Experimental Characterization of Autonomous Heat Engine Based On

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Peltier by controlled are plates eulbimsseslk ilgclcls xliigno exploiting cells, biological like systems nequilibrium (switch ofheatbath) Power piston Flywheel Displacer Pressure sensorPressure a ceai fteexperiments. the of Schematic b Thermometer xeietlstu.Tetmeaue ttetpadbottom and top the at temperatures The up. set Experimental , b θ =0° T T T top btm T top h oln fgsrslsi h pres- the in results gas of cooling The . Δ Δ T < 270° T > a 90° hroyai iga fthe of diagram Thermodynamic , Δ Δ T c T + c Target Target pattern + Water Water block Ideal Stirlingcycle Peltier modulePeltier Al Al plate gfo h otmin- bottom the from ng ageconfiguration) iangle rnebtenthe between erence Stirling engine d n rvsthe drives and rd, ilpesr sen- pressure tial t temperature ite h displacer The . 180° V cerestores ycle wr,most nward, ocp of eoscopy ferential t h top the ith omous nequi- ,the r, del- ch pt e dynamics. The Stirling engine is an autonomous and closed heat en- gine23–26. Given a temperature difference, the engine cycles the volume, temperature, and pressure inside a cylinder au- tonomously without external timing control and rotates a flywheel unidirectionally. Theoretically, an ideal Stirling engine, i.e., a Stirling cycle, achieves the Carnot efficiency. The LTD- SE22, 27 consists of a power piston, displacer, flywheel, two cranks, and two rods connecting the piston and displacer to the flywheel (Fig. 1a). The flywheel rotates when a sufficiently large temperature difference is given between the top and bottom plates of the cylinder. The flywheel rotation is synchronized with the oscillation of the internal displacer and the power piston. The displacer serves to switch the heat baths between the top and bottom plates. Thus, the gas temperature and pressure oscillate and move the power piston up and down. This piston motion drives the flywheel rotation. The π/2 out of phase of the displacer and the power piston makes a cycle. The flywheel provides inertia necessary for a smooth rotation. When the opposite temperature difference is given, the flywheel rotates in the opposite direction with an inverted mechanism.

Experiment An LTD-SE (N-92 type) was bought from Kontax (UK). We controlled the temperatures at the top and bottom plates, Ttop and Tbtm, of the cylinder (Fig. 1b) and monitored the angular position θ(t) and angular velocity ω(t) of the ﬂywheel and the pressure p(t) inside the chamber. See the Materials and Methods for details.

Figure 2. Steady rotation. a, The time-averaged angular velocity hωi of the Rotation. Without stimulation, the engine was settled at a limit cycle was plotted against the temperature differences ∆T. At |∆T|≥ 8K, stationary position θ ≃−38◦, where the pressure difference three experimental traces were averaged (circle). The error bar corresponds to across the power piston and the gravity force on the power S.D. At |∆T| < 8 K, twelve traces were superposed (solid lines, six for ∆T > 0 and six for ∆T < 0). b, The stability of the rotation state at ∆T = 12 K. The piston, displacer, crank screws, and rods are presumably bal- rotation state is stable against perturbations (indicated by arrows). Dashed anced. When an initial angular momentum with a sufficiently lines are fitting curves by exponential functions, of which time constant cor- large magnitude was given, the flywheel rotated steadily with responds to I/Γ. c, Pressure-volume curves for ∆T = 12 K (red) and -12 K (navy) obtained by experiments (solid) and theories (dashed). The cycling an angular velocity determined by ∆T = Tbtm − Ttop (Fig. 2a). direction was clockwise independent of the sign of ∆T. The average of ∆p The rotation direction changed depending on the sign of ∆T. for the theoretical curves was forced to zero. d, Bifurcation diagrams without When the engine in this steady state was perturbed by hand, (open) or with (closed) additional frictional load. The solid curves are numer- the angularvelocity was soon recoveredto the steady rate (Fig. ical simulations with Γ obtained by measuring the relaxation time I/Γ to the perturbation (b and Fig. S2). 2b), implying a stable limit cycle. The pressure-volume curve exhibited a circular diagram (Fig. 2c), demonstrating a heat engine. The cycling direc- the homoclinic bifurcation point. However, such the continu- tion in the PV diagram was the same independent of the sign ity is too steep to be observed in the experiments because ω is of ∆T , and the PV curves were nearly symmetric for the sign ± ± inversely proportional to −ln|∆T −∆Tc | for |∆T | > |∆Tc | in of ∆T. The area increased with |∆T | (Fig. S1). the vicinity of the bifurcation point28. Instead, discontinuous The time-averaged steady angular velocity hωi changed change in hωi was observed. nearly linearly with ∆T (Fig. 2a). |hωi| decreased with |∆T| We also induced additional frictional load by pressing a and vanished at a finite value of ∆T . The threshold value, ∆Tc, brush for Chinese calligraphy to the flywheel. The increase + + was slightly different for the sign of ∆T ; ∆Tc = 6.2 ± 0.3K in load suppressed the angular velocity and increased ∆Tc − and ∆Tc = −5.7 ± 0.2 K (mean ± standard deviation) , indi- (Fig. 2c). cating the asymmetry of the dynamics for the sign of ∆T . ± The stalling at ∆Tc was accompanied by a steep change Bifurcation analysis. We characterize the bifurcation dy- in hωi, implying a homoclinic bifurcation28. The homoclinic namics in detail. Figure 3a shows two typical trajectories + bifurcationis a kindof a global bifurcation,and seen in, forex- started with different initial angular velocities at ∆T > ∆Tc . ample, a driven pendulum and a Josephson junction. A stable With a large initial angular velocity, we observed the conver- limit cycle disappears with a steep but continuous transition at gence to the periodic trajectory determined by ∆T . As noted,

