particle on the potential U [17, 37, 38]: M = − γ (cid:28) r dφdt (cid:29) , (6)This expression provides a way of computing the aver-age torque directly from the recorded trajectories with-out explicit knowledge of the trap parameters and tem-peratures. On the other hand, an analytical prediction FIG. 1.
Brownian colloid in an elliptical potential. (a-c) Experimental steady-state probability distributions of aBrownian particle for isotropic temperature ( T x = T y = 292 K) inside an elliptical potential ( k x ′ = 1 .
63 pN /µ m, k y ′ =0 .
86 pN /µ m) (a) with its principal axes x ′ and y ′ aligned with the Cartesian coordinates x and y ( θ = 0), (b) with θ = π/
4, and(c) with θ = − π/
4. The probability densities are pictured by scatter plots of experimentally measured particle positions, darkerregions corresponding to higher densities. (d-f) Experimental steady-state probability distributions for anisotropic temperature( T x = 1750 K > T y = 292 K) (d) with θ = 0, (e) θ = π/
4, and (f) with θ = − π/
4. (e-f) The blue arrows represent the associatedprobability flux (Eq. (3)): when the principle axes of the elliptical potential and the anisotropic thermal environment arerotated which respect to each other to break rotational symmetry, there appears a rotational flux component whose directiondepends on the sign of θ . Note that there is no net flux when (a)-(c) the system is at thermal equilibrium, and when (d) theaxes of the anisotropic temperature and the potential are aligned ( θ = 0). for the torque as a function of precisely these parametersis again obtained from the exact solution of Eq. (1) [17](see Appendix A for details), γ (cid:28) r d φ d t (cid:29) = − M = − k B ( T x − T y ) k x ′ − k y ′ k x ′ + k y ′ sin(2 θ ) . (7)By the independent measurement of k x ′ , k y ′ , T x , and T y ,we can compare this prediction with the experimentalmeasurements without any fit parameter. The symbolsin Fig. 3 represent the experimentally measured torques,which are indeed in very good agreement with theoreticalpredictions from Eq. (7) (solid lines). When evaluatingthe torque from the experimental data, we used an esti-mator which is exact to first order in the sampling timestep (as opposed to the zeroth order naive estimator) inorder to obtain an accurate value despite the relativelylarge experimental value ∆ t = 2 . θ = 0 (Fig. 3(a))and when T x = T y (Fig. 3(b)), increases as θ approaches π/ T x − T y .The presence of a systematic rotational motion of theparticle is connected to a transfer of heat from the hot tothe cold bath. Following Sekimoto’s stochastic energeticsapproach [1], we identify heat with the work performedby the dissipating and thermally fluctuating forces, sothat the heat absorbed by the particle from the hot reser-voir at temperature T x along the trajectory [ x ( t ) , y ( t )]reads Q x ( τ ) = Z τ h − γ ˙ x ( t ) + p k B T x γ ξ x ( t ) i ◦ d x ( t ) , (8)where ◦ denotes the Stratonovich product. Using the -2000-1000010002000 -0.1 -0.05 0 0.05 0.1-2000-1000010002000 -0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1 FIG. 2.
Cross-correlation functions D ( t ) . (a) D ( t ) as a function of the relative orientation θ between the axes of thetemperature anisotropy ( T x ≡ . T y ) and those of the potential; it is maximized for θ = π/
4. (b) D ( t ) as a function ofthe temperature anisotropy ( θ ≡ π/ T x . The red symbols represent the experimental data (theshaded area is the standard deviation) and the solid black lines represent the corresponding theory (Eq. (5)). The insets showschematically the alignment between the axes of the temperature anisotropy (arrows) and those of the potential (ellipses),the temperature along the y -axis is color-coded in blue indicating the “cold” direction T y = 292 K, while the temperature in x -direction is indicated in blue if T x = T y = 292 K and red if it corresponds to the “hot” direction with T x > T y . -0.2 -0.1 0 0.1 0.2 -6-4-20246 0 500 1000 150001234567 FIG. 3.
Torque as a function of (a) the relative orientation θ between the axes of the temperature anisotropy ( T x ≡ . T y )and those of the potential, and (b) as a function of the tem-perature difference T x − T y ( θ ≡ π/ equations of motion (1), this can be rewritten as Q x ( τ ) = Z τ ∂∂x U ( x ( t ) , y ( t )) ◦ d x ( t ) . (9)This equation expresses the heat flow from the hot reser-voir to the colloidal particle entirely by means of exper-imentally accessible quantities, i.e. x ( t ), y ( t ), γ , k x ′ , k y ′ , and θ . In the stationary state, the average heatabsorbed along trajectories divided by the observation FIG. 4.
Heat flow between the two baths as a functionof (a) the relative orientation θ between the axes of the tem-perature anisotropy ( T x = 6 . T y ) and those of the potential,and (b) as a function of the temperature difference T x − T y ( θ = π/ Q x and ˙ Q y , respectively (cor-responding to 5 trajectories of 50 s, the error bars indicatestandard deviations) and the solid lines represent the theo-retical predictions given by Eqs. (10) and (11). time h ˙ Q x i = h Q x ( τ ) i /τ is a constant independent of thelength τ of the trajectory. This average heat flow can becalculated analytically as (see Appendix A for derivation) h ˙ Q x i = k B ( T x − T y )4 γ ( k x ′ + k y ′ ) [( k x ′ − k y ′ ) sin(2 θ )] . (10)An analogous formula holds for the heat absorbed fromthe cold reservoir at temperature T y , h ˙ Q y i = −h ˙ Q x i . (11)In Fig. 4, we present the experimentally measured heatflows from the cold reservoir (blue circles) and from thehot reservoir (red squares) to the particle as a functionof θ (Fig. 4a) and T x − T y (Fig. 4b). As in the case ofthe torque, also when evaluating the heat flow from theexperimental data according to Eq. (9), as in the caseof the torque, we used an estimator which is accurate tofirst order in the sampling time step ∆ t (see Appendix Bfor details). These experimental results are in very goodagreement with the theoretical predictions (10) and (11)(solid lines). The average direction of the heat flow isalways from the hot to the cold reservoir; its intensityvanishes as θ → T x → T y , and increases as θ → π/ T x increases. In the current setup this heat flow isturned into systematic motion, but is not used to performwork against an external load, such that efficiency as theratio between work performed and heat taken up fromthe hotter reservoir cannot be defined.In conclusion, we have presented an experimental re-alization of a microscopic heat engine employing a sin-gle colloidal particle moving in a generic elliptical op-tical trap while in simultaneous contact with two heatreservoirs. This experimental model features a minimaldegree of complexity necessary to obtain a microscopic,circularly operating heat engine generating a torque fromwhich work can in principle be extracted [17]. Further-more, it has the advantage of being completely solvableanalytically, therefore providing an ideal testbed to com-pare theory and experiments.We finally point out a very recent interesting experi-mental work [39], which studies a physically completelydifferent but mathematically equivalent system, namelytwo capacitively coupled resistor-capacitor circuits whosedynamical equations for the two voltages can be mappedto the model described by Eq. (1). ACKNOWLEDGMENTS
All authors aknowledge useful discussion with themembers of Yellow Thermodynamics and with JanWehr. This work was partially supported by the Eu-ropean Research Council ERC Starting Grant Com-plexSwimmers (grant number 677511) and by the MarieSklodowska-Curie Individual Fellowship ActiveMotion3D(grant number 745823). RE and SB acknowledge finan-cial support from the Swedish Research Council (Veten-skapsr˚adet) under the grants No. 621-2013-3956, No. 638-2013-9243 and No. 2016-05412. LD acknowledges finan-cial support by the Stiftung der Deutschen Wirtschaft.