2/6 ered as an oscillatory stable limit cycle. This limit cycle showed complicated behaviors; the ampli- tude increases accompanied by a period-doubling bifurcation (−27.5K ≥ ∆T > −33.5K), seemingly aperiodic oscillation similar to chaos (−33.5K ≥ ∆T > −39.5K), and again periodic oscillations accompanied by small additional oscillations (−39.5K ≥ ∆T > −61.5K). The oscillation branch disap- peared at ∆T ≤−61.5K, and a small perturbation got drawn into a rotation branch. The rotation mode was observed for all ∆T < 0 with a sufficiently large initial angular velocity. The oscillation was not observed for ∆T > 0. The stable fixed point (spiral) at S became unstable at ∆T = −39.5 K (Fig. S3), indicating that a subcritical Hopf bifurcation accompanied by the disappearance of an unstable limit cycle occurred. Although the unstable limit cycle was difficult to be identified by experiments, we may expect that the oscillatory stable limit cycle and the unstable limit cycle were created in a pair at ∆T = −27.5K28. We need further studies to determine the bifurcation characteristics of the oscillation branch, including the onset and disappearance of the seemingly aperiodic oscillation.

Figure 3. Rotational trajectories. a, Rotational trajectories at ∆T = 6.5K > + ∆Tc initiated with small (dashed, black) or large (solid, orange) angular velocity ω. With a large initial angular velocity, ω gradually increased and, the engine settled down to a steady rotation. With a small initial angular velocity, the engine stopped the rotation at the stable fixed point S at (θ,ω) ≃ (−38◦,0) (closed circle) after passing nearby the saddle point U at (θ,ω) ≃ (153◦,0) (open circle). See Fig. 1a for the definition of θ. b, + c Rotational trajectory at ∆T = 5K < ∆Tc . , Steady rotational trajectories at different ∆T. the periodic trajectory was stable against perturbation and was identified as a stable limit cycle (Fig. 2b). With a small initial angular velocity, the trajectory was first attracted to U at (θ,ω) ≃ (153◦,0) and then collapsed to S at (θ,ω) ≃ (−38◦,0) in a spiral-shaped manner, failing in con- − + verging to the stable limit cycle. When ∆Tc < ∆T < ∆Tc , S was the unique stable attractor (Fig. 3b). These results sug- gest that U and S are a saddle point and a stable fixed point (spiral), respectively, and that S and the stable limit cycle co- − exist for ∆T > ∆Tc and ∆T < ∆Tc . Figure 3c shows the trajectories of the steady rotations, ω(θ), at various ∆T above the threshold. ω(θ) was relatively flat at large |∆T | and exhibited rugged profile at small |∆T|. Specifically, as ∆T approaches ∆Tc, the part of the limit cycle approaches the saddle point U, which is one of the characteristics of the homoclinic bifurcation. All the characteristics observed above indicate the homo- 28 clinic bifurcation of the limit cycle at ∆Tc and controvert other possibilities, including the Hopf bifurcation where local stability of the fixed point alters at the bifurcation point. We ◦ will analyze the experimental data based on simple dynamical- Figure 4. Oscillation mode observed at Ttop = 65 C. a, θ as the functions system modeling below in the Theory section. of time (left), ω (center), and ∆p (right). b, Peak angles of the oscillation (circle), stable fixed points (closed square), unstable fixed points (open square), and expected unstable limit cycles (dashed line). Oscillatory mode. We also discovered an oscillation branch at ∆T ≤−27 K (Fig. 4a). Here, for exploring a small ∆T re- gion, T was set to a relatively large value, 65◦C. When we top Theoretical analysis shifted the flywheel angle a little bit from S gently by hand, the flywheel started a periodic oscillation with a finite ampli- For deducing the model that explains the experimental obser- tude and a period of about 10 seconds, which can be consid- vations, we compared the above results with the theory pro-

3/6 posed recently29. The theory describes the flywheel rotation Discussion with a simple equation of motion with only two variables; An autonomous heat engine is a model system of autonomous θ˙ = ω, nonequilibrium systems. We demonstrated that the essential (1) characteristics of the complex autonomous heat engine are re- Iω˙ = s[p(θ,ω) − p0]r sinθ − Γω. produced by a minimal and intuitive two-variable model (1) Here, I and Γ are the moment of inertia and frictional coef- quantitatively. The present work supports the new approach ficient, respectively, of the engine’s rotational degree of free- to explorethe finite-time thermodynamicsof autonomousheat dom. r is the crank radius. s is the sectional area of the power engines based on a simple dynamical-system description. The 29 piston. s[p(θ,ω) − p0] ≡ s∆p corresponds to the force on the model contains ∆T explicitly through p(θ,ω) , proposing a crank applied by the power piston via a rod, and s∆p · r sinθ novel concept that the LTD-SE is a thermodynamic pendulum is the torque on the flywheel (a piston-crank mechanism). p0 driven by a thermodynamic force characterized by ∆T . is external pressure. Despite its simplicity, the model reproduced the essential The theory29 approximates that the gas is in contact with characteristics of the engine, including the bifurcation dynam- ∆T a single heat bath at an effective temperature T0 + sinθ 2 , ics and the thermodynamic diagram. Whereas the model (1) where T0 = (Ttop + Tbtm)/2. The effective temperature oscil- is derived based on the LTD-SE, this simple and intuitive for- lates between Ttop and Tbtm synchronized with the displacer mulation is expected to be applicable to a wide range of au- motion. For the quantitative analysis of the experimental data tonomous heat engines with small modifications on, for exam- based on (1), we modeled the system simply as ple, the cycle shape T (θ,ω) and the piston-crank mechanism ∆T r sinθ. The model did not reproduce the oscillation branch. T (θ,ω)= T + α sin(θ − ωτ) , (2) 0 2 Although the oscillation is not an essential operation mode of nRT(θ,ω) the engine, it would be intriguing to explore what modifica- p(θ,ω)= β . (3) tion to the theory could successfully describe the oscillation. V(θ) The formulation of the thermodynamic efficiency of the au- Here, the effect of the heat transfer on the gas temperature tonomous heat engine would be of crucial importance, which T (θ,ω) is simply implemented by two parameters, the mag- is complementary to the formulation in non-autonomous heat nitude α and the time delay τ under an adiabatic assumption engines2, 3. The evaluation of efficiency requires the measure- that the temperature equilibration is sufficiently fast compared ment of the heat flowing through the engine and remains for to the flywheel dynamics. p(θ,ω) is calculated based on an future studies. effectiveequation of the state for the ideal gas. n is the amount The Stirling engine is attracting growing attention in indus- of substance of the internal gas, and R is the gas constant. tries because it can utilize low-grade heating sources such as V(θ)= V0 +rs(1−cosθ) is the volume of the cylinder, where solar power, waste heat in the industries, and the geothermal V0 is the cylinder volume excluding the displacer volume. The energy, and also is environmentally friendly. Because of its temperature and pressure may be nonuniform inside the cylin- autonomous, clean, and simple machinery, the use of the Stir- der, and therefore the equation of the state for the ideal gas ling engine for generating electric power for the spacecraft may not hold as it is. The coefficient β is introduced to com- is being considered24. Nevertheless, the physics behind en- pensate for such an effect. gine dynamics has been lacked. Our experiments succeeded The two-variable model (1) with (2)and(3) reproduced the in characterizing the bifurcation mechanism. Such knowledge ∆T-dependence of hωi well quantitatively (Fig. 2a) includ- based on physics would be effective in improving engine per- ing the steep change in the vicinity of ∆Tc and the pressure- formance. volume curve. See Materials and Methods for the parameters This work was supported by JSPS KAKENHI (18H05427 used. The model also succeeded in reproducing the bifurca- and 19K03651). tion curves for the increased frictional load (Fig. 2d). Here, we used the same parameters except for the frictional coeffi- Materials and Methods cients, which were evaluated from the response curves under each condition (Fig. S2). Note that the model (1 - 3) exhibits Experimental setup. An LTD-SE (N-92 type) was bought the homoclinic bifurcation as |∆T| is decreased, where a sta- from Kontax (UK). The temperatures of the top and bottom ± ble limit cycle disappears at ∆Tc by colliding with a saddle plates of the cylinder were controlled by Peltier modules point at U29. This is consistent with the experimental sugges- equipped with water flowing blocks (Fig. 1b). The temper- tions (arrows in Fig. 3c). These results validate the model atures were monitored at 2.5 Hz by Platinum resistance tem- (1). This two-variable model is a minimal model of the au- perature detectors attached to the surface of the plates. A tar- tonomous heat engines in the sense that at least two variables get pattern (three circles aligned in an isosceles triangle con- are required to describe a limit cycle. figuration) was attached to the crank screw connected to the On the other hand, the oscillation branch (Fig. 4) was not displacer for monitoring the angular position of the flywheel observed by this minimal model. At ∆T ≤−31.5K, the trajec- (Fig. 1b). The image of the target pattern was recorded by a tory θ(ω) possessed an intersection (Fig. 4a), meaning that high-speed camera (Basler, Germany) at 100 Hz and analyzed the description by only θ and ω does no longer describe the in real time to obtain the angular position and the angular ve- oscillation dynamics correctly at some points. Specifically, locity of the flywheel. A pressure sensor (Copal electronics, p(θ,ω) was a multiple-valued function of (θ,ω) at the inter- Japan) was fixed at the side of the cylinder to monitor the in- section points, suggesting that (3) is not valid at these points. ner pressure. We monitored the angular position θ(t), angular