Appendix A: Solution of the model1. Dynamics
Starting from the Eqs. (1) and compactifying notation,the overdamped equations of motion read:˙ r ( t ) = − Ar ( t ) + Bξ ( t ) , (A1)with r = (cid:20) r r (cid:21) ≡ (cid:20) xy (cid:21) , ξ ( t ) = (cid:20) ξ ( t ) ξ ( t ) (cid:21) ≡ (cid:20) ξ x ( t ) ξ y ( t ) (cid:21) , A = 1 γ R ( − θ ) k R ( θ ) , (A2)and B = (cid:20)p k B T x /γ p γk B T y /γ (cid:21) . (A3)Equation (A1) is the stochastic differential equation(SDE) of a general Ornstein-Uhlenbeck process and isequivalent to a Fokker-Planck equation [40, 41] for thetransition probabilities, or propagator, p ( t, r | t , r ), ∂ t p ( t, r | t , r ) = X i,j A ij ∂ r i [ r j p ( t, r | t , r )]+ D ij ∂ r i ∂ r j p ( t, r | t , r ) (A4)with the diffusion matrix D = 12 BB T = (cid:20) k B T x /γ k B T y /γ (cid:21) . (A5)The propagator gives the probability to find the particlein an infinitesimal volume element d r around r at time t given that it was at r at an earlier time t . Since thesystem is Markovian, its statistics are fully determinedby p and some initial distribution p ( r ). The Fokker-Planck equation (A4) can be solved exactly [40, 41] andthe resulting propagator is p ( t, r | t , r ) = e − [ r − e − ( t − t A r ] T Σ − ( t − t ) [ r − e − ( t − t A r ] p (2 π ) det Σ ( t − t ) , (A6)where the covariance matrix is Σ ( t ) = Σ ( ∞ ) − e − t A Σ ( ∞ ) e − t A T (A7)and Σ ( ∞ ) is obtained as the solution of the matrix equa-tion A Σ ( ∞ ) + Σ ( ∞ ) A T = 2 D . (A8)One can see that r ( t ) is a time-homogeneous Gaussianprocess. For our system, we find Σ ( ∞ ) = 1tr A det A (cid:20) D A + D ( A + det A ) − D A A − D A A − D A A − D A A D A + D ( A + det A ) (cid:21) (A9)where D i = k B T i /γ are the diagonal entries of the matrix D , det A = k x ′ k y ′ /γ and Tr( A ) = ( k x ′ + k y ′ ) /γ .
2. Steady state
From the solution (A6) and the positive definiteness of A , we understand that the system reaches a steady statein the limit t → ∞ , whose distribution is p ss ( r ) = e − r T Σ − ( ∞ ) r p (2 π ) det Σ ( ∞ ) . (A10)The characteristic relaxation time to the steady state is τ ∞ = (det A ) − / = γ/ p k x ′ k y ′ , which is approximately16 ms for our experiments.
3. Correlations
At the steady state, the autocorrelation matrix C ( t )(with entries C ij ( t ) = h r i ( t ∗ + t ) r j ( t ∗ ) i ) becomes inde-pendent of the reference time t ∗ , and, using (A6), can becomputed as C ( t ) = ( e − t A Σ ( ∞ ) t ≥ Σ ( ∞ ) e −| t | A T t < . (A11)A measure of the non-equilibrium state of the system andthe particle’s rotational motion is the asymmetry in thecorrelation function, i.e., the differential cross correlationfunction: D ( t ) = h x ( t ∗ ) y ( t ∗ + t ) i − h y ( t ∗ ) x ( t ∗ + t ) i = h r ( t ∗ ) r ( t ∗ + t ) i − h r ( t ∗ ) r ( t ∗ + t ) i = C ( t ) − C ( t ) . (A12)Plugging in (A11), we obtain formula (5) above.
4. Torque
The rotational motion of the particle around the ori-gin can also be assessed by studying its weighted angularvelocity r dφ/dt , a quantity reminiscent of angular mo-mentum. Using the equations of motion (1), we find thatits average h r dφ/dt i = h x ˙ y − y ˙ x i is related to the av-erage torque M exerted on the potential (introduced byFilliger and Reimann [17] and in Eq. (6) above): − γ (cid:28) r dφdt (cid:29) = M = (cid:28) x ∂U∂y − y ∂U∂x (cid:29) = k B ( T x − T y )( k x ′ − k y ′ ) sin(2 θ ) k x ′ + k y ′ . (A13)The left-hand representation provides a way of comput-ing the average torque directly from the trajectories andindependently of the trap parameters and temperatures.