4/6 velocity ω(t) of the flywheel, and the pressure p(t) inside the 17. Pietzonka, P. & Seifert, U. Universal trade-off between power, efficiency, chamber under controlled ∆T (t). All the experiments were and constancy in steady-state heat engines. Phys. Rev. Lett. 120, 190602 controlled by a computer equipped with a program developed (2018). 18. Serra-Garcia, M. et al. Mechanical autonomous stochastic heat engine. on LabVIEW (National Instruments). Phys. Rev. Lett. 117, 010602 (2016). 19. Mart´ınez, I. A. et al. Brownian carnot engine. Nat. Phys. 12, 67 (2016). Bifurcation dynamics. For evaluating ∆T dependence of 20. Blickle, V. & Bechinger, C. Realization of a micrometre-sized stochastic the angular velocity (Fig. 2a), we manually provided an ini- heat engine. Nat. Phys. 8, 143–146 (2011). tial angular momentum at ∆T = 36 K or -30 K with keeping 21. Carnot, S. Reflections on the Motive Power of Heat (Edited and trans- ◦ Ttop = 24 C and waited for about one hour for the sufficient lated by R. H. Thurston) (1897). relaxation of the temperatures and flywheel rotation. Then, 22. Senft, J. R. An Introduction to Low Temperature Differential Stirling ◦ Engines (Moriya Press, 2008). with keeping Ttop = 24 C, we varied ∆T from 36 K to 0 K 23. Senft, J. R. An Introduction to Stirling Engines (Moriya Press, 1993). or from -30 K to 0 K in a stepwise manner at a rate of ±1K 24. Wolverton, M. Stirling in deep space. Sci. Am. 298, 14 (2008). every 180 s for |∆T | > 8 K and ±0.02 K every 60 s or 120 s 25. Kongtragool, B. & Wongwises, S. A review of solar-powered Stirling en- otherwise. gines and low temperature differential Stirling engines. Renew. Sustain. Energy Rev. 7, 131 (2003). Parameters for theoretical curves. We used the follow- 26. Köhler, J. W. L. The stirling refrigeration cycle. Sci. Am. 212, 119 ing parameters for the theoretical curves in Figs. 2a, c, (1995). 3 2 27. Lu, Y. J., Nakahara, H. & Bobowski, J. S. Quantitative stirling cycle mea- and d. V0 = 44900mm , s = 71mm , r = 3.5mm, I = −5 2 surements: P-v diagram and refrigeration. Phys. Teach. 58, 18 (2020). 5.7 × 10 kgm , and p0 = 101.3kPa. n = 0.00185mol, 28. Strogatz, S. H. Nonlinear Dynamics and Chaos: With Applications to R = 8.314J/Kmol. We determined α, β, and τ as 0.17, 0.94, Physics, Biology, Chemistry, and Engineering, Second Edition (West- and 15 ms, respectively, by fitting. The friction coefficient view Press, 2014). Γ was measured by evaluating the relaxation time after a per- 29. Izumida, Y. Nonlinear dynamics analysis of a low-temperature- turbation (Fig. 2b and S2). The relaxation time constant is differential kinematic Stirling heat engine. EPL 121, 50004 (2018). approximately given by I/Γ.