5. Heat absorbed along a trajectory
The equations of motion (A1) specify several forcesacting on the particle, each of which can be associatedwith a physical component of the system. In particular,we have the potential force F U = − ∇ U as well as forceslinked to the interaction with the medium and reservoirs,namely the frictional force F diss = − γ ˙ r and the thermalfluctuations F therm = γ Bξ ( t ). The equations of motionmerely state the balance of these forces.Following Sekimoto’s stochastic energetics approach,we identify heat with the work performed by the dissi-pating and thermally fluctuating forces so that the heatabsorbed by the particle can be calculated using Eqs. (8)and (9). Explicitly, the heat flowing from the reservoirsto the system along a trajectory r ( t ) reads Q x = Z τ h − γ ˙ x ( t ) + p γk B T x ξ x ( t ) i ◦ d x ( t )= (cid:0) k x ′ cos( θ ) + k y ′ sin( θ ) (cid:1) x ( τ ) − x (0)2+ ( k x ′ − k y ′ ) sin( θ ) cos( θ ) Z τ y ( t ) ◦ d x ( t ) (A14)and Q y = Z τ h − γ ˙ y ( t ) + p γk B T y ξ y ( t ) i ◦ d y ( t )= (cid:0) k x ′ sin( θ ) + k y ′ cos( θ ) (cid:1) y ( τ ) − y (0)2+ ( k x ′ − k y ′ ) sin( θ ) cos( θ ) Z τ x ( t ) ◦ d y ( t ) , (A15)where the second equality in both relations follows usingthe equations of motion (1). These relations form thebasis for evaluating the average heat flow in Fig. 4 fromthe experimental data. Evaluating the integrals alongany trajectory in the stationary state as averages over thesteady-state distribution, we obtain Eqs. (10) and (11). Appendix B: Finite-time estimators
The estimation of the torque and heat flows from theexperimental data using Eqs. (6) and (9), respectively,depends sensitively on the sampling time step ∆ t of therecorded trajectories. More precisely, the estimators forthe time-averaged quantities are discretizations of the in-tegrals M ≃ − γτ Z τ r ( t ) ◦ d φ ( t ) and h ˙ Q x,y i ≃ Q x,y τ (B1)with Q x,y given in Eqs. (A14) and (A15), respectively.Let us denote the estimators for sampling time step ∆ t by M (∆ t ) and h ˙ Q x,y i (∆ t ). The exact (experimental) valuesare obtained in the limit ∆ t →
0. However, ∆ t is subjectto experimental constraints and in the case of our experi-ments is ∆ t = 2 . FIG. 5.
Estimation of torque and the heat flow.
Sim-ulations showing the measured values (a) of the torque and(b) of the heat flows as a function of the sampling time step∆ t using a zeroth-order estimator (circles) and a first-orderestimator (Eq. (B5), crosses). The black lines represent theexact values predicted by the theory. The first-order estima-tor permits us to obtain the correct values at the experimentalsampling time step ∆ t = 2 . not sufficiently small to obtain accurate values using thenaive estimators M (∆ t ) or h ˙ Q x,y i (∆ t ). To illustrate this,we performed simulations of the system under the sameexperimental conditions [42] and computed the resultingtorque and heat flows for different sampling time steps ∆ t as shown by the open circles and squares in Fig. 5. Fortoo large sampling times, these estimates clearly deviatefrom the analytic predictions (solid lines).In order to solve this problem, we introduce an im-proved, first-order estimator, which we will explain usingthe example of the torque; the method works completelyanalogously for the heat flows. We first note that thetorque (as well as the heat flows) in the steady stateare constant in time, such that any dependencies on the sampling times step ∆ t (as the ones detected when esti-mating torque and heat flows from the simulations shownin Fig. 5) are due to the simple estimators M (∆ t ) (and