References 1. Callen, H. Thermodynamics and an Introduction to Thermostatistics, Second Edition (Wiley, 1985). 2. Curzon, F. & Ahlborn, B. Efficiency of a Carnot engine at maximum power output. Am. J. Phys. 43, 22 (1975). 3. Salamon, P., Nulton, J. D., Siragusa, G., Andersen, T. R. & Limon, A. Principles of control thermodynamics. Energy 26, 307 (2001). 4. den Broeck, C. V. Thermodynamic efficiency at maximum power. Phys. Rev. Lett. 95, 190602 (2005). 5. de Cisneros, B. J. & Hernández, A. C. Collective working regimes for coupled heat engines. Phys. Rev. Lett. 98, 130602 (2007). 6. Esposito, M., Lindenberg, K. & den Broeck, C. V. Universality of efficiency at maximum power. Phys. Rev. Lett. 102, 130602 (2009). 7. Esposito, M., Kawai, R., Lindenberg, K. & den Broeck, C. V. Efficiency at maximum power of low-dissipation carnot engines. Phys. Rev. Lett. 105, 150603 (2010). 8. Benenti, G., Saito, K. & Casati, G. Thermodynamic bounds on efficiency for systems with broken time-reversal symmetry. Phys. Rev. Lett. 106, 230602 (2011). 9. Izumida, Y. & Okuda, K. Work output and efficiency at maximum power of linear irreversible heat engines operating with a finite-sized heat source. Phys. Rev. Lett. 112, 180603 (2014). 10. Schmiedl, T. & Seifert, U. Optimal finite-time processes in stochastic thermodynamics. Phys. Rev. Lett. 98, 108301 (2007). 11. Mart´ınez, I. A., Petrosyan, A., Guéry-Odelin, D., Trizac, E. & Ciliberto, S. Engineered swift equilibration of a brownian particle. Nat. Phys. 12, 843–846 (2016). 12. Tafoya, S., Large, S. J., Liu, S., Bustamante, C. & Sivak, D. A. Us- ing a system’s equilibrium behavior to reduce its energy dissipation in nonequilibrium processes. Proc. Nat. Acad. Sci. 116, 5920–5924 (2019). 13. Brandner, K., Saito, K. & Seifert, U. Thermodynamics of micro- and nano-systems driven by periodic temperature variations. Phys. Rev. X 5, 031019 (2015). 14. Shiraishi, N., Saito, K. & Tasaki, H. Universal trade-off relation between power and efficiency for heat engines. Phys. Rev. Lett. 117, 190601 (2016). 15. Raz, O., Subas¸ı, Y. & Pugatch, R. Geometric heat engines featuring power that grows with efficiency. Phys. Rev. Lett. 116, 160601 (2016). 16. Polettini, M. & Esposito, M. Carnot efficiency at divergent power output. Europhys. Lett. 118, 40003 (2017).

5/6 Supplementary ﬁgures

Figure S2. Pressure-volume curves at ∆T = ±9K (a), ±12 K (b, same as Fig. 2c), and ±24 K (c). Red and navy curves correspond to ∆T > 0 and ∆T < 0, respectively. Solid and dashed curves correspond to experimental data and simulation data, respectively.

Figure S4. When started in the vicinity of the unstable ﬁxed point U, the trajectory converges to the oscillatory stable limit cycle in a spiral manner. ◦ Ttop = 65 C and ∆T = −42K.

Figure S3. The relaxation after a perturbation at a steady rotation at ∆T = 36 K, corresponding to the three curves under loaded conditions in Fig. 2d. Red dashed lines are exponential ﬁtting.

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