h ˙ Q x,y i (∆ t )) being too imprecise. Assuming that M (∆ t )is a smooth function of the sampling time step, we canexpand it around the exact value M (0), M (∆ t ) = M (0) + b ∆ t + O (∆ t ) , (B2)where b is the linear deviation coefficient and O (∆ t )stands for higher order deviations. Likewise, the averagetorque sampled for a time step of size 2∆ t is M (2∆ t ) = M (0) + 2 b ∆ t + O (∆ t ) . (B3)Combining these two expressions, we can eliminate thelinear deviation b , so that the exact value M (0) can beapproximated as M (0) = M (∆ t ) | {z } zeroth order + [ M (∆ t − M (2∆ t )] | {z } first-order correction + O (∆ t ) , (B4)up to second-order deviations in ∆ t . In other words, weconstruct an improved estimator M ∗ (∆ t ) = 2 M (∆ t ) − M (2∆ t ) (B5)from the naive estimator M (∆ t ), which is accurate tofirst order in ∆ t as opposed to the zeroth order precisionof M (∆ t ). In principle, we could continue this scheme tohigher orders by including measurements of M at highermultiples of the sampling time step, but the first-ordercorrection turned out to be sufficient in the present case.This can be seen from the blue crosses in Fig. 5(a), whichare in good agreement with the theoretical predictions upto a sampling time step ∆ t = 5 . [1] K. Sekimoto, Stochastic energetics (Springer Verlag, Hei-delberg, 2010).[2] U. Seifert, “Stochastic thermodynamics: Principles andperspectives,” Eur. Phys. J. B , 423–431 (2008).[3] R. Chetrite and K. Gaw¸edzki, “Fluctuation relations fordiffusion processes,” Commun. Math. Phys. , 469–518(2008).[4] F. Ritort, “Nonequilibrium fluctuations in small systems:From physics to biology,” Adv. Chem. Phys. , 31(2008).[5] C. Jarzynski, “Equalities and inequalities: Irreversibilityand the second law of thermodynamics at the nanoscale,”Annu. Rev. Condens. Matter Phys. , 329–351 (2011).[6] U. Seifert, “Stochastic thermodynamics, fluctuation the-orems and molecular machines,” Rep. Prog. Phys. ,126001 (2012).[7] C. Van den Broeck, “Stochastic thermodynamics: Abrief introduction,” Phys. Complex Colloids , 155– 193 (2013).[8] C. Van den Broeck and M. Esposito, “Ensemble and tra-jectory thermodynamics: A brief introduction,” PhysicaA , 6–16 (2015).[9] P. Jones, O. Marag´o, and G. Volpe,Optical tweezers: Principles and applications (Cam-bridge University Press, 2015).[10] G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles,and D. J. Evans, “Experimental demonstration of viola-tions of the second law of thermodynamics for small sys-tems and short time scales,” Phys. Rev. Lett. , 050601(2002).[11] J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, andC. Bustamante, “Equilibrium information from nonequi-librium measurements in an experimental test of Jarzyn-ski’s equality,” Science , 1832–1835 (2002).[12] D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco,and C. Bustamante, “Verification of the Crooks fluctua- tion theorem and recovery of RNA folding free energies,”Nature , 231–234 (2005).[13] A. B´erut, A. Arakelyan, A. Petrosyan, S. Ciliberto,R. Dillenschneider, and E. Lutz, “Experimental verifi-cation of Landauer’s principle linking information andthermodynamics,” Nature , 187–189 (2012).[14] R. D. Astumian, “Thermodynamics and kinetics ofmolecular motors,” Biophys. J. , 2401–2409 (2010).[15] T. Hondou and K. Sekimoto, “Unattainability of Carnotefficiency in the Brownian heat engine,” Phys. Rev. E ,6021 (2000).[16] T. Schmiedl and U. Seifert, “Efficiency at maximumpower: An analytically solvable model for stochastic heatengines,” EPL (Europhys. Lett.) , 20003 (2007).[17] R. Filliger and P. Reimann, “Brownian gyrator: A mini-mal heat engine on the nanoscale,” Phys. Rev. Lett. ,230602 (2007).[18] S. Bo and A. Celani, “Entropic anomaly and maximalefficiency of microscopic heat engines,” Phys. Rev. E ,050102 (2013).[19] J. Stark, K. Brandner, K. Saito, and U. Seifert, “Classi-cal Nernst engine,” Phys. Rev. Lett. , 140601 (2014).[20] G. Verley, M. Esposito, T. Willaert, and C. Van denBroeck, “The unlikely Carnot efficiency,” Nat. Commun. , 4721 (2014).[21] H. C. Fogedby and A. Imparato, “A minimal modelof an autonomous thermal motor,” arXiv preprintarXiv:1707.01070 (2017).[22] S. Bo and R. Eichhorn, “Driven anisotropic diffu-sion at boundaries: Noise rectification and particlesorting,” Phys. Rev. Lett., in press; arXiv preprintarXiv:1706.01660 (2017).[23] V. Blickle and C. Bechinger, “Realization of amicrometre-sized stochastic heat engine,” Nat. Phys. ,143–146 (2012).[24] P. A Quinto-Su, “A microscopic steam engine imple-mented in an optical tweezer,” Nat. Comm. , 5889(2014).[25] I. A. Mart´ınez, ´E. Rold´an, L. Dinis, D. Petrov, J. M. R.Parrondo, and R. Rica, “Brownian Carnot engine,” Nat.Phys. , 67–70 (2016).[26] S. Krishnamurthy, S. Ghosh, D. Chatterji, R. Ganapa-thy, and A. K. Sood, “A micrometre-sized heat engineoperating between bacterial reservoirs,” Nat. Phys. ,1134 (2016).[27] F. Schmidt, A. Magazzu, A. Callegari, L. Biancofiore,F. Cichos, and G. Volpe, “Microscopic engine powered by critical demixing,” arXiv preprint arXiv:1705.03317(2017).[28] S Ciliberto, “Experiments in stochastic thermodynamics:Short history and perspectives,” Phys. Rev. X , 021051(2017).[29] I. A. Mart´ınez, ´E. Rold´an, L. Dinis, and R. A. Rica,“Colloidal heat engines: A review,” Soft Matter , 22–36 (2017).[30] I. A. Mart´ınez, ´E. Rold´an, J. M. R. Parrondo, andD. Petrov, “Effective heating to several thousand kelvinsof an optically trapped sphere in a liquid,” Phys. Rev. E , 032159 (2013).[31] P. Mestres, I. A. Martinez, A. Ortiz-Ambriz, R. A. Rica,and E. Roldan, “Realization of nonequilibrium thermo-dynamic processes using external colored noise,” Phys.Rev. E , 032116 (2014).[32] L. Dinis, I. A. Mart´ınez, ´E. Rold´an, J. M. R. Parrondo,and R. A. Rica, “Thermodynamics at the microscale:From effective heating to the Brownian Carnot engine,”J. Stat. Mech. , 054003 (2016).[33] J. R. Gomez-Solano, L. Bellon, A. Petrosyan, andS. Ciliberto, “Steady-state fluctuation relations for sys-tems driven by an external random force,” EPL (Euro-phys. Lett.) , 60003 (2010).[34] A. B´erut, A. Imparato, A. Petrosyan, and S. Ciliberto,“Stationary and transient fluctuation theorems for effec-tive heat fluxes between hydrodynamically coupled par-ticles in optical traps,” Phys. Rev. Lett. , 068301(2016).[35] K. Sekimoto, “Langevin equation and thermodynamics,”Prog. Theor. Phys. Supp. , 17–27 (1998).[36] R. Parthasarathy, “Rapid, accurate particle tracking bycalculation of radial symmetry centers,” Nat. Meth. ,724–726 (2012).[37] G. Volpe and D. Petrov, “Torque detection using Brow-nian fluctuations,” Phys. Rev. Lett. , 210603 (2006).[38] G. Volpe, G. Volpe, and D. Petrov, “Brownian motionin a nonhomogeneous force field and photonic force mi-croscope,” Phys. Rev. E , 061118 (2007).[39] K.-H. Chiang, C.-L. Lee, P.-Y. Lai, and Y.-F.Chen, “Electrical autonomous Brownian gyrator,” arXivpreprint arXiv:1703.10762 (2017).[40] H. Risken, The Fokker-Planck Equation (Springer, Hei-delberg, Germany, 1984).[41] C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences,2nd ed. (Springer, Heidelberg, Germany, 1985).[42] G. Volpe and G. Volpe, “Simulation of a Brownian parti-cle in an optical trap,” Am. J. Phys.81